¨
OBSTRUCTION AND EXISTENCE FOR TWISTED KAHLER-EINSTEIN METRICS
AND CONVEXITY
By
Ambar Rao

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mathematics–Doctor of Philosophy
2013

ABSTRACT
¨
OBSTRUCTION AND EXISTENCE FOR TWISTED KAHLER-EINSTEIN
METRICS AND CONVEXITY
By
Ambar Rao
Let L → X be an ample holomorphic line bundle over a compact K¨hler manifold
a
(X, ω0 ) with c1 (L) represented by the K¨hler form ω0 . Given a semi-positive real (1, 1) form
a
η representing −c1 (KX ⊗ L), one can ask whether there exists a K¨hler metric ω ∈ c1 (L)
a
that solves the equation Ric(ω)−ω = η. We study this problem by twisting the K¨hler-Ricci
a
flow by η , that is evolve along the flow ωt = ωt + η − Ric(ωt ) starting at ω0 . We prove that
˙
n
n
such a metric exists provided ωt ≥ Kω0 for some K > 0 and all t ≥ 0. We also study a

twisted version of Futaki’s invariant, which we show is well-defined if η is annihilated under
the infinitesimal action of η(X), in particular η is Aut0 (X) invariant. Finally, using Chens
-geodesics instead, we give another proof of the convexity of Lω along geodesics, which
plays a central in Berman’s proof of the uniqueness of critical points of Fω .

ACKNOWLEDGMENTS

I would like to thank my advisor Prof. Xiaodong Wang for all the help I received over the
years.

iii

TABLE OF CONTENTS

Introduction

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Chapter 1 Convexity of some functionals and consequences
1.1 Description of Functionals . . . . . . . . . . . . . . . . . . .
1.2 Geodesic Equation . . . . . . . . . . . . . . . . . . . . . . .
1.3 -Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Convexity and eigenvalue estimate . . . . . . . . . . . . . .
1.5 Convexity of Lω0 . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Maximizers using -geodesics . . . . . . . . . . . . . . . . . .
1.7 Uniqueness smooth case . . . . . . . . . . . . . . . . . . . .
1.8 Berndtsson argument setup . . . . . . . . . . . . . . . . . .
1.9 Generalized Gradient Vector Field . . . . . . . . . . . . . . .
1.10 Non-smooth case . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 2 Obstruction and existence for twisted K¨hler-Einstein
a
2.1 Twisted K.E scalar Equation . . . . . . . . . . . . . . . . . . . . . .
2.2 Various Estimates (Toy version) . . . . . . . . . . . . . . . . . . . .
2.3 Twisted Mabuchi functional . . . . . . . . . . . . . . . . . . . . . .
2.4 Twisted Futaki type invariant . . . . . . . . . . . . . . . . . . . . .
2.5 Convexity of the Twisted Mabuchi functional . . . . . . . . . . . .
2.6 Application of Twisted Mabuchi Energy: Existence . . . . . . . . .
2.7 Perelman’s estimates twisted setting . . . . . . . . . . . . . . . . .
2.8 Twisted Perelman entropy . . . . . . . . . . . . . . . . . . . . . . .
2.9 Extracting canonical metric . . . . . . . . . . . . . . . . . . . . . .
2.10 C 3 Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5
5
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equation 54
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Chapter 3 Future direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.1 Coupled system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
BIBLIOGRAPHY

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iv

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Introduction
The study of extremal K¨hler metrics has generated a lot of work. A major theme centers
a
around the equivalence of special metrics and stability in various senses. In the K¨hler setting
a
the manifold version, conjectured by [Y1], [Tian00] and [Do] has taken time to handle. In
one direction, when Aut(X)0 is trivial, a refinement of an argument, due to [Sto] of [Do5]
shows existence of a constant scalar curvature metric in c1 (L) implies (X, L) is K-stable.
See [Ber12] for generalizations to Fano varieties and other improvements: conditions on the
group of automorphisms Aut(X) is dropped while the constant scalar curvature assumption
is strengthened to admitting K¨hler-Einstein metric. For the correct notion of stability there
a
have been several candidates.
While existence of K¨hler-Einstein metrics in the C1 (−KX ) positive case comes with
a
obstructions, existence has been shown to be equivalent to properness of Jω functional on
Hω , this is analytic stability [Ti97]. Various notions of stability have been introduced by
Yau, Tian, Donaldson and others, and progress to various degrees have been made. [Do3] has
introduced a notion of B-stability from which existence of K¨hler-Einstein metrics was dea
ducible granted additional hypotheses some of which are removable. In [Sz1] on a Fano manifold X under the additional assumptions that both Riemann curvature tensor and Mabuchi
functional are bounded below along K¨hler-Ricci flow on X, it is shown K-polystability is
a

1

enough to obtain the existence of a K¨hler-Einstein metric. Recently [CDS] and [Ti13] have
a
given solutions of Yau-Tian-Donaldson conjecture for K¨hler-Einstein metrics.
a
Study of the twisted case appear in various settings see [Sto09], [SzCo], [Kel]. This was
preceded by [Fi]. See also [Ber10] and [Bo] for more recent work. In [Sto09] a moment map
description of constant scalar curvature equation (cscK) S(ω) − S = Λω α is available when α
is a symplectic form, and its shown there is an obstruction and a stability condition in terms
of the Ross-Thomas polynomial for the equation. [Do4] showed scalar curvature comes up as
an equivariant moment map for the action of Symp(B, ω) on the space of integrable complex
structures J .The second term Λω α in the twisted equation can be viewed as an equivariant
moment map of the action of Symp(B, ω) on the space of diffeomorphism f : B → M ,
M, due to [Do7]. The full twisted cscK equation comes as a moment map for the diagonal
action of Symp(B) on a new space S ⊂ M × J [Sto09]. From these considerations Stoppa
is motivated to introduce twisted K-energy a quantity which we use in arguments below.
Further in this connection, provided that η is annihilated under the infinitesimal action of
η(X), the Futaki type invariant for the twisted equation that we study below, can be shown
to be well-defined. So in certain situations the equation in our setting does come with
classically motivated obstructions.
We also study existence in the twisted setting below, and inspired by [Pa], we establish
an existence result in the twisted setting under similar assumptions:
Proposition 1. Let L → X be holomorphic ample line bundle polarizing (X, ω) (c1 (L) = ω).
Prescribe 0 ≤ η ∈ −c1 (KX ⊗ L) then there is a K¨hler metric such that ω ∈ c1 (L) solving
a
the twisted K¨hler Einstein equation Ric(ω) − ω = η provided ωt deforms according to the
a
ωn

twisted K¨hler-Ricci flow starting at ω while satisfying the uniform estimate ωt ≥ K for
a
n
t ∈ [0, ∞) and some constant K > 0.
2

Recalling that η(X) is the lie algebra of holomorphic vector fields on X, the following
holds
Proposition 2. When η is annihilated under the infinitesimal action of η(X) the corresponding Futaki-type invariant for the twisted equation is well defined.
So as one expects from the corresponding monge-ampere equations, which are not solvable
in general, the twisted K¨hler-Einstein equation comes with obstructions.
a
Moving in another direction, the problem of existence of smooth geodesics in the space
of K¨hler metrics and their properties are useful for the study of special metrics [Do6].
a
Using various methods: continuity method [Chen00], quantization [PS] only the existence
of geodesics with C 1,1 regularity have been established. On the other hand one can obtain
C 0 regularity geodesics directly by an envelope construction see [Bo]. Even with this weaker
result progress on Bando-Mabuchi like theorems can still be made see [Ber10a], [Bo]. Despite
the existence of smooth geodesics in the space of K¨hler metrics being considered a dubious
a
problem(they don’t exist in general see [LV] and C 1,1 regularity is the best you can expect in
general see [TL]), it morally clarifies the role of convexity in the infinite dimensional analysis.
In fact with these considerations various generalized Moser Trudinger type inequalities are
obtainable. [PSSW] have verified a conjecture of Tian that on a K¨hler-Einstein manifold
a
(X, ωKE ) properness of FωKE on HωKE (X) can be upgraded to coerciveness with respect
to JωKE . In a similar direction on a integral K¨hler manifold with fixed smooth volume
a
form using geodesics, Bergmann kernel asymptotics and convexity properties of log Kφt a
moser trudinger inequality conjectured by Aubin is established in [BeBo12], although in the
K¨hler-Einstein setting this is the Moser Trudinger inequality first proved by [DT]. It might
a
be worth checking if the Moser-Trudinger inequality corresponding to the coercive estimate

3

can be obtained using these considerations in the K¨hler-Einstein setting, and naturally
a
the next step would be to see if the quantitative versions holds beyond the K¨hler-Einstein
a
setting.
An important feature in analyzing the Bando Mabuchi type theorem in [Ber10a], [Bo09],
[Bo] is the convexity of the L functional. Below we study this in a special case and then
in general and obtain that it is convex along geodesics (in the sense of X.X Chen) using
methods from complex geometry and -geodesics. In [Ber10a], [Bo09] estimates involved
rely on the Hormander ∂ estimates and the setup is more sophisticated. We also study the
uniqueness issue, but from an elementary point of view provided the geodesics are smooth.
So in particular i∂∂ut > 0. Again we rely on a complex geometry inequality crucial to
obtain convexity and analyze the equality case. In the setting when L = −KX , [Bo] shows
the scope of the result can be improved by establishing uniqueness using only sub-geodesics
and bypasses difficulties introduced by the degeneracies i∂∂ut ≥ 0. Because of its relevance
we describe this but suppress his bundle theoretic set-up in the discussion.

4

Chapter 1
Convexity of some functionals and
consequences

1.1

Description of Functionals

Let (L, h0 ) −→ X be a hermitian holomorphic line bundle over a compact complex manifold
X so that L is ample. A K¨hler form ω0 is given by the curvature (1,1) form, that is,
a
√
√
set ω0 = −(2π)−1 −1∂∂ log h0 in c1 (L). Write ω0 = (2π)−1 −1∂∂ψ0 , where locally the
background hermitian structure h0 is represented as h0 = e−ψ0 . This data is manufactured
by using an embedding determined from H 0 (Lk ) (k >> 0) pulling back the Fubini study
metric on O(1) → Pn and taking k-th roots gives a hermitian metric on L with the required
properties. Now set V = X dvolg . Consider the functional defined on the K¨hler potentials,
a
√
the open convex subset Hω0 = {u ∈ C ∞ (X)|ωu = ω0 + −1∂∂u > 0} ⊂ C ∞ (X), as
1
Eω0 (u) =
(
(n + 1)! V

n
n−j

i=0 X

i
uωu ∧ ω0

Proposition 1. The functional Eω0 on Hω0 has differential

(dEω0 )u (v) =

5

1
vω n
n! X u

)

for u ∈ Hω0 and v ∈ C ∞ (X)
Proof.

d
(Eω0 (u + tv))|t=0
dt
n
1
d
n−i
i
(
(u + tv)ωu+tv ∧ ω0 ))|t=0
= (
dt (n + 1)! V
X

(dEω0 )u =

i=0

=

1
[
(n + 1)! V

n
j

j=0 X
n

1
[
=
(n + 1)! V
=

1
[
(n + 1)! V

i=0 X
n

j=0 X
n

=

1
[
(n + 1)! V

j=0 X

n−j

(vωu+tv ∧ ω0

j−1

n−j

+ j(u + tv)ωu+tv ∧ ddc v ∧ ω0

)]|t=0

n
i
vωu

n−i
∧ ω0

+
i=1 X
n

n−i
i
vωu ∧ ω0 +
n−i
i
vωu ∧ ω0 +

i=1 X
n
i=1 X

n−i
i−1
iuωu ∧ ddc v ∧ ω0 ]
n−i
i−1
ivωu ∧ ddc u ∧ ω0 ]
n−i
i
ivωu ∧ ω0

n
j−1

−
i=1 X

ivωu

1
=
[
(n + 1)! V

n+1−j

∧ ω0

]

n

j=0 X

n
i
vωu

n−i
∧ ω0

+
i=1 X

n−i
i
ivωu ∧ ω0

n−1

−
i=0 X

n−i
i
(i + 1)vωu ∧ ω0 ]

1
=
[
(n + 1)! V

n
i=0 X

n−1
i
vωu

n−i
∧ ω0

−
i=1 X

n−i
i
vωu ∧ ω0 +

X

n
v(n)ωu −

X

n
vω0 ]

1
n
n
n
n
[
vω0 +
vωu +
v(n)ωu −
vω0 ]
(n + 1)! V X
X
X
X
1
=
vω n
n! V X u
=

Remark 1. This is true in much lower regularity settings in fact Eω0 extends to C 0 (X) by
composing with a nonlinear projection. The extension is gateaux differentiable and has the
6

same differential with u, v ∈ C 0 (X) see [Ber10a].
A similar calculation can be made for a path ut ∈ Hω0 that is at time t, v corresponds
to ut .
˙
1
1
0
n
Note that −Eω0 (u) = n!V (Jω0 (u) − X uω0 ) = n!V Fω0 where Jω0 is Aubin’s energy
0
functional. Note whereas Fω0 is convex, Eω0 is concave. Recall Jω0 (u) has derivative
1
n
− V X u(ωu − ω0 ) since this induces a closed 1-form on Hω0 its primitive is taken to be Jω0
˙ n

after fixing the correct normalization. The differential dEω can also be calculated in terms
of the differential of Jω . Also given a constant c, Eω0 (u + c) = Eω0 (u) + c follows from the
1
0
0
formula. Since Eω0 = − n!V Fω0 the functional has the same cocycle property that Fω0 has.

Recall s, s ψ0 = in

2
X

s ∧ se−ψ0 for s ∈ H 0 (X, L ⊗ KX ).

Another functional of importance is

Lω0 (u) := −

1
log det(T (u))
N

where T (u) = [ si , sj ψ0 +u ] and {si } is a basis of H 0 (X, L⊗KX ) orthogonal with respect
to ·, · ψ0 . This is independent of the {si } basis orthogonal with respect to ·, · ψ0 , since any
two bases of this type are related by a unitary transformation and the effect of the change is
to conjugate T [u] by this. So Lω0 remains unchanged. The property Lω0 (u+c) = Lω0 (u)+c
follows from the definition.
Finally define Fω0 := Eω0 − Lω0 . This is a functional on Hω0 . Recall the natural
action of R∗ on hermitian metrics e−u h0 on L by multiplication by e−c descends to additive
action of R on Hω i.e by addition by c. Under the action the functionals Eω0 , Lω0 have
values translated by c it follows Fω0 is constant under this action so that it descends to a
functional on the space of all K¨hler metrics in c1 (L).
a
7

Also note that the natural action of Aut0 (X, L) on the space of metrics on L corresponds
to the action (u, F ) → v := F ∗ (ψ0+u ) − ψ0 so that ωv = F ∗ ωu . So a statement P holds
for ω1 in the space of K¨hler metrics up to automorphism means that P also holds for any
a
K¨hler forms ω in the orbit of ω1 under the action of Aut0 (X, L).
a
Proposition 2. Given any orthogonal basis {si } of (H 0 (X, L ⊗ KX ), ·, · ψ0 )
• Lω0 is a well defined functional on Hω0 . The differential takes the form
−1 n2
i
(dLω0 )u (v) =
N

N
i=1 X

vsi ∧ si e−(ψ0 +u)

in any basis {si } orthonormal with respect to ·, · ψ0 +u with u, v as before.
• The functional Fω0 = Eω0 − Lω0 defined on Hω0 is translation invariant and the
differential at u is given by the second equality in (1.1) in any basis {si } orthogonal
with respect to ·, · ψ0 +u . In particular critical points of Fω0 are smooth solutions of
(1.2).
Proof. The first item is essentially the discussion above and the differential (dLω0 )u may be
computed similarly as in the previous proposition: take {si } orthonormal with respect to
·, · ψ0 +u then
d
d −1
Lω0 (u + tv)|t=0 = (
log det( si , sj ψ0 +u+tv ))|t=0
dt
dt N
−1
= [ (T ij T˙ )(u + tv)]|t=0
ij
N
−1
−1 n2
˙
=
T r[T ] =
i
N
N

8

N
i=1 X

vsi ∧ si e−(ψ0 +u)

Critical points u of Fω0 are smooth solutions of
2 N

1 n in
ω +
0 = (dFω0 )u =
v(
n! V u
N
X

si ∧ si e−(ψ0 +u) ) v ∈ C ∞ (X)

(1.1)

i=1

Equivalently u is a smooth solution of the monge-ampere equation
2 N

1
in
(ddc u + ω0 )n = −
n! V
N

1.2

si ∧ si e−(ψ0 +u)

(1.2)

i=1

Geodesic Equation

The space of K¨hler potentials associated to a k¨hler manifold (X, ω) is
a
a

Kω = {ωφ |ωφ = ω + i∂∂φ > 0, φ ∈ C ∞ (X)}

(1.3)

This may be identified with Hω = {φ|φ ∈ SP SH(X, ω) ∩ C ∞ (X)} which is open in C ∞ (X).
For each point of Hω one can associate to it a measure on X, dµφ =

n
ωφ
n! .

With this the

metric on Hω is given by specifying the L2 norm on functions i.e.

||ψ||2 =
φ

X

ψ 2 dµφ

where ψ ∈ Tφ Hω ∼ C ∞ (X).
=
So for a path φ(t) in Hω0 , parametrized by the unit interval, length is given by
1

l(φ) :=
0

X

9

˙
(φ(t))2 dµφ(t) dt

(1.4)

1
˙
By taking the first variation of the energy functional 0 X |φ(t)|2 dµφ(t) dt, since the critical

points define geodesics in the space of k¨hler metrics, the smooth geodesic equation is
a

1
˙
¨
φ(t) − | φ(t)|2 = 0
φ(t)
2

(1.5)

A path ut in Hω0 is viewed as a function U on X × [0, 1]. The following was first noticed
by Donaldson and Semmes.
Proposition 3. A path ut satisfying the geodesic equation is the same as looking for solutions
∗
of Ωn+1 = 0 with U (·, 1) = u0 and U (·, e) = u1 , where ΩU = (ddc U + πX ω0 ).
U

Proof.

∗
0 = (ddc U + πX ω0 )n+1
∗
= ((∂X ∂t + ∂t ∂ X + ∂t ∂t + ∂X ∂ X )ut + πX ω0 )n+1

= ((∂X ∂t + ∂t ∂ X + ∂t ∂t )ut + ωut )n+1
= P (ωut )

where P is a polynomial of degree at most n + 1 i.e
n+1
i
a(n+1−i) ∧ ωut

P (ωt ) =
i=0

n+1
Clearly we may assume a0 = 0 since ωut = 0 does not contribute. From binomial expansion

the terms an+1−i for i ≥ 3 are terms with forms containing at least three ”dt’s”( dt ∧ dt ∧ dt

10

etc), so they can also be assumed to vanish. For similar reasons we may assume

an+1−1 = an = (n + 1)∂t ∂t ut
an+1−2 = an−1 =

n
since (∂X ∂t ut ) ∧ ωut = 0

(n + 1)n
2∂X ∂t ut ∧ ∂t ∂ X ut
2

1
∗
(ddc U + πX ω0 )n+1
(n + 1)!
n
ωut
1
n−1
= ∂t ∂t ut ∧
−
∂ ∂ ut ∧ ∂ X ∂t ut ∧ ωut
n!
(n − 1)! X t

=⇒ 0 =

A local calculation shows

∂φ ∧ ∂φ ∧ ω n−1 =

1
| φ|2 ω n
g
2n

It follows

0=

1
1
n
(ut − | ∂t ut |2 t )ωut ∧ dt ∧ dt
¨
u
n!
2

11

1.3

-Geodesics

By considering the boundary value problem involving a degenerate Monge-Ampere equation
instead in [Chen00] C 1,1 geodesics are found i.e. solutions of

0 = Ωn+1
φ

(1.6)

= det(gαβ + φαβ ) on X × R
φ0 = φ on ∂(X × R)

(1.7)
(1.8)

where R is a riemann surface with boundary which can be taken to be a cylinder.
Solutions are extracted by running a continuity method. Adjustments at t = 1 are
made so the corresponding equation is elliptic. In other words its solution defines a K¨hler
a
potential on V ×R not just on each slice V ×{w}, w ∈ R. C 0 estimates can be obtained using
the boundary data and the maximum principle. An application of Yau’s C 2 estimate yields
the alternative that either the laplacian is uniformly bounded from above or the maximum
occurs on the boundary. So to obtain a pointwise bound on the maximum of the laplacian
in terms of maximum of the gradient a boundary estimate is needed. This is achieved from
the maximum principle applied to a barrier function construction and the structure of the
equation over the continuity path. So uniform C 2 bounds for t > 0 are obtained by obtaining
point-wise bounds on the gradient. This is done through a blowup analysis. This furnishes
a sequence of regular solutions φi to elliptic equations corresponding to a sequence ti

0.

Since at t = 0 the equation is degenerate, one passes only to a subsequence extracting a
weak C 1,1 solution. An application of maximum principle shows this limit is unique. The
details are the main content of [Chen00].

12

As a consequence the following also holds. Another application of the maximum principle
is required to get estimates on solutions with respect to the s parameter again see [Chen00]
for details.
Lemma 1 (Geodesic Approximation Lemma). Let Ci : φi (s) : [0, 1] → H (i = 1, 2) be
smooth curves in H. For 0 > 0 sufficiently small there exist two parameter smooth family
of curves C(s, ) : φ(t, s, ) : [0, 1] × [0, 1] × (0, 0 ] → H satisfying
1. For any fixed s, , there is an epsilon approximate geodesic C(s, ) joining φ1 (s) and
φ2 (s) i.e. φ(z, t, s, ) solves

V ×R

(1.9)

φ(z , 0, s, ) = φ1 (z , s)

(1.10)

φ(z , 1, s, ) = φ2 (z , s)

(1.11)

det(gαβ + φαβ ) = det(g)

where zn+1 = t +

√

−1θ and φ has trivial dependence on θ.

2. There exists a uniform constant C(which depends only on φ1 , φ2 ) satisfying

|φ| + |∂s φ| + |∂t φ| < C
2
0 ≤ ∂t φ < C

(1.13)

2
∂s φ < C

3. For fixed s let

(1.12)

(1.14)

→ 0, then the convex curve C(s, ) converges to the unique geodesic

between φ1 (s) and φ2 (s) in the weak C 1,η topology(0 < η < 1).

13

4. The energy element along C(s, ) is given by

E(t, s, ) =
V

|∂t φ|2 dg(t, s, )

(1.15)

where g(t, s, ) is the K¨hler metric corresponding to φ(t, s, ). Then there exists a
a
uniform constant C such that

max |∂t E| ≤ · C · M
t,s

(1.16)

So the energy/length element converges to a constant along each convex curve as → 0.

1.4

Convexity and eigenvalue estimate

In this section we study convexity of Lω0 along smooth geodesics in the setting when
dim H 0 (X, KX ⊗ L) = 1 and L ⊗ KX is globally generated. So L = −KX . For example this happens when X = P1 since L ∼ OP1 (m) for some m ∈ Z from an application of a
=
theorem of Grothendieck. So m = 2 since L ⊗ KP1 = O(0).
Let (X, L, ω0 ) be given as in §1 with the above restrictions. Since N = 1 for any
s ∈ H 0 (X, KX ⊗ L), det(T (u)) simplifies to

2

n
in s ∧ se−ψ0 = eθω0 ω0

14

which is basically (using L = −KX )
e−ψ0
= eθω0
det g

(1.17)

We may write s|Uα = φ ⊗ t locally where φα ∈ Γ(Uα , KX ) and tα ∈ Γ(Uα , L) holomorphic.
Note that

φα ⊗ tα = φβ det(ψαβ )−1 ⊗ tβ det(ψαβ ) = φβ ⊗ tβ

(1.18)

where {ψαβ , Uαβ := Uα ∩Uβ } is the cocycle determining TX . Denote by ||·||2 the fiber length
induced by hermitian metric h0 . Let θα : LUα ∼ Uα × C be the associated trivialization
=
induced by tα so |θα (tα )| = 1.

2

2

2

2

(in s ∧ ∗s)|Uα = in φα ∧ φα ||tα ||2 = in φα ∧ φα |θ(tα )|2 e−ψ0 = in φα ∧ φα e−ψ0

(1.19)

2

from (1.18) we may glue the local versions together and view in s ∧ se−ψ0 as a global section
−ψ
n2
of KX ⊗ KX just like ω n and so i s∧se 0 defines a global function. Note also that from
ωn

the last equality in (1.19), after shrinking Uα to a coordinate chart, with respect to the
coordinates there choosing φα = dz1 ∧ . . . ∧ dzn gives that the global function is of the form
given in (1.17).
Remark 2. L = −KX is not necessary all thats needed is 0 = s ∈ H 0 (KX ⊗ L) i.e
2

holomorphic global sections of it. That is in s∧se−ψ0 transforms as sections of KX ⊗K X just
as volume forms do so the ratio is a global function. To see this note that two trivialization
θ, θ are related by θ = gθ on the overlap and similarly ψ0 = ψ0 + log |g|2 on the overlap.

15

So θω0 ∈ C ∞ (X). Applying ∂∂log to (1.17)
√

√
−1
−1
θω0
∂∂ log(e ) =
(∂∂ log(e−ψ ) − ∂∂ log det(g))
2 √
2
−1
∂∂θω0 = Ric(ω0 ) − ω0 (∗)
2

Equation ∗ above is essentially

[F

L⊗KX

] = c1 (L ⊗ KX ) = c1 (O(0)) = 0

with c1 (KX ) represented by the negative of the ricci form and c1 (L) by the curvature of a
hermitian metric given locally by e−ψ0 . So it follows that ωt − Ric(ωt ) =

√

−1
2 ∂∂(−θωt ).

The negative sign is benign and chosen only to suit our conveniences.
√

−1
Recall Ric(ω0 ) = − 2 ∂∂ log det(g). Also

√
ωt = ω0 +

−1
∂∂φt
2

where φt ∈ Hω0 is an arbitrary path. So
√

−1
∂∂θωt =Ric(ωt ) − ωt
2
√

=Ric(ωt ) − Ric(ω) + ω0 + −1∂∂θω0 − (ω0 +
√
−1
det(g)
=
(log(
) + θω0 − φt ) (∗t )
2
det(g )

16

√

−1
∂∂φt )
2

and

θωt = log(

det(g)
det(g )

) + θω0 − φt + ct

(1.20)

θωt is clearly globally defined. To pin down the constant we choose normalization

X

n
e−φt +ct +θω0 ω0 = 1

(1.21)

Henceforth we abuse notation and denote φt + ct by φt . We conclude after exponentiating
(1.20)
ωn n
n
n
e−φt +θω0 ω0 = e−φt +θω0 0 ωt = eθωt ωt
n
ωt

(1.22)

That is φt moves along continuity path (1.22). Restricting further to smooth geodesics
we have
Proposition 4. When L ⊗ KX is globally generated and dim(H 0 (X, L ⊗ KX )) = 1 the
functional Lω0 is convex along smooth geodesics.
To check convexity along these geodesics use second variation of the Lω functional. This
n
simplifies to − log X e−φt +θω0 ω0 from the discussion above. So taking t derivatives two

times we obtain the quantity
n
n
˙
¨
˙ n
( X e−φt +θω0 ω0 ) X ((φ)2 − φ)e−φt +θω0 ω0 − ( X eθωt (φt )ωt )2
n
( X e−φt +θω0 ω0 )2

from which convexity of Lω is determined provided the following inequality( Lω has a neg-

17

ative sign):

(
X

n
e−φt +θω0 ω0 )

X

n
˙
¨
e−φt +θω0 ((φt )2 − (φt ))ω0 − (

X

˙ n
e−φt +θω0 (φt )ω0 )2 ≤ 0

Which simplifies using (1.22) to

(
X

n
eθωt ωt )

n
˙
¨
eθωt ((φt )2 − (φt ))ωt − (

X

X

˙ n
eθωt (φt )ωt )2 ≤ 0

˙ n
We may without loss assume X eθωt (φt )ωt = 0 which follows from differentiating the normalization condition chosen. Hence the inequality desired is

X

⇒

n
˙
¨
eθωt ((φt )2 − (φt ))ωt ≤ 0

1
n
˙
˙
eθωt ((φt )2 − (| φt |2 ))ωt ≤ 0
g(t)
2
X
⇒
X

n
˙
eθωt (| φt |2 )ωt ≥ 2
g(t)

X

n
˙
eθωt ((φt )2 )ωt

√

So we need to show whenever Ric(ω) − ω =

eθω (f )2 ω n ≤
X

−1
2 ∂∂θω

and X f eθω ω n = 0

1
(| f |2 ))eθω ω n
g
2
X

⇒ λ1 (−∆ −
holds.

18

θω ·) ≥ 2

(1.23a)

The first eigenvalue estimate in (1.23a) translates to, in the K¨hler case,
a

1
g αβ fα fβ eθω ω n
| f |2 eθω ω n =
2
X
X
= −
X

( f + g αβ fα (θω )β )eθω f ω n

That is the corresponding first eigenvalue estimate (1.23a) in this setting is µ1 (−
1
2

−

·, θω ) ≥ 1.

Remark 3. Just as with eigenvalue estimates for µ1 (− ) we may similarly consider using
1. Rαβ = gαβ + θαβ
2. 1 | f |2 = |fαβ |2 + |fαβ |2 + g αβ (fα ( f )β + ( f )α fβ ) + Rαβ f α f β
2
3. − f = µ1 f + 1
2

θω , f

However this point view encounters problematic pure type terms which cannot be controlled
by (1).
Instead consider u : X → C. Set u, v θ =

X

uveθω ω n hermitian weighted scalar

product. Using an orthonormal frame we have

|∂u|2 eθω ω n =
X

X

=
X

=
X

where Lu :=

uα uα eθω ω n
−(uαα + θα uα )ueθ ω n
−( u + θα uα )ueθ ω n = − Lu, u θ

u + θα uα .

Applying to first eigenfunctions u with eigenvalue µ1 obtain µ1 ||u||2 > 0 so µ1 > 0.
θ
19

In the case where Ric(ω) ≥ ω it follows λ1 ( ) ≥ 1(using ·, · 0 restricted to real valued
functions u):

0≤

|uαβ |2 ω n =

uαβ uαβ ω n

= −

uβα,β uα ω n

= −

(uββ,α uα + Rαβ uα uβ )ω n

Using Ric(ω) ≥ ω it follows that µ1 ( ) ≥ 1.
Similarly consider the quantity X |uαβ |eθ ω n ≥ 0 in the weighted setting. Also note
that since the operator L is essentially

up to lower order terms so it is elliptic: symbol

is determined by highest order terms. We wish to apply the following lemma to the first
eigenfunction which is a priori smooth.
Lemma 2. Let u ∈ C ∞ (X, C) then

X

(−(Lu)α uα − |∂u|2 )eθ ω n ≥ 0

Proof. Following the discussion above

0≤
X

|uαβ |eθ ω n =

X

=
X

=
X

=
X

uαβ uαβ eθ ω n
−(uβαβ uα + uαβ θβ uα )eθ ω n
(−( u)α uα − Rsβαβ uα us − uαβ θβ uα )eθ ω n
(−(Lu)α uα + θβα uβ uα
+ θβ uαβ uα − uαβ θβ uα )eθ ω n

− Rsα uα us
20

(1.24)

√

Since Ric(ω) − ω =

−1
2 ∂∂θ

in coordinates is equivalent to Rαβ − θαβ = gαβ = δαβ , it

follows

(Rβα − θβα )uβ uα = uα uα = |∂u|2

So (1.24) simplifies to

X

(−(Lu)α uα − |∂u|2 )eθ ω n ≥ 0
−(Lu)α uα eθ ω n ≥

⇒
X

|∂u|2 eθ ω n
X

proof of proposition. Suppose u is a first eigenfunction of the operator L with eigenvalue µ1
(i.e. −Lu = µ1 u) then by the lemma

µ1

X

uα uα eθ ω n ≥

|∂u|2 eθ ω n
X

=⇒ µ1 ≥ 1

1.5

Convexity of Lω0

Let ut be a path in Hω0 . Recall N := dim H 0 (X, KX ⊗ L) < ∞ by hodge theory. Also
recall that Lω0 (ut ) = −1 log det(T (ut )) where T (ut ) = [ si , sj ψ0 +ut ].
N

21

Proposition 5. When ut ∈ Hω0 is an arbitrary smooth path, the first and second variations
of Lω0 are given by
d
Lω (ut ) =
dt 0
d2
Lω (ut ) =
dt2 0

−1 ij ˙
(T Tij )
N
−1
˙
¨
[T r(−(T −1 T )2 ) + T r(T −1 T )]
N

(1.25)
(1.26)

Proof. This follows by direct computation recalling that for square matrices

d
˙
det(T (t)) = det T (t)T r(T −1 (t)T (t))
dt

p
p
Let s ∈ H 0 (X, ΩX ⊗ L) ∼ H p,0 (X, L) using resolution of ΩX ⊗ L by sheaves Ap,· (L)
=

which are acyclic (this is Dolbeaut’s theorem asserting Dolbeaut cohomology is isomorphic
to sheaf cohomology of holomorphic differential forms).
In this setting, via Hodge theory using hodge decomposition for holomorphic hermitian
vector bundles on compact hermitian manifolds and type considerations in this range we
obtain the decomposition

∗

Ap,0 (X, L) = ∂ Ap,1 (X, L) ⊕ Hp,0 (X, L)

∗

Note that ∂ Ap,1 is orthogonal to Ker∂. Let α ∈ Ap,1 , then since

∗

∗

(∂∂ α, α) = ||∂ α||2 > 0

we can specialize to p = n to obtain that H n,0 (X, L) ∼ Hn,0 (X, L). Hence s ∈ H 0 (X, KX ⊗
=
22

L) ∼ Hn,0 (X, L) is harmonic. So for s ∈ A0 (X, Ωn ⊗ L) satisfying s ⊥ H 0 (X, KX ⊗ L) we
=
X
∗

have by hodge decomposition that s = ∂ σ + ∂β where σ ∈ An,1 and β ∈ An,−1 = 0.
Lemma 3. With s as above, orthogonal to global holomorphic sections of KX ⊗ L, the
following estimate holds
||∂s||2 ≥ ||s||2

(1.27)

Remark 4. Restricted to the orthogonal complement of H 0 (X, KX ⊗ L), ∂ K ⊗L operator
X
∗

has no kernel (on the orthogonal complement, where ∂ K ⊗L = 0, ∂ K ⊗L is a restriction
X
X
∗

of the elliptic ∂ + ∂ operator) so should be invertible and thus satisfy an inequality of the
type (1.27) with perhaps better constants.
∗

Proof. Since s = ∂ σ, (1.27) is equivalent to

∗

∗

||∂∂ σ||2 ≥

||∂ σ||2

An application of cauchy-schwartz gives that

∗

∗

∗

||∂ σ||2 = ∂∂ σ, σ ≤ ||∂∂ σ||||σ||

(1.28)

Thus it suffices to show
Claim 1.
∗

||σ||2 ≤ ||∂ σ||2
∗

(1.29)

The Hodge decomposition σ = α + ∂β + ∂ γ simplifies to σ = ∂β where β ∈ An,0 (L).

23

∗

∗ ∗

Since 0 = ∂ α = ∂ ∂ γ

∗

∗

s=∂ σ=∂ β

so we may take σ = ∂β.
A version of the Bochner-Kodaira-Nakano identity simplifies using [Λ, θ(L)] = [Λ, ω] =
[Λ, L] to

D

=

D

+ (p + q − n) · I

(1.30)

So we may obtain the L2 identity from (1.30) applied to σ

∗
||∂σ||2 + ||∂ σ||2 =||D σ||2 + ||(D )∗ σ||2 + (n + 1 − n)||σ||2
∗

=⇒ ||∂σ||2 + ||∂ σ||2 ≥||σ||2

(1.31)

But σ is a holomorphic section so we obtain

∗

||∂ σ|| ≥ ||σ||

So the claim follows and hence the lemma.
Now given ||∂s|| ≥ ||s|| for s ⊥ H 0 (X, KX ⊗ L) we can determine the shape of the
inequality for s ∈ An,0 (L). To do this consider

P : An,0 (L) → H 0 (X, KX ⊗ L)

24

the projection of s ∈ An,0 (L) to its holomorphic part that can be viewed as a holomorphic
global section of KX ⊗ L.
Proposition 6. Given s ∈ An,0 (L)

||∂s||2 ≥ ||s||2 − ||P (s)||2

(1.32)

Proof. This follows immediately from the lemma applied to s − P (s) which is orthogonal to
H 0 (X, KX ⊗L), and using that the holomorphic projection P is an orthogonal projection.
Now having obtained the inequality (1.32) we can proceed to the the main business of
this section
Proposition 7. Assume H 0 (X, KX ⊗ L) = 0 then Lω0 is convex along smooth geodesics
(when they exist).
Remark 5. We could have replaced the assumption with the globally generated condition
appearing in [Ber10a] but that assumption is really cooked up for the critical points equation;
so that it is elliptic.
Proof. Let h be an arbitrary metric on L deformed from the background metric on L, h0 ,
related by h = h0 e−φ . Taking a basis {si } ⊂ H 0 (X, KX ⊗ L) orthogonal basis with respect
˙
to ·, · h , applied to φsi we have in (1.26) that T −1 = Id. So

˙
T r(−(T −1 T )2 ) = −

(

˙
φ(si , sj )h )2

ij

X

i

˙
| φ|2
˙
− φ2 )(si , si )h
2
X

¨
T r(T −1 T ) = −

(

25

(1.33)

Also note

˙
||P (φsi )||2 =

(
j

˙
||∂(φsi )||2 =

X

˙
φ(si , sj )h )2

˙
| φ|2
(si , si )h
2
X

(1.34)

˙
Applying (1.32) to sections φsi with respect to the metric h on L and summing over 1 ≤ i ≤
dim(H 0 (KX ⊗ L)) gives

˙
˙
(||φsi ||2 − ||P (φsi )||2 )

˙
||∂(φsi )||2 ≥
i

⇒
i

i

˙
| φ|2
˙
˙
(
(
φ(si , sj )h )2
− φ2 )(si , si )h ≥ −
2
X
X
ij

(1.35)

The proposition follows.
Together with -geodesics it can be shown that Lω0 is convex along C 1,1 geodesics. This
is verified in §7.

1.6

Maximizers using -geodesics

Recall from section §3 there is a smooth path ut , an -geodesic, connecting a critical point
u0 to another point u1 of Hω0 . It satisfies
| ut |2
˙ g(t)
(ut −
¨
) det g(t) = det g > 0
2
| ut |2
˙ g(t)
det g
=⇒ ut =
¨
+
det g(t)
2

26

where g(t) = gαβ + (ut )αβ ( 1 ≤ α, β ≤ n)
From proposition 1 along smooth paths ut ∈ Hω0
n

ωu
1
ut t
˙
dEω0 (ut ) =
V X
n!
n
| ut |2
˙ g(t) ωu
d2
1
=⇒ 2 Eω0 (ut ) =
) t
(ut −
¨
V X
2
n!
dt

so along an -geodesic it follows
d2
Eω (ut ) =
V
dt2 0

n

det g ωut
=
X det g(t) n!

(1.36)

Take an orthonormal basis {si } ⊂ H 0 (X, KX ⊗ L) with respect to ·, · h and using the
-geodesic equation we get
−1
d2
Lω0 (ut ) =
[−
N
dt2

(
ij

X

ut si , sj h )2
˙

i

| ut |2
˙ g(t)
det g
(ut −
˙
−
)(s , s ) ]
2
det g(t) i i h
X

i

det g
(si , si )h ≥ 0
X det g(t)

2

+
≥

1
N

(1.37)

The last inequality follows from (1.35). In other words

−

(
ij

X

ut si , sj h
˙

)2

| ut |2
˙ g(t)
(ut −
˙
)(si , si )h ≤ 0
2
X
2

+
i

The right hand side of (1.37) is positive and by item 3 of lemma 1 -geodesics converge in
C 1,1 topology to the C 1,1 geodesic connecting the points. In particular sending

→ 0, we

obtain f (t) := Lω0 (t) the function obtained by restricting Lω0 along C 1,1 geodesic, is the
27

uniform limit of similarly defined functions f (t) that are convex. So f (t) is convex. In
summary we have
Proposition 8. If dim H 0 (X, KX ⊗ L) ≥ 1 then Lω0 is defined and is convex along C 1,1
geodesics.
Proof. See discussion above.
Similarly e(t) := Eω0 (t) restricted to the C 1,1 geodesic is a uniform limit of functions
e (t), those function that arise from restricting Eω0 to

geodesics. Recall Eω0 is continuous

under uniform limits in Hω0 ∩ C 0 (X) (the L1 (X, ω0 ) closure), a result of Bedford-Taylor, in
particular with respect to uniform limits of -geodesics. From (1.36) the second derivative
of Eω0 along -geodesics goes to zero. Since the second derivative of Eω0 is the integral of
the geodesic equation in the sense of Donaldson and Semmes, the convergence to zero is in
the sense of Chen. Thus e(t) is affine.
Corollary 1. The functional
Fω0 := Eω0 − Lω0
is concave along C 1,1 geodesics.
Proof. This follows since Eω0 is affine while −Lω0 is concave along C 1,1 geodesics.
The result of Berman [Ber10a] follows:
Corollary 2. Critical points (when they exist) are maximizers of Fω0
Proof. Let u0 be a critical point of Fω0 and u1 ∈ Hω0 any other point connected by the
C 1,1 geodesic ut . Since Fω0 is concave along ut , dt Fω0 decreases along ut . It follows u0 is a
maximum in Hω0 for Fω0 . So an absolute maximum of Fω0 on Hω0 obtains at any critical
point (when it exists).
28

Also for future use we record
d2
1
F (u ) ≤ (1 −
2 ω0 t
N
dt

i

det g
(si , si )h ) ≤
X det g(t)

along -geodesics.

1.7

Uniqueness smooth case

Let ut be a smooth geodesic in Hω0 so ωut > 0 connecting critical points of Fω0 . A (1, 0)
vector field Vt can be defined by
ωut (Vt , ·) = ∂ ut
˙
We abuse language and call Vt a gradient vector field of u(otherwise we need to fit the
˙
fixed complex structure J into expressions when making the reference).
The main objective in this section is
Proposition 9. When there is a smooth geodesic connecting critical points ω0 := ωu0 , ωu1
of Fω0 in Hω0 the critical points are related by an automorphism φ of (X, L) i.e ωu1 = φ∗ ω0 .
1
Remark 6. Really there is always a C 1,1 geodesic connecting the critical points. The proposition applies with this path smooth.
In this direction note that expression for a gradient vector field in the adjoint setting can
be written as
Lemma 4. For s ∈ An,0 (L), ωut > 0 and Vt the gradient vector field of ut
˙

−∂ ut ∧ s = ωut ∧ (Vt s) = Lωu (Vt s)
˙
t

29

(1.38)

where Lωu is the lefschetz operator defined by ωut .
t
Proof. Simply wedge the equation defining the gradient vector field with s ∈ An,0 (L). Use
that ωut ∧ s is of type (n + 1, 1) and therefore vanishes. Then conclude using the elementary
calculation

0 = ωut ∧ s
0 = Vt (ωut ∧ s)
= (Vt ωut ) ∧ s + ωut ∧ (Vt s)
=⇒ −∂ ut ∧ s = ωut ∧ (Vt s)
˙

Corollary 3.
˙
−Λωt ((∂ ut ) ∧ s) = Λωt Lωu (Vt s)
t

(1.39)

Proof. Just take traces of (1.38). That is, act on it by Λωu .
t

Lemma 5. As above all operators are defined with respect to ωut > 0

˙
˙
−D (Vt s) = ΛD (∂ ut ∧ s) + i(D )∗ (∂ ut ∧ s)

(1.40)

Proof. To see this recall in the K¨hler setting
a

[L, Λ] = H

where L, Λ, H are the Lefschetz, dual Lefschetz, and counting operator which act on the
30

form part of the section.
Note that LΛ(Vt s) = 0 since Vt s is an L valued (n − 1, 0) form and the action of Λ
reduces type by (1, 1).
Claim 2.
ΛL(Vt s) = (Vt s)
Proof. Indeed,

ΛL(Vt s) = −[L, Λ](Vt s) = −H(Vt s) = −(n − 1 − n)I(Vt s) = (Vt s)

From the K¨hler identity
a
[Λ, D ] = −i(D )∗

(1.41)

noting that the left side of (1.41) is a commutator it follows directly that

D Λ(∂ ut ∧ s) = ΛD (∂ ut ∧ s) + i(D )∗ (∂ ut ∧ s)
˙
˙
˙

(1.42)

Now (1.40) follows immediately from the corollary and claim.
Proposition 10. Vt is holomorphic for each fixed t.
Proof. It is enough to show by lemma 5 (1.40) that D (Λ(∂ ut ) ∧ s) = 0 (∂ = D
˙

for

unitary connections compatible with the holomorphic structure). This follows because from
holomorphicity of s we obtain Vt is away from the zero’s of the vector field, and conclude Vt
is holomorphic by Riemanns extension theorem(Vt smooth).

31

Start by analyzing the equality case. Since Fω0 is concave it is affine along smooth
geodesics connecting any two of its critical points. Eω0 is affine along geodesics so Lω0 is
too. By smoothness this means (1.26) and identities (1.33), (1.34) yield the equalities

0=

d2
−1
(
L (u ) =
2 ω0 t
N
dt

(||ut si ||2 − ||P (ut si )||2 − ||∂(ut si )||2 )) ≥ 0
˙
˙
˙

(1.43)

i

which is equivalent to

||∂(ut si − P (ut si ))||2 = ||ut si − P (ut si )||2
˙
˙
˙
˙

(1.44)

∗

Recalling (1.28), (1.29) and from the discusion in section §5 solving ∂ σi = ut si − P (ut si )
˙
˙
with σi holomorphic we have

∗

∗

∗

∗

∗

∗

||∂ σi ||2 ≤ ||∂∂ σi ||||σi || ≤ ||∂∂ σi ||||∂ σi || ≤ ||∂ σi ||||∂ σi ||

∗

(1.45)

∗

The last inequality is a consequence (1.44) from which one obtains ||∂∂ σi || = ||∂ σi ||.
Since (1.45) is really a string of equalities it follows that the inequality (1.29) is an equality
∗
||σi || = ||∂ σi ||. Since (1.31) is equality precisely when the terms ||D σi ||, ||(D )∗ σi || vanish

in the Bochner Kodaira type identity we have

D

σi = 0,

∂σi = 0

(1.46)

Claim 3.
∂ ut ∧ si = σi
˙

32

(1.47)

Proof. This a consequence of the equality case :

∂ ut ∧ si = ∂(ut s)
˙
˙
= ∂(ut si − P (ut si ))
˙
˙
∗

= ∂∂ σi
= σi

The last equality follows from type considerations and equations in(1.46) (essentially the
content of (1.30), (1.31)). So

∗

∂∂ σi =

D

σi = (

D

+ I)σi = σi

As a consequence of the claim and (1.46)

(D )∗ ∂ ut ∧ si = (D )∗ σi = 0
˙

˙
D ∂ ut ∧ si = D ∂(φsi ) = 0;
˙

So by the lemma 5 (1.40) Vt is holomorphic for each fixed t.
Proposition 11. Vt is static.

33

Pass to co-ordinates and make the local calculation for Vt ωut = ∂ ut
˙
√

−1gmn Vtm X n = (ut )q X q
˙
p

Vt = g pq gmq Vtm
= −

√
−1g pq (ut )q
˙

The following is well known see [Bo09], except here we operate directly on the manifold.
Lemma 6. Along smooth geodesics

1
˙
Vt ωut = ∂(¨t − | ut |2 u ) = 0
u
˙ ω
t
2

(1.48)

˙
Proof. Differentiate Vt ωut = ∂ ut to obtain

˙
Vt ωut + Vt ∂t ωut = ∂ ut
¨
˙
Vt ωut = ∂ ut − Vt ∂t ωut
¨

Using Vt ωut = ∂ ut again obtain
˙
√

−1∂Vt ωut =

√

−1∂∂ ut = ∂t ωut
˙

Equivalently (1.50) is
˙
Vt ωut = ∂ ut −
¨

√
−1Vt (∂(Vt ωut ))

34

(1.49)
(1.50)

Computing

√

−1Vt ∂(Vt ωut ) locally:
√

−1Vt ∂((ut )j dz j ) =
˙
=

√
√

−1Vt ((ut )ij dz i ∧ dz j )
˙
−1Vti (ut )ij dz j
˙

= g is (us )(uij )dz j
˙ ˙
1
= ∂ | ut |2 u − ∂(Vti )(ut )i
˙ ω
˙
t
2
1
= ∂( | ut |2 u )
˙ ω
t
2

proof of prop. 11. By the lemma

1
˙
Vt ωut = ∂(¨t − | ut |2 u ) = 0
u
˙ ω
t
2

So conclude δVt = 0
+V = −ImV genproof of prop. 9. Now we are ready to conclude with proposition 8. −V2

erates the flow φt (φt biholomorphism). Since V above is static so is iV . Abusing notation
denote this by V and then φt is the flow generated by ReV . Locally V ∂∂(ut + ψ0 ) = ∂ ut .
˙
Virtues of compactness grant a uniform r > 0 , with ut + ψ0 given on some Bω0 (r, pi0 ) and
r
r
φt (Bω0 ( 2 , pi )) ⊂ Bω0 (r, pi ) for |t| < 2 and all i. A consequence of chain rule is

1
˙
˙
((ut + ψ0 ) · φt ) = (V ∂(ut + ψ0 ) + V ∂(ut + ψ0 )) · φt + (ut + ψ0 ) · φt
2

35

(1.51)

taking

√

−1∂∂ of (1.51) obtain
√

˙
−1∂∂((ut + ψ0 ) · φt ) =

+

√

√
1
−1∂∂ (V ∂(ut + ψ0 ) + V ∂(ut + ψ0 )) · φt
2

˙
−1∂∂ (ut + ψ0 ) · φt

√
d ∗
1
φt ωut = φ∗ ( −1∂∂(V ∂(ut + ψ0 ) + V ∂(ut + ψ0 )))
t
dt
2
√
˙
+ φ∗ ( −1∂∂ (ut + ψ0 ))
t
√
√
d ∗
1
φt ωut = φ∗ (∂( −1V ∂∂(ut + ψ0 )) + ∂(− −1V ∂∂(ut + ψ0 )))
t
dt
2
√
˙
+ φ∗ ( −1∂∂ (ut + ψ0 ))
t
=

√
1√
˙
˙
˙
−1φ∗ (−∂∂ (ut + ψ0 ) + ∂∂ (ut + ψ0 )) + φ∗ ( −1∂∂ (ut + ψ0 )) = 0
t
t
2

d
˙
So dt φ∗ (ωut )|B( r ,p ) = ∂∂((ut + ψ0 ) · φt ) = 0. Although the proposition now follows alt
2 i
0

most directly, we may also conclude by partitioning [0, 1] into sufficiently small sub-intervals
depending on the cover and the fact that the time one map is a composition of the maps
corresponding to each subinterval.
Remark 7. Note the above is essentially a manifestation of

d ∗
d
∗
φ−t ωut = φ∗
−t ωut + φ−t L (V −V ) ωut
dt
dt
−
2

where V is the gradient vector field originally defined and φt is the flow generated by ImV .
Use Cartans formula to obtain cancellations.
Since φ∗ ωut − ωu0 = 0 this means at the level of potentials φ∗ (ut + ψ0 ) − ψ0 = Ct .
−t
−t
V can be lifted to a holomorphic vector fields on the total space L so that action by φt
is induced from Aut0 (M, L) by lemma 13 in [Ber10a].

36

1.8

Berndtsson argument setup

Recall (X, ω) is K¨hler and that Fω is concave. Another advantage is that it behaves nicely
a
in the low regularity setting. In fact, for low regularity purposes, when L = −KX the
functional simplifies to negative of the Ding-Tian functional and this has better regularity
properties than the Mabuchi functional . Berndtsson shows by a direct envelope construction
critical points of the Ding-Tian functional can be connected by a C 0 sub-geodesic (see §11,
[Bo] and [BerDe]). These two inputs (Ding-Tian functional and C 0 sub-geodesics) can be
used to obtain the Bando-Mabuchi uniqueness theorem.
More precisely Berndtsson obtains Bando-Mabuchi uniqueness theorem by deducing
Proposition 3. Let L = −KX be semi-positive and assume H n,1 (X) = 0 . Let φt be C 0
sub-geodesic such that φt does not depend on Imt, then L(t) = − log in

2
X

e−φt dz ∧ dz is

convex. Further if L(t) is affine in a neighborhood of 0 then there is holomorphic vector field
V (perhaps time dependent) on X with flow Ft such that Ft∗ ∂∂φt = ∂∂φ0 .
As in the previous section one needs to analyze the smooth case. The final stage involves
approximation.
Let ut smooth but i∂∂ut ≥ 0. Then consider solvability of

∂ ut v = π⊥ (ut s) =: η
˙

(1.52)

s ∈ H 0 (X, Ωn ⊗ L) as before. From the consequence of the Lefschetz decomposition that
X
Λn−1 V ∼ Λn+1 V write α = Lω v = v ∧ ω where v ∈ An−1,0 (X) when α ∈ An,1 (X, L). Then
=

37

Proposition 12. Solvability for v in (1.52) is equivalent to solvability of

∗

∂ ut α = η

for α ∈ An,1 (X, L) when η is orthogonal to H 0 (X, KX ⊗ L).
Proof. The equivalence is a calculation using the following facts:
• Note that ∂ut is the (1, 0) part of the Chern connection i.e

(1,0)

:= ∂ + ∂ log(e−ut ) = ∂ − ∂ut ∧ = ∂ut

• Recall we have for α ∈ P k (primitive elements of Λk )

∗Lj α = (−1)

k(k+1)
2

j!
Ln−k−j I(α)
(n − k − j)!

∗, L determined by the structure from (X, ω).
∗
• [∂ , L] = i∂. For L|U ∼ OU with hermitian structure depending on ut as above, the
=

adjoint operator is given by

∗

∂ t = −∗∂ut ∗ = − ∗ ∂ut ∗

38

then

∗

∗

∂ ut α = ∂ ut v ∧ ω
= − ∗(∂ − ∂ut ∧ ∗(v ∧ ω))
∗

= [∂ , L]v + ∗∂ut ∗ Lv

But v is primitive and in v ∈ P n−1 . It follows ∗Lv = (−1)

(n−1)n
2
in−1 v,

and using that

∂φ ∧ v ∈ P n we get

∗

∗

∗

∂ ut α = ∂ Lv + ∗∂ut ∗ Lv = [∂ , L]v + (−1)

(n−1)n
2
in−1

∗ ∂ut ∧ v

= i∂v + (−1)

(n−1)n
2
in−1

= i∂v + (−1)

(n−1)n
n(n+1)
2
in−1 (−1) 2 in ∂ut

∗ ∂ut ∧ v
∧v

2

= i∂v + (−1)n (−1)n−1 i∂ut ∧ v
= i(∂ − ∂ut ∧)v
= i∂ut v
∗

=⇒ ∂ ut α = η = ∂ut v

The solvability of these equations also comes with estimates.
Fact 1. Recall for ∂ and its adjoint (Von Neuman’s sense is the relevant one)
∗

• Ker∂ = (Im∂)⊥
• (Ker∂)⊥ = Im∂

∗

39

∗

In particular when ∂ is a surjection, ∂ is injective. In fact when H n,1 (X) = 0 we have
a surjection on ∂ closed forms and the adjoint is injective. When ∂ has closed range the
adjoint has closed range equal to (Ker∂)⊥ . In this setting closedness of range boils down to
the estimates.
If ∂ has closed range then solutions to the equation ∂f = α comes with the estimate

||f || ≤ C||α||

(1.53)

From here it follows that solvability of ∂ut v = η comes with the estimates

||v|| ≤ C||η||

To elaborate, (1.53) is similar to §6 where solving σ1 = ∂σ0 for σ0 orthogonal to holomorphic (n, 1) L-valued sections comes with estimate ||∂σ0 || ≥ C0 ||σ0 ||, C0 = 1. In §6 we
specialized the hermitian metric to the one whose curvature is −iω to conclude. However,
in the K¨hler setting the Akizuki-Nakano identity applies for any unitary connection coma
patible with the holomorphic structure. So more generally ||∂σ0 || ≥ C||σ0 || (adjusting to an
auxiliary hermitian metric h changes the constant arising from X [θh , Λ]σ0 , σ0 dV ). So for
∗

our purposes proceed trivially obtaining the estimate ||∂ ut σ0 ||2 + ||∂σ0 ||2 ≥ C||σ0 ||2 , adding
∗

the extra nonnegative term ||∂ ut σ0 ||2 . From here conclude through a functional analysis
∗
˜
argument (see [De]) to obtain the estimate ||∂ ut σ0 || ≥ C||σ0 ||.
∗

In particular we obtain η = ∂ ut α and α = ∂f
∗

˜
||∂ ut α|| ≥ C||α||

40

Next we claim
Claim 4. For α = v ∧ ω as above, ||α|| = ||v||.
Proof.

α, α = Lv, Lv = v, ΛLv
ΛLv = −[L, Λ]v = −Hv = −(n − 1 − n)v = v
=⇒ α, α = v, v

As a consequence
∗

˜
˜
||∂ ut α||C ≥ ||α|| ⇐⇒ ||∂ut v|| ≥ C||v||
It follows that if vt solves
∂ut vt = π⊥ (us)
˙
the following estimate holds

||vt || ≤ ||π⊥ (ut s)|| ≤ ||ut s|| ≤
˙
˙

1
||u ||||s||
˙
˜ t
C

(1.54)

From the properties of C 0 sub-geodesics i.e its Lipschitz, it is known that ||ut || is bounded.
˙

41

1.9

Generalized Gradient Vector Field

Since L ⊗ KX ∼ OX we can take 1 = s ∈ H 0 (X, L ⊗ KX ) and define vt by ∂ ut vt = π⊥ (ut s).
˙
=
In turn, since s does not vanish, define Vt by

vt = −Vt s

(1.55)

Recall that curvature of a hermitian connection on a holomorphic vector bundle has no (0, 2)
part so it is given by

[Dut , ∂] = ∂ ut ∂ + ∂∂ ut = ∂∂ut

(1.56)

Lemma 7. For solutions ∂ut vt = π⊥ (ut s) the following identity holds
˙

˙
∂∂ut ∧ v = ∂(ut s) + ∂ ut ∂v

Proof. Using (1.56) and the definition of ∂ ut vt

∂∂ut ∧ vt = [Dut , ∂]vt = ∂ ut ∂vt + ∂π⊥ (ut s)
˙

From (1.55) obtain

∂∂ut ∧ v = ∂∂ut ∧ (−Vt s)
= (Vt ∂∂ut ) ∧ s
42

(1.57)

Lemma 8. If ∂v = 0 then Vt defined in (1.55) satisfies

∂ ut = Vt ωut
˙

(1.58)

Proof. For s as given above

∂ ut ∧ s = (Vt ∂∂ut ) ∧ s
˙
˙
=⇒ i∂ ut = Vt i∂∂ut
= Vt ωut

Remark 8. Vt as defined above is a generalized gradient vector field(referred to as a gradient
vector field for convenience since it behaves similarly).
Note that
e−ut = − log ||s||2

F(t) := − log

(1.59)

X

where ||s||2 = X cn s ∧ se−ut .
Remark 9. ||˜||2 can be viewed as integration along fibers. Exactly as Berndtsson considers
s
˜
K¨hler fibrations with compact fibers, p : X → Y . Here, 0 ∈ U 0 = U = Y ⊂ C, the fibers
a
˜
are copies Xt := X = p−1 (t) and we may think of X = U × X (we suppress the other
structures since we wish to discuss this naively. Involved is the introduction of a structure
E := ∪t∈U {t} × Et with Et := H 0 (Xt , L|Xt ⊗ KXt ). E → U is naturally a holomorphic
vector bundle from semi-positivity of L and that X is K¨hler using an Ohsawa-Takegoshi
a
extension type theorem: an element of Et needs to extend to a section of E locally in a
43

holomorphic way with estimates. Elements of Et which are L valued (n, 0) forms on Xt can
be viewed as sections taking values in KX over Xt (by wedging with dt) or as the restriction to
˜
˜
Xt of sections over X with values in KX see [Bo09] and [Bo07] for details. Granted elements
˜
of Et extend to local sections of E, take a basis of Et that extends to a local frame of E.
These can be viewed as a collection of (n, 0) forms over the preimage under the projection p
of an open set in U in the base. Denote one such by u. Its restriction to each Xt defines an
element of Et . u defines holomorphic section of E if it defines a holmorphic section of KX
˜
i.e u ∧ dt is a holomorphic section of KX , that is ∂u ∧ dt = 0. Finally E has a naturally
˜
defined hermitian metric coming from that on L(fiber-wise this is the usual hermitian metric
on L ⊗ KX ) allowing to define the Chern connection operator on E.)
Recall
||s||2 = π∗ (cn s ∧ se−ut )
by definition of integration along fibers obtain
Claim 5. Let v ∈ An−1,0 (X) and s as above. Set s = s − dt ∧ v then
˜

s
∂∂ t ||˜||2 = ∂∂ t ||s||2

This can also be calculated directly viewing X as integration along fibers since the
fibration is trivial.
First note
s ∧ s = s ∧ s − s ∧ dt ∧ v − dt ∧ v ∧ s + dt ∧ v ∧ dt ∧ v
˜ ˜

44

Proof. In the following calculation the type I forms have no contribution so

s
∂∂ t ||˜||2 =

X

=
X

cn ∂∂ t (˜ ∧ se−ut )
s ˜
cn ∂∂ t (s ∧ s − s ∧ dt ∧ v − dt ∧ v ∧ s + dt ∧ v ∧ dt ∧ v)e−ut

= −
X

=
X

cn s ∧ s ∧ ∂∂ t ut e−ut +

X

(cn s ∧ s ∧ ∂t ut ∧ ∂ t ut e−ut )

cn ∂∂ t (s ∧ se−ut ) = ∂∂ t ||s||2

Proposition 13. Given s as above we have
˜

∂∂ t ||˜||2 =(−1)n (
s
+
X

cn ∂ ut s ∧ ∂ ut se−ut +
˜
˜
X

cn ∂˜ ∧ ∂˜e−ut )
s
s
X

cn s ∧ s ∧ ∂∂ut e−ut
˜ ˜

(1.60)

cn ∂˜ ∧ ∂˜e−ut +
s
s

=(−1)n
X

X

cn s ∧ s ∧ ∂∂ut e−ut
˜ ˜

˜ ˜
=(||∂v||2 dt ∧ dt + π∗ (cn ∂∂ut ∧ s ∧ se−ut ))

where X is interpreted as integration along fibers.
Proof. (1.60) simplifies to (1.61) since ∂ ut s vanishes:
˜

∂ ut vt = π⊥ (us)
˙
=⇒ dt ∧ ∂ ut vt = dt ∧ π⊥ (ut s)
˙
= dt ∧ (ut s + h)
˙
= ut dt ∧ s + dt ∧ h
˙

45

(1.61)
(1.62)

= π⊥ (∂ut ∧ s)
= − π⊥ ((∂ − ∂ut )s)
= − ∂ ut s

Where in the third line h subtracts out the holomorphic part of ut s. The last line follows
˙
∗
⊥
since Im∂ u ⊆ Im∂ = Ker∂ . So ∂ ut s = 0.
˜

(1.62) follows since s = s − dt ∧ v so
˜

∂˜ = dt ∧ ∂v
s
=⇒ ∂˜ ∧ ∂˜ = (−1)n ∂v ∧ ∂v ∧ dt ∧ dt
s
s

To obtain (1.60) some pre-computation is necessary. Observe (1.63), (1.64), and (1.65) hold:

X

cn ∂ t s ∧ se−ut = 0
˜ ˜

(1.63)

since this involves integration along fibers of a type I form

X

cn ∂ t s ∧ se−ut =
˜ ˜

X

cn dt ∧ ∂ t v ∧ se−ut = 0
˜

Similarly
s
cn s ∧ ∂˜e−ut = 0
˜

(1.64)

X

from (1.63) and (1.64) it follows:

0 = ∂t

cn s ∧ ∂˜e−ut =
˜
s
X

cn ∂˜ ∧ ∂˜e−ut + (−1)n
s
s
X

cn s ∧ ∂ ut ∂˜e−ut
˜
s
X

46

(1.65)

(1.60) follows from the following computation:

u

s
˜
∂∂ t ||˜||2 = (−1)n ∂t π∗ (cn s ∧ ∂t t se−ut )
˜
u

u

u

˜
˜
˜
= (−1)n π∗ (cn ∂t t s ∧ ∂t t se−ut ) + π∗ (cn s ∧ ∂∂t t se−ut )
˜
= (−1)n π∗ (cn ∂ ut s ∧ ∂ ut se−ut ) + π∗ (cn s ∧ ∂∂ ut se−ut )
˜
˜
˜
˜
= (−1)n π∗ (cn ∂ ut s ∧ ∂ ut se−ut ) + π∗ (cn s ∧ ∂∂ut ∧ se−ut )
˜
˜
˜
− π∗ (cn s ∧ ∂ ut ∂˜e−ut )
˜
s

Further, applying (1.65)

s
˜
∂∂ t ||˜||2 = (−1)n (π∗ (cn ∂ ut s ∧ ∂ ut se−ut )
˜
+ (−1)n π∗ (cn ∂˜ ∧ ∂˜e−ut )) + π∗ (cn s ∧ ∂∂ut ∧ se−ut )
s
s
˜
˜

and the proposition follows.
Proposition 14.
||s||2 ∂∂ t F(t) = θt s, s = −∂∂ t ||s||2
where θt is the curvature of E.
Proof. Recall ||s||2 = ||˜||2 . For the second equality see [Bo09]. In the first equality note
s
that

∂t ||˜||2 = ∂ t s, s + (−1)n s, ∂ ut s = 0
s
˜ ˜
˜
˜

47

so

∂t ∂t F(t) = −

s
s
s
s
∂∂ t ||˜|| ∂t ||˜||∂t ||˜||
∂∂ t ||˜||
∂∂ t ||s||
+
=−
=−
2
4
2
||s||
||s||
||s||
||s||2

where the last equality follows from the claim.
Remark 10. Granted the necessary regularity, from convexity of F along ut connecting
two K¨hler-Einstein metrics ∂∂ t F ≡ 0 i.e F is linear on ut . (1.62) and the subsequent
a
˜ ˜
˜
proposition obtain ||∂v|| = 0. It also follows ∂∂ut ∧ s ∧ s = 0 from which ∂∂ut ∧ s = 0 since
i∂∂ut ≥ 0. Since ∂v = 0 we see from (1.58) that Vt as defined is a gradient vector field.
˜
Lemma 9. Suppose ut is smooth and ∂∂ut ∧ s = 0 then

(

∂ut
∂ut
− ∂X (
)(Vt )) = 0
∂t∂t
dt

(1.66)

Proof. Since ∂∂ut ∧ s = 0 it follows that the coefficient of dt ∧ dt vanishes:
˜
∂ut
∧ s) − ∂X ∂t ut ∧ dt ∧ v
∂t∂t

(1.67)

∂ut
∂X ∂t u ∧ dt ∧ v = ∂X
dt ∧ dt ∧ Vt s
∂t

(1.68)

0 = dt ∧ dt(

but since vt = −Vt s

and

∂ut
)∧s
∂t
∂ut
∂ut
=⇒ 0 = Vt ∂X (
) ∧ s − ∂X (
) ∧ Vt s
∂t
∂t
0 = ∂X (

48

(1.69)

So from (1.68) and (1.69)

∂ut
∂X ∂t u ∧ dt ∧ vt = dt ∧ dt ∧ s(Vt ∂X (
))
dt
∂ut
∂ut
=⇒ 0 = dt ∧ dt ∧ s(
− ∂X (
)(Vt ))
∂t∂t
dt

This concludes the calculation.
∂u

∂u

t
Set µ := ( ∂t∂t − ∂X ( dtt )(Vt )).

Remark 11. When i∂∂ut > 0, we have 0 = µ = c(φ) satisfies the geodesic equation because
˙
∂ φ(Vt ) = |Vt |2 with Vt the gradient vector field.
t
Since Vt satisfies the equation vt = −Vt s, the condition that Vt is static translates to

0=

∂Vt
∂vt
=−
s
∂t
∂t

since s does not vanish.
∂v

Proposition 15. If H n,1 (X, L) = 0(vanishes if i∂∂ut > 0) then ∂tt = 0
Proof. Recall we have
∂ ut v = ut s + ht
˙

(1.70)

where ht is holomorphic for each fixed t.
Since ∂ ut = ∂ − ∂ut ∧
∂v
∂ut
∂ ut
∂ v = ∂ ut
− ∂X
∧v
∂t
∂t
∂t
∂v
∂ut
= ∂ ut
+ Vt ∂X (
)s
∂t
∂t
49

(1.71)

from (1.69).
Similarly the right hand side of (1.70) becomes
∂
∂ 2 ut
∂ht
(ut s + ht ) =
˙
s+
∂t
∂t∂t
∂t

(1.72)

Combining (1.71), (1.72)
∂ ut

∂ht
∂v
= µs +
∂t
∂t

but ∂ ut ∂v is orthogonal to holomorphic forms so
∂t

∂ ut

∂v
∂ht
= π⊥ (µs +
) = π⊥ (µs) = 0
∂t
∂t

since µ = 0. Note ∂v ∧ ω is ∂ X closed. This entails ∂v = 0 because the assumption
∂t
∂t
H n,1 (X) = 0 gives that ∂ X is surjective so the adjoint is injective i.e let v belong to the
kernel of the adjoint. Then

u, v = ∂ X γ, v = (−1)n γ, ∂ ut v = 0 =⇒ v = 0

So the generalized gradient vector field as defined is static and the proposition follows.

1.10

Non-smooth case

This section overviews the last part of Berndtssons argument. See [Bo] for further details.
In general a singular metric φ with i∂∂φ ≥ 0 cannot be approximated by a decreasing
sequence of smooth metrics with nonnegative curvature. However, this is possible if the line

50

bundle has some smooth metric of strictly positive curvature.
In fact it is known that one can approximate a singular metric on L with nonnegative
curvature by a decreasing sequence of smooth metrics such that

i∂∂φν > − ν ω

(1.73)

where ω is some K¨hler form.
a
This proceeds by considering the line bundle L + F where F is positive. Then L + F
admits hermitian metric ut + ψ and this can be approximated with smooth metrics χν
of positive curvature (see [ZBSK]). Then uν = χν − ψ approximates ut satisfying (1.73).
t
Further the sequence may be arranged to be decreasing.
Recall given ui where i = 0, 1 such that i∂∂ui ≥ 0 there is a bounded geodesic ut defined
for the real part of t ∈ [0, 1] where ut is given by

ut = sup{ψt }

and the supremum is taken over all ψt with limt→i ψt ≤ ui . Note that the following barrier
function participates in the supremum

χt = max{φ0 − A (t), φ1 + A( (t) − 1)}

for A > 0 sufficiently large because χt satisfies the boundary conditions and is plurisubharmonic.
˙
So it suffices to restrict to competitors larger the χ. But then we have −A ≤ limt→0+ ψ
˙
and limt→1− ψ ≤ A. Since ψ is independent of the imaginary part of t, i∂∂ψ ≥ 0 gives that
51

ψ is convex hence

˙
−A ≤ ψ ≤ A

(1.74)

=⇒ φ0 − A (t) ≤ ψ ≤ φ1 + A (t)

So the same inequality holds for the majorant ut and in fact its upper semicontinous regularization participates in the supremum so that ut is plurisubharmonic. Since its maximal, it
solves the monge-ampere equation with given boundary values. Inequality (1.74) gives that
ut is Lipschitz. Solutions ut arising in this way are called C 0 sub-geodesics.
Obtain Fν from F in (1.59) by replacing ut with uν approximating ut as above. The
t
loss in positivity i∂∂uν ≥ − ν ω is notational and one can instead proceed as if i∂∂uν ≥ 0.
t
t
Then i∂∂ t Fν goes to zero weakly. Corresponding to the smooth metrics uν solutions of
t
ν ν
∂ ut vt = π⊥ (u˙ν )
t

satisfy
ν
||vt || ≤ C||π⊥ (uν s)|| ≤ C||s||||uν || ≤ C A < ∞
˙t
˙t
ν
So we may extract a subsequence of vt that weak converges to a form v ∈ L2 . Proposition
ν
13 and 14 give that ||∂vt || → 0 on X × K where K ⊂ Ω compact. So weak converges to an

element w ∈ L2 . This is ∂v in the distributional sense. It follows ∂v = 0 since

ν
ν
w, w = lim ∂vt , w ≤ lim inf ||∂vt ||B = 0

from cauchy-schwartz and definition of weak convergence.

52

v also satisfies
∂ ut v = π⊥ (ut s)
˙
in the weak sense i.e

dt ∧ dt ∧ v ∧ ∂W e−ut = (−1)n
X×Ω

X×Ω

dt ∧ dt ∧ π⊥ (ut s) ∧ W e−ut
˙

where W is a smooth form of appropriate degree.
When there are no nontrivial holomorphic vector fields then v = 0, and hence π⊥ (ut s) =
˙
0. So ut is holomorphic and constant since it depends only on the real part of t. Otherwise
˙
one needs to show ∂t v = 0 in a weak sense. Following the smooth case one needs to obtain a
∂v

the distributional formulation of ∂ ut ∂tt = π⊥ (µs) and then conclude using the cohomological
assumption. There is some difficulty in doing this since care is needed taking limits because
it is only known that vt ∈ L2 . Some work is also required in deriving the distributional
formulation. However the starting point is the use of proposition 13 and 14 to get that

ν

X×Ω

i∂∂uν ∧ u ∧ ue−ut → 0
ˆ ˆ
t

See the latest version of [Bo] for details.

53

Chapter 2
Obstruction and existence for twisted
K¨hler-Einstein equation
a

2.1

Twisted K.E scalar Equation

Let L be an ample holomorphic hermitian line bundle on a K¨hler manifold X. Given
a
[η] = −c1 (KX ⊗ L) ∈ H 2 (M, R) ∩ H 1,1 (M, C) it is natural to seek a corresponding K¨hler
a
metric ω with [ω] = c1 (L) satisfying the twisted K¨hler-Einstein equation
a

Ric(ω) − ω = η

(2.1)

In the K¨hler-Einstein setting for Fano manifolds, where η = 0(L = −KX ), it is known this
a
is not always solvable. Similarly extra conditions are needed here.
A flow version of (2.1) can be written as

˜
∂t g˜ = −Rij + g˜ + ηij
ij
ij

(2.2)

g˜ (0) = gij
ij

This flow is now known to be called Twisted K¨hler-Ricci flow (Tkrf) (see [SzCo]).
a
Heuristically, if we had long time existence and convergence in C ∞ topology for time
54

derivatives included, then as t → ∞

˜
0 = lim ∂t g˜ = lim(−Rij + g˜ + ηij )
ij
ij
= − (R∞ )ij + (g∞ )ij + ηij

In other words we obtain a metric g∞ satisfying the twisted K¨hler-Einstein equation. §20
a
makes this precise.
But to even consider the flow above, short time existence must be clarified when starting
the Tkrf flow at any initial metric g0 . To do this it is enough to notice that this can be
written as a scalar flow very similar to the K¨hler-Einstein setting for Fano manifolds where
a
scalar flow is a parabolic flow for which short time existence is well known.
Write g˜ = gij + uij where u ∈ C ∞ (M × [0, T )), 0 < T < ∞. Set Tij = gij + ηij . Since
ij
√

√
−1
−1
Rij dz i ∧ dz j , 2π Tij dz i ∧ dz j
2π

∈ C1 (M ), 0 = [T − Ric] ∈ H 2 (M, R) ∩ H 1,1 (M, C), there

is f ∈ C ∞ (M ) unique up to a constant such that Tij − Rij = fij .
Proposition 16. (2.2) can be written as the scalar equation
∂u
ωm
= log u + u + f + φ(t)
∂t
ωm

(2.3)

where φ(t) comes from the ambiguity in constant on each time slice that is fixed by the
normalization
∂u −(u+f )

e ∂t

dV = eφ(t) V ol(M )

M

55

(2.4)

Proof. Rewrite (2.2)

˜
∂t g˜ = − Rij + g˜ + ηij
ij
ij
˜
= − Rij + (Rij − Rij ) + g˜ + ηij
ij
˜
= − Rij + Rij + (−Rij + g˜ + ηij )
ij

Using the formula for ricci curvatures obtain

=
=

∂2
∂z i ∂z j
∂2
∂z i ∂z j

ωm
log u + uij + (−Rij + (gij + ηij ))
ωm
ωm
log u + uij + (fij )
ωm

This is equivalent to
∂u
ωm
− log u − (u + f )) = 0
∂t
ωm
∂u
ωm
=⇒
= log u + u + f + φ(t)
∂t
ωm
∂∂(

Recall [ωu ] = [ω] so we obtain the normalizations in (2.4).
Remark 12. As a monge-ampere equation the twisted K¨hler-Einstein equation is the same
a
as that arising in the K¨hler-Einstein Fano case i.e.
a

Ae−φ+hω ω m = (ω )m

56

(2.5)

where A is given by
e−φ+hω dV

dV = A

V olg (M ) =
M

M

Using the following
1. g = gij + φij
ij
2. Ric(ω ) − ω = η ⇔ R − g = ηij
ij
ij
3. (Rij ), (ηij + gij ) with associated forms in C1 (M )
4. Tij − Rij = (−hω )ij
obtain

(Rij − Rij ) + Rij − gij = ηij
Rij − Rij − φij = Tij − Rij = (−hω )ij

This is equivalent to
(ω )m
− φ + hω ) = 0
ωm
(ω )m
=⇒ − log m − φ + hω = − log A where A > 0
ω
∂∂(− log

=⇒ Ae−φ+hω ω m = (ω )m

Since [ω ] = [ω], A is normalized to (2.6).

57

(2.6)

2.2

Various Estimates (Toy version)

Adjust the scalar equation (2.3) of the twisted K¨hler-Ricci flow by dropping the term φ(t)
a
(referred to ct in subsequent sections) and study
ωn
∂t u = log u + u + f
ωn

(2.7)

temporarily to gain experience (c(t) adds complications so is relegated to later sections).
Thanks to short time existence there is a T > 0. Choose 0 < < T < ∞.
Proposition 17. C 2 estimates of u on M × [0, T − ] depend on oscillation of u and boundedness of ∂t u. Explicitly the following estimates are available
• |∂t u| ≤ maxM |f |
• n + ∆u > 0 and ∆u < C. The reduction to estimates on ∆u comes from well known
point-wise calculation. See [Jo].
Proof. (2.7) can be written as
ωn
et ∂t (e−t u) = log u + f
ωn

Differentiate in time to obtain

∂t (et ∂t (e−t u)) = ∆g (∂t u)
˜
et [∂t ∂t (e−t u) + ∂t (e−t u)] = ∆g (∂t u)
˜
∂t [∂t (e−t u) + e−t u] = ∆g (e−t ∂t u) = ∆g (∂t (e−t u) + e−t u)
˜
˜
58

(2.8)

Parabolic maximum principle entails (really quantities need to be adjusted by ± t and let
run to zero.)

max

(∂t + I)(e−t u) = max e−t ∂t u = max ∂t u ≤ max f
M

M ×[0,T − ]

M

M

The last inequality follows from (2.8) at t = 0 (replace u with −u to get the bound from
below). So u is bounded.
˙
Lemma 10. Along Tkrf on [0, T − ], ωu > 0. In particular n + ∆u > 0
n
˙
Proof. Since u is bounded the monge-ampere equation corresponding to (2.7) ωu = eu−u−f ω n
˙
n
gives ωu > 0(ω > 0) so it follows its eigenvalues are nonzero real. At a minimum point p ∈ M ,

uαβ has positive eigenvalues so that ωu has positive eigenvalues near p. By connectedness
and covering appropriately this holds globally(change in sign of eigenvalues would require
n
ωu to degenerate somewhere). Really this just says that if ωu is positive at a point of Mt it

is positive everywhere on Mt . So T rg (ωu ) = n + ∆u > 0 on [0, T − ].
To get an upper bound on ∆u use C 2 inequality obtained by Yau:

∆g (e−C0 (n + ∆g u)) ≥ e−C0 u (∆F − Cn2 )
˜
− C0 e−C0 u n(n + ∆g u)
F

n

−
+ (C0 + C)e−C0 u e n−1 (n + ∆g u) n−1

where C = inf i=k Riikk and C0 + C > 0.
Rewriting (2.7) we set
ωn
∂u
F := log( u ) =
−u−f
n
ω
∂t

59

(2.9)

Inserting into estimate (2.9) and evaluating at a maximum point of the quantity e−C0 u (n +
∆g u), (p0 , t0 ) ∈ M × [0, T − ] obtain

0 ≥ ∆g (e−C0 u (n + ∆g u)) ≥ − e−C0 u (∆g f + Cn2 )
˜
− C0 e−C0 u n(n + ∆g u) + e−C0 u ∆g (
+ (C0

−
+ C)e−C0 u e

∂u
− u)
∂t

(−u−f + ∂u )
n
∂t
n−1
(n + ∆g u) n−1

(2.10)

(2.10) becomes after multiplying by ec0 u and rearranging

C ≥ ∆g f + Cn2 ≥ −C0 n(n + ∆g u) − ∆g u + ∆g

∂u
∂t

∂u −f
n
−( ∂t
) u
+(C0 + C)e n−1 e n−1 (n + ∆g u) n−1

(2.11)

Claim 6. At (p0 , t0 ) when 0 ≤ t0 ≤ T −

∆g

∂u
∂u
≥ C0 (n + ∆g u)
∂t
∂t

Proof. Indeed

0≤

∂ −C u
∂u
∂u
(e 0 (n + ∆g u)) = −C0 e−C0 u (n + ∆g ) + e−C0 u ∆g u
∂t
∂t
∂t
∂u
∂u
=⇒ ∆g
≥ C0 (n + ∆g u)
∂t
∂t

60

(2.12)

So (2.11) becomes

C (n) ≥ − (C0 n + 1)(n + ∆g u) + C0

∂u
(n + ∆g u)
∂t

∂u −f
n
−( ∂t
) u
+(C0 + C)e n−1 e n−1 (n + ∆g u) n−1

(2.13)

at t0 = 0 we have

e−C0 u(p0 ,0) n ≥ e−C0 u(p0 ,0) (n + ∆g u(p0 , 0)) ≥e−C0 u (n + ∆g u)
C (u−inf M ×[0,T − ] u)
=⇒ (n + ∆g u) ≤ ne 0

−C inf
u
using that e−C0 u(p0 ,t0 ) ≤ e 0 M ×[0,T − ] .

When 0 ≤ t ≤ T − reduce (2.13) further

C (n) ≥ − (C0 n + 1 − C0
+ (C0 + C)e

∂u
)(n + ∆g u)
∂t

−|f |∞ +f
u
n
n−1 e n−1 (n + ∆g u) n−1
−||u|| 0
C

n

˜
≥ − C0 (n + ∆g u) + Ce n−1 (n + ∆g u) n−1
f −|f |∞

˜
here C0 = C0 n + 1 + C0 |f |∞ , C = e n−1 (C0 + C) and n + ∆g u > 0 holds.
So the inequality takes the form
||u|| 0
C

n

˜ −
C2 (1 + n + ∆g u) ≥ Ce n−1 (n + ∆g u) n−1

61

(2.14)

Claim 7. For x > 0 and positive constants a, b inequalities of the form

n

(1 + x)a < (x) n−1 b

hold whenever
x > (2k )n−1
provided k > log2 ( a ) + 1 i.e for x > (2 a )n−1
b
b
n

1

Proof. Clearly x n−1 grows faster than x. Taking x n−1 > 2k

(1 + x)a
n
x n−1 b

=

a
n
x n−1 b

+

a
1
x n−1 b

a 1
1
a
< ( nk + k ) < k−1 < 1
b 2
2
b2

for k an integer bigger than log2 ( a ) + 1.
b
By the claim, since (2.14) is the reverse inequality it follows that there is a 0 < C :=
2C

( ˜2 )n−1 so that
C
||u|| 0
C

(n + ∆g u)(p0 , t0 ) ≤ Ce

=⇒ 0 < e−C0 u (n + ∆g u) ≤ e−C0 u(p0 ,t0 ) (n + ∆g u(p0 , t0 ))
≤ Ce||u||0 e−C0 u(p0 ,t0 )
=⇒ 0 < (n + ∆g u) < Ce||u||0 eC0 (u−u(p0 ,t0 ))
(supM ×[0,T − ] u−inf M ×[0,T − ] u)(C0 +1)

< Ce

62

note that u(0) = 0 so

||u||0 ≤ max{

u, −

sup
M ×[0,T − ]

2.3

inf

M ×[0,T − ]

u} ≤

sup

u−

M ×[0,T − ]

inf

M ×[0,T − ]

u

Twisted Mabuchi functional

In this section consider the K-energy functional in the twisted setting. This involves adjusting the K-energy functional so that its critical points satisfy the twisted K¨hler-Einstein
a
equation (2.1).
Definition 1.

η
νω (φ)

=−

1
0 M (Ric(ωφ ) − (ωφ

ω n−1
φ
+ η)) (n−1)!

∧ dt

Proposition 18. The twisted Mabuchi functional defines a closed 1-form

˜
Bφ (ψ) =

M

ψ(ωφ + η − Ric(ωφ )) ∧

n−1
ωφ

(n − 1)!

.

where ψ ∈ Tφ Hω
Recall that
˜
dβφ (u, v) = δu β(v) − δv β(u)
where u, v ∈ Tφ Hω . Mabuchi energy being understood, it suffices to study the differential
of

n
ωφ

ψtrφ η n! . This is given by

Claim 8.
δu

vtrφ η

n
ωφ

n!

=

v(trφ η φ u −
63

u, η )

n
ωφ

n!

(2.15)

Proof. Clearly the first term in (2.15) comes from differentiating the volume form. The
second term comes from differentiating the trφ η term which is locally given by

−g αt g sβ (ust )ηαβ = −

u, η

Really a factor is suppressed but its harmless.
The proposition follows from the next lemma.
Lemma 11.

˜
dβ(u, v) = δu
=

vtrφ η

n
ωφ

n!

− δv

utrφ η

[trη(v φ u − u φ v) + u

n
ωφ

n!
v, η − v

u, η ]

n
ωφ

n!

=0

(2.16)

Proof. Note the second equality follows from the claim. Involved is integration by parts to
obtain cancellation to zero. We suppress all integrals and divergences involved and focus on
the integrands.
By performing integration by parts on trη(v φ u − u φ v) in (2.16) we obtain four terms
that are given by

(−trη v ·

u − v trη ·

u) + (trη u ·
= −v trη ·

v + u trη ·
u + u trη ·

v)
v

(2.17)

The formulas may be verified through a local calculation. For example an integration by

64

parts on a term with integrand trηvg pq upq yields

−(trη)p vg pq uq − trηvp g pq uq = (−v trη ·

u − trη v ·

u)

similarly for the other term.
An integration by parts on the third and fourth terms in (2.16) gives

−g αt g sβ vs ut ηαβ − g αt g sβ vs ηαβ,t u + g αt g sβ vs ut ηαβ + g αt g sβ ut ηαβ,s v
= −g αt g sβ vs ηαβ,t u + g αt g sβ ut ηαβ,s v

using that dη = 0 we obtain

−g αt g sβ vs ηαβ,t u = −g αt g sβ vs ηαt,β u = −g sβ (trη)β vs u = −u trη ·

v

(2.18)

similarly
g αt g sβ ut ηαβ,s v = vg αt (trη)α ut = v trη ·

u

˜
Pairing up corresponding terms in (2.17) with (2.18) and (2.19) we obtain dβ = 0.
Recall the Cech 2-differential is given by

(δf )(ω1 , ω2 , ω3 ) = f (ω1 , ω2 ) + f (ω2 , ω3 ) + f (ω3 , ω1 )

where f (ωα , ωβ ) = −f (ωβ , ωα ). When f defines a Cech cocycle we have

f (ω1 , ω2 ) − f (ω3 , ω2 ) = f (ω1 , ω3 )

65

(2.19)

Letting ω1 = ω, ω2 = ω + i∂∂φ, and ω3 = ω + i∂∂ψ =: ω we can recover how Tian writes
the cocycle condition
fω (φ) − f (φ ) = fω (ψ)
ω

η

By the lemma dνω (φ) defines a closed 1 form. Since Hω is contractible the one form is
in fact exact and thus can be integrated to give the functional
1

η

νω (φ) = νω (φ) +

0

ω n−1
˙
φt η ∧ t
(n − 1)!
X

(2.20)

η

From the discussion in the previous paragraph νω also satisfies the cocycle condition.

2.4

Twisted Futaki type invariant

For this section we denote the manifold by M so we may notate holomorphic vector fields
by X. Set G := Aut0 (X). Let η(M ) denote the lie algebra of holomorphic vector fields on
M . Given a smooth differential form α we say that the infinitesimal action of X ∈ η(M )
annihilates α if LX α = 0. We say η(M ) annihilates α under the infinitesimal action if
LX α = 0 for each X ∈ η(M ).
Here we see that the Futaki invariant can be adapted to the twisted setting under the
condition that η is annihilated by η(M ). With this taken for granted it can be seen why the
non collapsing condition introduced in subsequent sections guaranteeing existence cannot
hold if the twisted Futaki invariant does not vanish. Though the following argument is
an explicit calculation it seems possible also to conclude through using a moment map
interpretation appearing in [Sto09].

66

η

Proposition 19. Provided η is annihilated by η(M ), FM : η(M ) → C given by (2.21) is
well-defined.

η

FM (X, [ω]) = −

M

θX (Ric(ω) − ω) ∧

ω n−1
ω n−1
+
θX η ∧
(n − 1)!
(n − 1)!
M

(2.21)

where θX + α = iX ω, α a harmonic 1-form, and X ∈ η(M ).
Remark 13. Since η is a real (1, 1) form the condition that it is annihilated by the infinitesimal action of η(M ) means, using that the lie algebra of G is generated by the real
holomorphic vector fields of M , since LX+X η = 0 for each X ∈ η(M ) we conclude that η is
G-invariant.
The first term above is the usual Futaki invariant FX so is independent of the choice of
metric in [ω]. However the second term can potentially destroy the independence.
Following the classical argument there is no loss in assuming holomorphic vector fields
satisfy
iX ω = ∂θX

(2.22)

since the harmonic piece has no contribution after an integration by parts (see [Tian00] and
(2.26)).
In co-ordinates (2.22) reads

X i = g ij (θX )j = (θX )i

When the metrics vary over any family ωt = ω + ∂∂φt in a fixed K¨hler class
a

θX,t = θX + X(φt ) + ct
67

(2.23)

since
∂(θX,t ) = iX ωt = iX (ω + ∂∂φt ) = ∂(θX + X(φt )).
Now deduce as in (2.23)
ij

i
X i = gt (θX,t )j = θX,t .

(2.24)

Using (2.22), that X ∈ η(M ) and the definition of Ric

∂ t θX,t = −iX Rict

see [Tian00] for details. Since η ∈ −c1 (L ⊗ KM ) we have

Ric(ω) − ω = η + ∂∂ψ

for some ψ ∈ C ∞ (M ). Varying over the family {ωt } we get

Rict − (ωt + η) = ∂∂ξt

(2.25)

where Rict = Ric(ωt ).
proof of proposition. From (2.25) we may simplify to get

η

FM (X, ωt ) = −

M

θX,t ∂∂ξt ∧

n−1
ωt
ωn
=
Xξt t
(n − 1)!
n!
M

(2.26)

Since the space K¨hler metrics is affine it is enough to check the variation over any family
a

68

of metrics in the fixed K¨hler class vanishes. So start by computing
a
ωn
d η
˙
FM (X, ωt ) =
(X ξ˙t + t φt Xξt ) t
dt
n!
M

(2.27)

Recall we obtain the deformation of the scalar curvature St by differentiating

kl
St = −gt

∂2
log det((gt )ij )
∂zk ∂zl

to get
˙
˙
˙
St = − 2 φ − Rαβ φαβ

(2.28)

Tracing Rict − (ωt + η) = ∂∂ξt we obtain

St − n − trt η =

t ξt

(2.29)

Differentiating (2.29) and applying (2.28) we obtain

˙
˙
˙
˙
− 2 φ − Rαβ φαβ + ηαβ φαβ = ˙ t ξt + ξt
t

(2.30)

˙
˙ t ξt = −(ξt ) φαβ
αβ

(2.31)

˜
Rαβ := Rαβ − ηαβ − ξαβ = gαβ

(2.32)

Recall that

Set

∗
˜
˜
then Ric is harmonic since ΛRic = cnst and ∂ = −i[Λ, ∂] on a K¨hler manifold.
a

69

˜
In terms of Ric from (2.30), (2.31)

ξ˙t = −

2 φ − R φαβ
˙
˙
t
αβ

= −

˙
˙
+ ηαβ φαβ + ξαβ φαβ

2 φ − R φαβ
˙
˜ ˙
t
αβ

(2.33)

So we obtain integrating by parts and using the identity (2.33) for

˙

t ξt

ωn
d η
˙
˙
(−θX,t ξt + t φt Xξt ) t
F0 (X, ωt ) =
dt
n!
M
n

˙ ω
˙
˜ ˙
(θX,t 2 φ + θX,t Rαβ φαβ + Xξt t φ) t
t
n!
M
n
n
˙ω
˜ ˙ ω
θX,t Rαβ φαβ t
( t θX,t + Xξt ) φ t +
=
n!
n!
M
M
=

(2.34)

The first term in (2.34) simplifies to
n

ω
αβ ˙
−gt φα (( t θX,t )β + X i (ξt )iβ ) t
n!
M

(2.35)

using ∂ t θX,t = −iX Rict and (2.32), (2.35) simplifies to
n

ω
αβ ˙
gt φα (Riciβ X i − (ξt )iβ X i ) t
n!
M

n

=

n

ω
ω
αβ ˙ ˜
αβ ˙
gt φα (Riβ )X i t +
gt φα ηiβ X i t
n!
n!
M
M

(2.36)

Performing by parts on the second term in (2.34) using (2.24) gives
n

=−

αβ ˙ ω
i ˜
θX,t Riβ gt φα t
n!
M

70

(2.37)

Putting (2.36) and (2.37) together gives
n

n

ω
ω
αβ ˙
αβ ˙ ˜
i
gt φα ηiβ X i t
gt φα Riβ (X i − θX,t ) t +
n!
n!
M
M
n
ω
αβ ˙
=
gt φα ηiβ X i t
n!
M

=

(2.38)

Compressing notation in (2.38) write

˙
∂ φ, iX η :=

M

n
αµ ˙
i ωt
gt φα ηiµ X
n!

To get a well defined invariant we need the last term to vanish. But η(M ) annihilates η so

0 = LX η = ∂iX η
(ηiµ X i )α dz α ∧ dz µ = 0
=⇒ (ηiµ X i )α = 0

From integration by parts and (2.39)

˙
∂ φ, iX η = −

ωn
αµ
gt θX,t (ηiµ X i )α t = 0
n!
X

71

(2.39)

2.5

Convexity of the Twisted Mabuchi functional

We restrict ourselves to the smooth setting and consider the twisted mabuchi functional.
Write the differential as

η

dνω (φ) = dE0 (X, ω) +

ω n−1
˙
φη ∧
(n − 1)!
M

where φ ∈ C ∞ ∩P sh(ω, X). Here we actually mean strictly ω-psh so ωφ > 0. It was observed
in [Sto09]
Proposition 20. Under the provision that η ≥ 0 the twisted mabuchi functional is convex
along smooth geodesics. The second variation of twisted mabuchi energy given by
d2 η
ω n−1
¨ 1 ˙
F0 (X, ω) = −
(φ − | φ|2 )(Ric(ωφ ) − ωφ − η) ∧
φ
2
(n − 1)!
dt2
M
˙
˙
˙
+ (∂ φ ∧ ∂ φ, η) + ||Lφ||2

where the operator L = ∂ ↑ ∂ (also denoted D).
No argument is given in the literature to the best of our knowledge so we suspect, although
its straightforward, there is an easier way to see this than the argument given below.
Proof. Mabuchi computed the second variation for E0 to be

−
M

¨
φ(Ric(ωφ ) − ωφ ) ∧

72

n−1
ωφ

(n − 1)!

˙
+ ||Lφ||2

see [Mab]. So it suffices to compute the first variation of

˙
φ∧η∧
M

n−1
ωφ

(n − 1)!

Differentiating we obtain

¨
φη ∧
M

n−1
ωφ

(n − 1)!

˙
˙
φη ∧ i∂∂ φ ∧

+
X

n−2
ωφ

(2.40)

(n − 2)!

Integrating by parts the second term may be written as

˙
˙
∂φ ∧ ∂φ ∧ η ∧

−
M

n−2
ωφ

(n − 2)!

In the following we abuse notation η ↔ η where η = η + ω > 0 since we can let
zero without trouble.
At a point p ∈ M by choosing normal co-ordinates we may arrange that

η = ηii dzi ∧ dzi ,
√

we omit the

−1
2

ωφ = dzj ∧ dzj
n

factor which is ultimately absorbed into ω .
n!

Claim 9. The (n, n) form in the second term of (2.40) at the point p is
n
ωφ

(−φ˙α φ˙β dz α ∧ dz β ) ∧ (ηii dzi ∧ dzi ) ∧

n! (dzp ∧ dzp ) ∧ (dzq ∧ dzq )
(pp=qq)
n

ωφ
1 ˙
˙ ˙
= (− | φ|2 + φp φp )ηpp
2
n!

73

run to

Proof. This follows because

ηii dzi ∧ dzi ∧

[. . .]
(pp=qq)

=

ηpp
p<q

=

ηpp
p=q

n
ωφ

n! dzq ∧ dzq

+

ηqq
p<q

n
ωφ

n! dzp ∧ dzp

n
ωφ

n! dzq ∧ dzq

Now

˙
˙
−(∂ φ ∧ ∂ φ) ∧

[. . .] = −
p=q

p=q

ωn
˙ q φq ηpp φ
˙
φ
n!
n

ωφ
1 ˙
˙ ˙
= (− | φ|2 + φp φp )ηpp
2
n!

Remark 14. Another point-wise calculation shows that

˙
˙
(∂ φ ∧ ∂ φ, η) =

ωn
˙ α φα ηαα φ
˙
φ
n!
M

So it follows from the remark and point-wise computations that

˙
˙
∂ φ ∧ η ∧ i∂∂ φ ∧
M

n
ωφ

(n − 2)!
n

ωφ
1 ˙2
˙
˙
| φ|φ trφ η
+ (∂ φ ∧ ∂ φ, η)
= −
n!
M 2

74

Modulo the second variation of Mabuchi energy, the second variation looks like
ω n−1
˙
˙
¨ − 1 | |2 )η ∧ φ
+ (∂ φ ∧ ∂ φ, η)
(φ
2 φ
(n − 1)!
M
so we obtain the second variation formula of Stoppa in untraced form along smooth geodesics:
d2 η
ω n−1
¨ 1 ˙
(φ − | φ|2 )(Ric(ωφ ) − ωφ − η) ∧
F0 (X, ω) = −
φ
2
(n − 1)!
dt2
M
˙
˙
˙
+ (∂ φ ∧ ∂ φ, η) + ||Lφ||2

Convexity along smooth geodesics follows immediately provided η ≥ 0.

2.6

Application of Twisted Mabuchi Energy: Existence

To attack the existence problem of solutions to the twisted ricci equation

Ric(ω) − ω = η

η≥0

we consider twisted K¨hler-Ricci flow starting at some K¨hler metric ω0 ∈ c1 (L) satisfying
a
a
the following condition along the flow

n
n
ωt ≥ Cω0

C > 0 cnst ∀t ≥ 0

(2.41)

Remark 15. Replacing the condition η ≥ 0 with ω0 + η represents a K¨hler class allows η
a
to be negative. Unfortunately reapplying arguments for η ≥ 0 don’t carry over in any obvious
fashion in regards to the C 0 estimate and the maximum principles for Perelmans estimates.
75

To establish existence one uses the parabolic formulation

d
η
ωt = Tt − Rict
dt
η

where Tt = ωt + η. Henceforth, as is common this will be referred to twisted K¨hler-Ricci
a
flow (Tkrf). Establishing convergence as t → ∞ is the goal. For this we use three ingredients
the twisted mabuchi functional introduced above, a modified version of Perelmans estimates
for K¨hler-Ricci flow, and a potential theory based C 0 estimate of Tian-Zhu [Ti07].
a
The relevant theorems are described below. First the Tian-Zhu estimate.
Proposition 4. Let (X, ω) be a compact K¨hler manifold of complex dimension n. Let
a
ωn

φ ∈ Hω solve ωφ = f . Then there is 0 , δ0 so that for
n

∈ (0, 0 ) and δ ∈ (0, δ0 ) there exists

constants C, C > 0 depending only on ω, 0 , δ0 such that

Osc(φ) ≤ C(

1 n+δ
)
||f ||δ 1+
+C
L
(X,ω)
δ

(2.42)

for non-negative f ∈ L1+ (X).
η

The twisted version version of Perelmans theorem can be stated by first setting Tt :=
η

ωt + η. Then in the situation η is chosen so that Tt ≥ 0
Proposition 5. Over a Fano manifold (X, ω) of complex dimension n, the twisted K¨hlera
Ricci flow
dωt
˙
= Tt − Ric(ωt ) = i∂∂ φt
dt
with η ≥ 0 satisfies uniform estimates on the following quantities

˙
˙
˙
|φ|, | t φ|t , |∆t φt |, Diamt (X)
76

(2.43)

n

ωt
Provided ω ≥ K0 , t ∈ [0, ∞) we also have

|Sct | ≤ C

(2.44)

˙
Here normalize the Ricci potential φt by the condition
1
˙ n
e−φ ωt = 1
V X

(2.45)

This proposition differs from the recent result [SzCo] only in that the extra condition on
the density of the volume of the Tkrf allows to bound scalar curvature.
The twisted K¨hler-Ricci functional is decreasing along the twisted K¨hler-Ricci flow
a
a
and there is a similar identity to the classical case from which we conclude this functional
is uniformly bounded from below. Extracting the canonical metric is described in §20. The
arguments follow similar lines as appearing in [Pa] .
Recall the at the level of potentials we have the equation

˙
φt = log

n
ωφ

t

ωn

+ φt + f + ct

ˆ
where f = −hω,η and let φ := φ + ct .
Also the differential of the twisted mabuchi functional looks like

η

dνω (φt ) =

ω n−1
˙
φt (Tt,η − Rict ) ∧ t
(n − 1)!
X

It is essential that our choice of functional enjoys the following property.
Proposition 21. If φt evolves according to the Tkrf flow above then the twisted mabuchi
77

functional decreases along the flow.
Proof. Indeed,
ω n−1
d η
˙
˙
νω (φt ) =
φt i∂∂ φt ∧ t
dt
(n − 1)!
X
n
˙ ω
= −
|∂ φt |2 t
t
n!
X

The following can be considered among the more important properties for the existence
result. Since extracting the limit in §20 crucially depends on this.
Proposition 22. The twisted mabuchi energy (2.20) is bounded along Tkrf provided the non
η

collapsing estimate (2.41) holds. If the flow exists for all time then limt→∞ vω (φt ) < ∞
Claim 10. Along Tkrf the following identity for the twisted K-energy is available.

η

νω (φ) =

1
1
1
˙ n
ˆ
φt ωt + Jω (φt ) −
φt ω n +
hω,η ω n
V X
V X
V X

(2.46)

The twisted ricci flow
dωt
= Tωt ,η − Ric(ωt )
dt
ˆ
at the level of potential is exactly (2.3)(let φ = φ + ct ), and is written as
n

ω
ˆ
˙
φ = φt − log t + hω,η
ωn

78

(2.47)

The corresponding monge-ampere equation is

˙

n
ehω,η −φ ω n = e−φ ωt

(2.48)

In the classical setting there is the following expression for mabuchi energy(see [Ru]):

νω (φ) =

−1
1
1
1
n
fωφ ωφ +
fω ω n + Jω (φ) −
φω n − log
efω −φ ω n
V X
V X
V X
V X

(2.49)

Adjusting the formula for twisted K-energy we obtain essentially the same formula.
Lemma 12. Along Tkrf we have the following identity:

η

νω (φ) =

1
1
−1
n
hωφ ,η ωφ +
hω,η ω n + Jω (φ) −
φω n
V X
V X
V X
1
ehω,η −φ ω n
− log
V X

(2.50)

In fact the argument is very similar to the untwisted version.
Proof. The log term in (2.50) vanishes because of normalization (2.45) using (2.48). Recall
√
√
˙
the twisted ricci potential is given by − −1∂∂ φ = Ric−Tt,η = −1∂∂hωt ,η which translates
at the level of potentials
ωn
hωt ,η = − log t − φt + hω,η
ωn
˙
˙
=⇒ hω˙t ,η = − ∆t φt − φt

d
˙ n
Again since the log term vanishes Fω (φ) = Jω (φ) − X φ ∧ ω n . So dt Fω (φ) = − X φωt =
X

n
hω˙t ,η ωt . The lemma follows from the following calculation using an integration by parts

79

in the second line

1
d 1
n
˙ n
hωt ,η ωt =
(hω˙t ,η + hωt ,η ∆t φ)ωt
dt V X
V X
d
1
n−1
˙
= Fω (φ) +
φ(i∂∂hωt , )n ∧ ωt
dt
V X
1
d
n−1
˙
= Fω (φ) +
φ(Ric(ωφ ) − Tωt ,η ) ∧ nωt
dt
V X

(2.51)

Notice that the second term in (2.51) is negative of the differential of twisted mabuchi
k-energy. Integrating in t from 0 to 1 obtains (2.50).
n
˙
proof of claim. To obtain the claim note that hωt ,η = φ − ct and since X ωφ = V we may

combine the ct term as in the statement of the claim.
n
proof of proposition. By hypothesis we have uniform noncollapsing estimate ωt ≥ kω n for
ωn

t ∈ [0, ∞). Applying the uniform lower bound on the ratio of the volume forms ωt ≥ k to
n
(2.47) translates into an upper bound

ˆ
φ = φt + ct ≤ C

˙
using the Perelman type uniform estimate |φ| ≤ c in (2.43). With this and that Jω (φ) ≥ 0
˜
on the K¨hler potentials we have by the claim
a

0 ≤ Jω (φ) = νω (φt ) −

1
1
η
˙ n 1
ˆ
φt ωt +
φt ω n −
hω,η ω n ≤ νω (φ) + C
V X
V X
V X

So
η

νω (φ) > −C

80

(2.52)

We can conclude since the twisted K¨hler ricci functional decreases along the flow and is
a
η

bounded from below limt→∞ νω (φt ) < ∞.
n

1
˙ ωt
Remark 16. We note that the term 0 X φη (n−1)! dt appearing in the definition of the

twisted mabuchi K-energy can be bounded from below in terms of the scalar curvature.
1
0

1
ω n−1
ωn
˙
˙
dt =
φtrωt η t dt
φt η ∧ t
(n − 1)!
n!
0 X
X
1
ωn
>−C
trωt η t
n!
0
1

=−C

η∧
0

X

n−1
ωt
dt
(n − 1)!

Since
trωt η ≥ 0,

1

η∧
0

X

˙
|φ| < C

n−1
1
1
ωt
ω n−1
ω n−1
(η − Ric(ωt )) ∧ t
Ric(ωt ) ∧ t
dt =
dt +
dt
(n − 1)!
(n − 1)!
(n − 1)!
0 X
0 X

Since η ∈ −c1 (KX ⊗ L), η − Ric(ωt ) ∈ −c1 (L) we may write η − Ric(ωt ) = −(ωt + ∂∂)f .
It follows
1
0

X

(η − Ric(ωt )) ∧

n−1
1
ωt
ω n−1
= −
(ωt + ∂∂f ) ∧ t
dt
(n − 1)!
(n − 1)!
0 X
1
ω n−1
dt = −nV
= − nV −
∂∂f ∧ t
(n − 1)!
0 X

where we used that
n−1
ωt
∂∂f ∧
=0
(n − 1)!
X

which follows from integration by parts since ωt is closed.
81

Since we have that |S(ωt )| < C along the flow
1
0

X

Ric(ωt ) ∧

n−1
1
ωt
ωn
=n
S(ωt ) t < nCV
(n − 1)!
n!
0 X

Remark 17. 0 < Tωt ∈ c1 (−KX ) so that X is Fano.
n
Proposition 23. For all times t ≥ 0 assume ωt ≥ kω n along the twisted ricci flow(non

collapsing estimate). Then this flow satisfies the uniform estimate |φt + ct | ≤ K0 , where
K0 > 0, k are independent of t.
Details differ marginally from [Pa] but we provide the argument below for convenience.
Claim 11. It suffices to obtain the estimate

ˆ
ˆ
||φt ||C 0 (X) ≤ Osc(φt ) + C

The propositions follows after applying the estimate in (2.42) and moser iteration.
Proof. The claim follows from

ˆ
ˆ
||hω,η − φt || ≤ Osc(hω,η − φt )

ˆ
which clearly holds if hω,η − φt changes sign. But this is true since along the twisted flow
˙

n
the volume forms deform according to ehω,η −φ ω n = e−φ ωt and the normalization condition

˙
on φt gives that
˙

ehω,η −φ ω n =
X

X

n
e−φt ωt = V =

82

ωn
X

(2.53)

So
ˆ

(ehω,η −φt − 1)ω n = 0
X

ˆ
so hω,η − φt changes sign.
˙
ˆ
From the estimate in (2.42) we need to bound X e(φt +hω,η −φt )(1+ ) ω n . In view of

˙
Perelman’s estimate on φ we can reduce to bounding

ˆ
e−(1+ )φt ω n
X

ˆ
ˆ
ˆ
Claim 12. The quantity maxX φt is bounded. So define θt := maxX φt − φt . It follows
ˆ
n
e−(1+ )φt ω n ≤ Ce θt ωt for t ≥ 0.
n

ω
˙
ˆ
Proof. Since ωt ≥ K from (2.47) and Perelman’s estimate |φ| < C it follows that φt < C1
n

ˆ
ˆ
. So it suffices to show that maxX (φt − hω,η ) ≥ 0 to conclude maxX φt is bounded for
ˆ
ˆ
ˆ
t ≥ 0. But note that if 0 > maxX (φt − hω,η ) ≥ φt − hω,η then ehω,η −φt > 1. In which case
ˆ
ˆ
we contradict X (ehω,η −φt − 1)ω n = 0. So max φt is bounded and θt is well-defined. The

ˆ
inequality for the volume forms is an easy calculation which involves writing −φt (1 + ) =
ˆ
ˆ
ˆ
ˆ
θ − max φt − φt and then use that max φt is bounded and φt is bounded from above (see
[Pa]) .
So from the volume estimate in the claim it follows it is sufficient to bound

X

n
e θt ωt

83

By power series expanding e θt it suffices to get the bound

p

X

n
θt ωt ≤ C p p!

1
for all integers p ≥ 1 and then take 0 < < C so that the geometric series converges.
p+1 n
ωt

1
n
Claim 13. The estimates 0 ≤ V X θt ωt ≤ C and X θt

p

n
≤ C(p + 1) X θt ωt hold

along the flow.
Note that by iterating the second estimate and combining with the first estimate we may
obtain the desired bound for all integers p ≥ 1.
n
Proof. For the first estimate apply inequality (2.52) , which uses the hypothesis ωt ≥ kω n .

Combined with the fact that the twisted mabuchi functional is decreasing along the flow, it
follows that Jω (φt ) is bounded along the flow.
So
0 < Iω (φt ) =

1
n
ˆ
φt (ω n − ωt ) ≤ (n + 1)Jω (φt ) ≤ C
V X

(2.54)

From (2.48) we have

ˆ

ˆ

ehω,η −φ ω n ≥ C

V =
X

e −φ t ω n
X

1
ˆ
e−φt ω n < C
V X
1
ˆ
=⇒ −
φt ω n < C
V X

=⇒

where the last line follows from Jensens inequality.
Remark 18. In fact (2.55) follows directly from (2.52)

84

(2.55)

From equation (2.54) and (2.55) obtain

−

1
1
1
ˆ n
ˆ
ˆ
φt ωt = Iω (φt ) −
φt ω n =≤ C −
φt ω n < 2C
V X
V X
V X

(2.56)

ˆ
Using (2.56) and that maxX φt is bounded, conclude

0≤
X

n
θt ωt ≤

1
ˆ
ˆ n
(max φt − φt )ωt < C
V X X

For the second estimate the starting point is the well known identity

p

p

X

n−1 ∧ ω) = −
n
θt (ωt − ωt

X

n−1 =
θt ∂∂θt ∧ ωt

p+1
4p
n
|∂θt 2 |2 ωt
t
n(p + 1)2 X

n−1 ∧ ω ≥ 0)
from which it follows (using ωt
p+1
n(p + 1)2
p n
n
|∂θt 2 |2 ωt ≤
θt ωt
t
4p
X
X

(2.57)

In [Pa] it is shown that
Lemma 13. On compact K¨hler manifolds (X, ω) of complex dimension n for any u, h ∈
a
C ∞ (X, R)

X

|∂ 1,0 u|2 eh ω n = −
ω

X

∂∆ω,h u, ∂u ω en ω n

(Ric(ω) − i∂∂h)( ω u, J ω u)eh ω n

−
X

from which a poincare-type inequality obtains.
85

Corollary 4. Let X be a Fano manifold of complex dimension n. Let ω ∈ c1 (L) be a K¨hler
a
metric, 0 ≤ η ∈ −c1 (L ⊗ KX ) , and let hω,η ∈ C ∞ (X, R) satisfy Ric(ω) − Tω,η = i∂∂hω,η .
Vhω,η := X ehω,η ω. Then for any φ ∈ C ∞ (X)

X

|∂φ|2 ω,η ehω,η ω n ≥
T

φ2 ehω,η ω n −
X

1
Vhω,η

φehω,η ω n )2

(
X

ωn

Note that when the uniform (in t) estimate ωt ≥ K > 0 is true
n

cωt ≥ Tωt ,η ≥ ωt

(2.58)

The second inequality follow directly since η ≥ 0, independent of the uniform estimate. The
first inequality is a consequence of the C 2 estimate and is discussed in §7.
Remark 19. Roughly the content of this lemma has appeared in §5. The identity in the
lemma after dropping the nonnegative term leads to an inequality, which for first eigenfunctions simplifies to

2
X

|∂u|2 ehω,η ω n =
ω

X

| u|2 ehω,η ω n ≤
ω

X

| u|2 ω,η ehω,η ω n
T

≤ λ1

X

|∂u|2 ehω,η ω n
ω

Note that (2.58) is applied in the first inequality and the lemma is used for the second
inequality. The variational characterization of the first eigenvalue
|∂u|2 ehω,η ω n
ω
= λ1 ≥ 2
inf X
2 ehω,η ω n
u
X

86

φehω,η ω n
can be used to obtain the corollary.
applied with φ − X V
hω,η

proof of proposition. Under the twisted K¨hler-Ricci flow
a

˙
∂∂ φ = Tωt ,η − Ric(ωt )
p+1

˙
So hωt ,η = −φt . Applying the corollary with metric ωt and function θt 2

gives

p+1
p+1
1
˙ n
˙ n
˙ n
p+1
|∂θt 2 |2 ω,η e−φ ωt ≥
(
θt 2 e−φ ωt )2
θt e−φt ωt −
T
Vhω,η X
X
X

Apply h¨lder’s inequality to
o

X

˙
p+1 −φ n
e ωt

θt

p+1
˙
θt 2 e−φt

=

˙
˙
1
φ p −φ
2 e− 2t θ 2 e 2 t
θt
t

p+1

≤
X

˙ n
|∂θt 2 |2 e−φt ωt + C
t

and using Tωt ,η ≥ ωt obtain

p

X

˙

n
θt e−φt ωt

˙

X

n
θt e−φt ωt

˙
Apply inequality (2.57) with the uniform estimate |φt | < C to obtain

X

˙
p+1 −φ n
e ωt

θt

p

≤ Cp
X

n
θt ωt + C

p

X

˙

n
θt e−φt ωt = C(p + 1)

p

X

n
θt ωt

ˆ
and so the second estimate of the claim follows: ||φt ||C 0 (X) ≤ C for t ≥ 0 with C independent
of t.

2.7

Perelman’s estimates twisted setting

In proposition (2.43) we need to check Perelman’s estimates carries over to the twisted
setting, avoiding any circularity in verifying the first inequality in (2.58) . That is, twisted
87

flows satisfying the non-collapsing estimate are uniformly equivalent to the initial metric.
Recall from the calculation

∂
˙
∂∂ φt = ωt
∂t
= Tt,η − Ric(ωt )
= Tt,η − Tω0 ,η + Ric(ω0 ) − Ric(ωt ) − Ric(ω0 ) + Tω0 ,η
ωn
= ∂∂(φt + log t − hω0 ,η )
n
ω0
the scalar equation for the potentials is
n

ω
˙
φt = φt + ct + log t − hω0 ,η
n
ω0

(2.59)

˙
Time differentiating and setting ut = φt we obtain the same equation appearing in the
K¨hler-Ricci flow setting
a
t ut

where at = c˙t and

t

= ut + at

1
= ∂t − 2 ∆. So apply the argument appearing in [Pa] directly to obtain

that at is uniformly bounded in t.
Lemma 14. The scalar curvature R is uniformly bounded from below along the twisted
K¨hler-Ricci flow.
a
Note this follows from the maximum principle:
Proof.
∂Λωt (Ric − η)
(ωt ) = |Ricαβ − ηαβ |2 + ∆t (Λωt (Ric − η))(ωt ) − Λωt (Ric − η)(ωt ) (2.60)
∂t
88

This simplifies to
∂et Λωt (Ric − η)
(ωt ) = et |Ricαβ − ηαβ |2 + et ∆t (Λωt (Ric − η))(ωt )
∂t

from which we get
Λωt (Ric − η)(ωt ) ≥ e−t Λω0 (Ric − η)(ω0 )

(2.61)

Since ωt > 0 and η ≥ 0
Λωt η = trt η ≥ 0
we obtain
S(ωt ) ≥ e−t (S(0) − trω η) ≥ min{0, S(0) − trω η}
To verify (2.60) note that the twisted K¨hler-Ricci flow differs from the standard K¨hlera
a
˙
˙
Ricci flow in that Ric is replaced by Q := Ric − η. Note Q = Ric. So the check for the
identity is no different from the K¨hler-Ricci flow setting:
a

dΛt Q
∂ αβ
αq pβ
αβ ˙
˙
= gt Qαβ = −gt gt gpq Qαβ + gt Qαβ
∂t
∂t
αβ

pq

˙
= − (gt − Q)pq (Q)pq − gt ∂α ∂β gt gpq
pq

pq

= − gt Qpq + |Qpq |2 − ∆t gt (gt − Q)pq
= |Qpq |2 − Λωt Q + ∆t Λωt Q

Corollary 5. ∆u is bounded from above.

89

Proof.

∆u =n − (S(ωt − T rt η))
≤n − e−t (S(ω0 ) − T r0 η)
≤n − max{0, S(ω0 ) − T r0 η}
<C

In the following lemma we sketch details in places since the argument is almost identical
with the untwisted setting in [ST].
˙
Lemma 15. The function u(t) := φ(t) is uniformly bounded from below.
Proof. Along the twisted flow we have from (2.59)

du
= n − (R − trt η) + u + a ≤ n + C + u
dt

(2.62)

where the second inequality follows from the lower bound on R − trt η obtained from the
lemma. The argument proceeds by contradiction as in [ST].
Start off assuming there is a point and time (t0 , y0 ) where u is very negative. Using
(2.62) gives that u(t) stays negative for t ≥ t0 in a neighborhood U of y0 . Obtain estimates

˜
u(t)(z) ≤ et−t0 (C + u(t0 )) ≤ −Cet

t ≥ t0

˜
φ(t)(z) ≤ φ(t)(z) − Cet−t0 ≤ −Cet

t >> 0 z ∈ U

z∈U
(2.63)

The first item follows from integrating out the differential inequality du ≤ C + u and using
dt
90

˙
that u(t0 ) can be made very negative. For the second integrate one more time using u = φ.
1
Using the normalization V M e−u(t) = 1 obtain that u(t) can’t be everywhere too neg-

ative. In particular we have the uniform estimate

max(u(t)) ≥ −C

(2.64)

d
(u − φ) < C
dt

(2.65)

˜
max φ(t) ≥ −C − Ct

(2.66)

M

With (2.62) rewritten as

obtain the estimate
M

after integrating and combining with (2.64). All constants are uniform.
We may obtain an upper bound on maxM φ(t) using the Green’s formula (see [ST])
applied to φ(t) and −∆0 φ(t) = −tr0 g(t) + n < n and (2.63). Obtain

˜
max φ(t) ≤ α max φ(t) − Cet + C
M

M

where 0 < α < 1 and t ≥ t0 . Then α < 1 gives the estimate

˜
max φ(t) ≤ −Cet + C
M

So taking large values of t it follows that maxM φ(t) decays no slower than a linear function whereas the upper bound gives that it decays at least as fast as an exponential so a
contradiction is obtained.

91

Remark 20. By the corollary
∆u < C(u + 2B)
for B sufficiently large, since u is bounded below by the lemma.
Proposition 24. Under the twisted ricci flow we check the evolution of | u|2 and ∆u satisfy

(∆u) = (∂t − ∆)∆u = −|

u|2 + ∆u

(| u|2 ) = ∂t | u|2 − ∆| u|2 = −|

(2.67)

u|2 − |

u|2 + | u|2 − η, u u g

(2.68)

Proof. Check (2.67) by direct calculation using gij = uij = Tij − Ricij and ∂t u = ∆u + u + a
˙

∂t ∆u =∂t g ij uij
˙
=∆u − 2g iq g pj gpq uij
˙
=∆(∆u + u + a) − 2g iq g pj upq uij
=∆(∆u + u) − |

u|2

and the identity follows.
For (2.68) we start with the Bochner formula

1
1
∆ | u|2 = |
2
2

1
u|2 + |
2

u|2 + Rαβ uα uβ + g αβ (uα (∆u)β + (∆u)α uβ )

92

Also note that

1
∂t | u|2 =g ij ui uj + g ij ui uj − g it g sj ust ui uj
˙
˙
2
=g ij (∆u + u + a)i uj + g ij ui (∆u + u + a)j + g it g sj (Ric − T )st ui uj

Putting this together

(∂t − ∆)| u|2 = −|

u|2 − |

u|2 + | u|2 − η ij ui uj

The identity follows.
Using these evolution equations applied to the same quantities appearing in [ST] obtain
by application of the maximum principle the following estimates
Claim 14.
| u|2 ≤ C(u + c)

(2.69)

−∆u ≤ C(u + C)

(2.70)

provided η ≥ 0.
So uniform bounds on |u| give uniform bounds on | u| and |∆u|.
Remark 21. After this is verified we follow arguments of [SzCo] where Proposition 7. in
the exposition of Sesum-Tian is replaced by a twisted entropy functional more appropriate to
the study of the twisted Ricci flow. At this point instead of following arguments in [ST] such
as Claim 8 where upper bounds on u, R are obtained in terms of the diameter, u is analyzed
just as the diameter is (by considering sub level sets of u) in the subsequent propositions. We

93

outline briefly the remaining arguments for the sake of exposition. Roughly the arguments
make essential use of the monotonicity properties of twisted Perelman entropy and that its
coercive (in the sense of Tao), in that it provides a scale invariant geometric control on the
flow known as κ-noncollapsing. Eventually to bound the scalar curvature we will see that the
n
assumption ωt ≥ Kω n for t ∈ [0, ∞) is used.

Remark 22. A priori its not clear what the effect of η ≥ 0 is on applying the maximum
principle as in original K¨hler setting. We will see the effect is benign. However, if η < 0
a
there are complications.
Proof. Just as in the case of K¨hler Ricci flow we consider the quantity
a

H=

| u|2
u + 2B

an application of the maximum principle will yield (2.69).
Using ∂t u = ∆u + u + a

∂t

| u|2
∂t | u|2 | u|2 (∆u + u + a)
=
−
u + 2B
u + 2B
(u + 2B)2

Similarly (we omit a factor of 2)

∆

| u|2
| u|2
=g pq ∂q ∂p (
)
u + 2B
u + 2B
| u|2
| u|2 up
p
=g pq (
−
)q
u + 2B (u + 2B)2
pq
2
2
g pq | u|2 up uq
∆| u|2 g pq | u|2 uq g | u|q up + | u| ∆u
p
−
−
+2
=
u + 2B
(u + 2B)2
(u + 2B)2
(u + 2B)3

94

So
2
pq
g pq | u|2 up uq
∂t | u|2 | u|2 (u + a) ∆| u|2 g pq | u|2 uq g | u|q up
p
−
+
H=
−
+
−2
u + 2B
u + 2B
(u + 2B)2
(u + 2B)2
(u + 2B)2
(u + 2B)3

=

| u|2 | u|2 (u + a)
+
−
u + 2B
(u + 2B)2

| u|2 , u + | u|2 , u
| u|4
−2
(u + 2B)2
(u + 2B)3

using the evolution identity (2.68) obtain

−|

u|2 − |

+

H=

u|2 − η, u u g | u|2 (2B − a)
+
u + 2B
(u + 2B)2

| u|4
| u|2 , u + | u|2 , u
−2
(u + 2B)2
(u + 2B)3

(2.71)

Note that

H

=g αβ (

| u|2 uα
| u|2
α
−
)
u + 2B (u + 2B)2

u· H
u + 2B
u, | u|2
| u|4
=(2 − )
− (2 − )
(u + 2B)2
(u + 2B)3

=⇒ (2 − )

(2.72)

Rewriting the last two terms in (2.71) using (2.72) obtain

2

u, | u|2
| u|4
u· H
−2
= (2 − )
+
u + 2B
(u + 2B)2
(u + 2B)3
−

95

| u|4
(u + 2B)3

u, | u|2
(u + 2B)2
(2.73)

Using an orthonormal frame and an application of Cauchy-Schwartz gives

| u·

| u|2 | = i u i ( j u j u)
= i u( i j u) j u +
≤| u|2 (|

u| + |

i u j u( i

j u)

(2.74)

u|)

Fix a constant C ≥ 1. Then apply (2.74) to the term in (2.73) gives
| u|2 (|
u|2 + |
u|2 )
| u · | u|2 |
≤C
(u + 2B)2
(u + 2B)3/2 (u + 2B)1/2
≤

C 2 (|
| u|4
+
4 (u + 2B)3
2

u|2 + |
u + 2B

u|2 )

(2.75)

From (2.75) obtain

2

u, | u|2
| u|4
u· H
+
−2
=(2 − )
u + 2B
(u + 2B)2
(u + 2B)3
≤(2 − )

Choose

u, | u|2
| u|4
−
(u + 2B)2
(u + 2B)3

|
u· H
+ 2C 2
u + 2B

u|2 + |
u + 2B

u|2

so that 2C 2 < 1 . Applying this inequality to the expression for
2

H=

−|

u|2 − |

+ (2 − )

−

3
| u|4
4 (u + 2B)3

H obtain

u|2 − η, u u g | u|2 (2B − a)
+
u + 2B
(u + 2B)2

u· H
|
+ 2C 2
u + 2B

u|2 + |
u + 2B

Note that since η ≥ 0

η, u u g ≥ 0

96

u|2

−

3
| u|4
4 (u + 2B)3

(2.76)

This can be checked through point-wise calculation simultaneously diagonalizing η with
respect g. In co-ordinates this looks like

η ij ui uj = λii δij ui uj = λii |ui |2 ≥ 0

So drop this term in (2.76) to obtain the inequality

H≤

(2C 2 − 1)(|
u|2 + |
u + 2B

u|2 )

+

u· H 3
| u|2 (2B − a)
| u|4
+ (2 − )
−
u + 2B
4 (u + 2B)3
(u + 2B)2

Since 2C 2 − 1 < 0 and u + 2B > 0 we may drop the first term on the right hand side to
obtain

H≤

| u|4
| u|2 (2B − a)
u· H 3
−
+ (2 − )
u + 2B
4 (u + 2B)3
(u + 2B)2

(2.77)

So even with the extra term η, u u g we apply the maximum principle to the quantity
H just as in the K¨hler-Ricci flow case. At a point where H achieves it maximum we have
a
H = 0 and ∆H ≤ 0. So

0 ≤ ∂t Hmax ≤

Hmax ≤

3 | u|2
| u|2
(2B − a −
)
4 u + 2B
(u + 2B)2

(2.78)

| u|2

If the inequality H = u+2B ≤ C fails we may produce a sequence of counter examples
given by the data {(xmax , tn ), Cn } where Cn → ∞. So that the parenthesis of the second
inequality in (2.78) is negative on the sequence for n >> 0. Contradiction. (2.69) follows.
The second inequality −∆u < C(u + 2B) obtains similarly. The dependency on η arises

97

as follows
u|2 (K + bH)(2B − a)
u|2 − (b − 1)|
+
u + 2B
u + 2B
u · (K + bH)
η, u u
+2
−
u + 2B
u + 2B

(K + bH) =

−b|

−∆u
where K = u+2B and b > 1.

Since

η, u u
u+2B

≥ 0 we may drop the last term. Set G = K + bH. Then the evolution

identity for Gmax is
|
u|2 Gmax (2B − a)
d
Gmax ≤ −(b − 1)
+
dt
u + 2B
u + 2B

This is also the inequality that one gets in the K¨hler-Ricci flow case so the remaining part
a
of this argument is identical.
Tracing the twisted K¨hler-Ricci flow equation obtain ∆u = T rωt Tt − R then R =
a
T rωt Tt − ∆u. Once we can establish

Tωt ,η ≤ cωt

(2.79)

where c is a constant independent of t, we can bound the scalar curvature (2.44) since

T rt Tωt ,η ≤ cn

and we can obtain the bound R < C(u + 2B).
Remark 23. However, proceeding exactly as in [ST] does not work since the κ-noncollapsing

98

property for the flow:
V olt (B(x, 1)) ≥ κ
for any metric g satisfying

|R − trt η| ≤ 1 on B(x, 1) ∂B(x, 1) = ∅

needs to be established.
Recently this has been verified in [SzCo]. This involves introducing the twisted entropy
functional:
Definition 2. On a compact K¨hler manifold let η be a closed nonnegative (1,1) form. The
a
twisted entropy functional
W : M et × C ∞ (R) × R>0 −→ R is given by

W η (g, f, τ ) :=
M

(τ (R − trg η + | f |2 ) + f − 2n)(4πτ )−n e−f dm
g

for the unnormalized twisted K¨hler ricci flow
a

ω = −2(Ric(ω) − η)
˙

(2.80)

This is Perelman’s entropy functional with R replaced by R − trg η; exactly the same adjustment needed to obtain the twisted Mabuchi functional. Similar to the twisted Mabuchi
functional the monotonicity property of the entropy functional carries over to the twisted
setting. The twisted entropy functional shares many other useful properties with the entropy
functional.
99

Proposition 6. For (g(t), f (t), τ (t)) ∈ M et × C ∞ (R) × R

∂t W(g(t), f (t), τ (t)) = τ

|Ric − η + Hess(f ) −
M

g 2
| (4πτ )−n e−f dm
2τ t

where the triple (g(t), f (t), τ (t)) satisfies the usual system of PDE’s with R, Ric replaced by
R − trt η, Ric − η.
Following a contradiction argument κ-noncollapsing in the formulation of [SzCo] is obtained by applying the twisted entropy to a test function. From its monotonicity properties
and effective estimates one can conclude. The softer version in the spirit of [ST] works too.
But first the flow needs to be reparametrized to work with the twisted entropy functional.
Claim 15. The twisted K¨hler ricci flow can be reparametrized to unnormalized twisted ricci
a
flow
ω = −2(Ric(ω) − η)
˙

(2.81)

Proof. Let g = ψ(t)(g) denote the reparametrized metric with respect t(s). To determine
˜
t(s) we need to solve ode’s:

˙
∂t g = ψ(g) + ψ(g)
˜
˙
˙
= ψ(g) + ψ(−Ric + g + η)
˙
= (ψ + ψ)(g) + ψ(η − Ric)

So
˙
2
ψ
∂t g = (2 + 2)(g) − 2Ric = −2(Ric − η)
˜
ψ
ψ
100

˙
∂t
2
provided ψ +1 = 0. Since ∂s = ∂s ∂t = ψ ∂t a choice of reparmetrization t(s) can be obtained
ψ

by solving
˙
2ψ
∂ψ
+2=
+2=0
ψ
∂s
dt
2
=
ds
ψ

Solving we obtain t(s) = −ln(C − 2s) and if we enforce that t(0) = 0 we can take C = 1.
Remark 24. Reparametrization allows to transfer the κ-noncollapsing property for unnormalized flow to normalized twisted K¨hler ricci flow. See [SzCo].
a
From the discussion above adjusting u by a constant appropriately the following uniform
estimates are in hand
|∆u|, | u|2 < Ku

(2.82)

So it suffices to bound u from above.
It was observed in [SzCo] that by considering sublevel sets of the form

M (a, b) = {x ∈ M |a < u < b}

instead of geodesic annuli u can be bounded directly without requiring a diameter bound.
Remark 25. If a < b < c < d then M (a, b) ∩ M (c, d) = ∅
For the purpose of bounding u a contradiction argument needs to be made and one starts
by assuming u grows without bound.

101

V = V ol(Mt ) is constant along the flow. Partition M using the range of u then
N

V ol(M (210

i−1 k

i
, 210 k )) < V

i=1

for k sufficiently large (depending on u) and taking N > V , there is an 1 ≤ i0 ≤ N for
which
0 < V ol(M (210

i0 −1 k

i0
, 210 k )) <

Note that i0 can be taken to be 1 at the cost of making k larger.
Like geodesic annuli considered in [ST], [SzCo] does the same for M (a, b) instead. In
particular:
Lemma 16. There is a point x ∈ M with u(x) = a + 1 and constants κ1 such that if
b − 2 > a > K then
V ol(M (a, b)) > κ1 a−n
Restricting (2.82) to M (a, a + 2) ⊂ M (a, b) gives estimates necessary to apply the κ
non-collapsing property to conclude.
After specifying a threshold that k above must exceed, since u is assumed unbounded we
may assume there is k so that V ol(M (2k , 210k ) < < 1 and we can always find an x ∈ M so
that u(x) = 25k + 1 say. Clearly for k1 , k2 ∈ [k, 10k], V ol(M (2k1 , 2k2 )) < holds. Moreover,
similar to Claim (10) in [ST]
−1

Lemma 17. Provided k exceeds the threshold max{log2 (κ n ), 2} and 0 < < 1, there exists

102

k1 , k2 ∈ [k, 10k] with k2 > k1 + 4 such that

V ol(M (2k1 , 2k2 )) <
V ol(M (2k1 +2 , 2k2 −2 )) > 2−3n V ol(M (2k1 , 2k2 ))

The second estimate above follows by iterating the reverse inequality starting with the
sublevel set M (2k , 29k+2 ). Finally to conclude one applies the previous lemma and uses the
threshold to obtain a contradiction.
The penultimate step is similar to Lemma to 11 in [ST] with −∆u = T rt (Ric − η) − n
replacing scalar curvature and provided k2 > k1 + 1 then
Lemma 18. There exists r ∈ [2k1 , 2k1 +1 ] and r2 ∈ [2k2 −1 , 2k2 ] so that

(−∆u)dm < CV ol(M (2k1 , 2k2 ))
M (r1 ,r2 )

As before one works with (2.82) on sublevel sets. An application of co-area formula allows
to pass to estimates on some smooth sets u = ri , i = 1, 2. Then conclude as in [ST].
Finally just as in Proposition 9 in [ST] we are in the setting of lemma (17) so
−1

Proposition 25. There is an > 0 such that if k > max{log2 (κ1n ), 2} and V ol(M (2k1 , 2k2 )) <
then u is bounded.
This proceeds by contradiction, when u grows without bound a cutoff function is constructed so that lemma’s 17, 18 may be used and fed into the twisted entropy functional
just as in [ST]. In fact the same argument in [ST] with use of the twisted entropy function
can be made. However [SzCo] proceeds using effective estimates to obtain the contradiction:
roughly by lemma (17) there is k1 , k2 such that V := V ol(M (2k1 , 2k )) < , whereas via the
103

twisted entropy functional we may obtain a choice for which V >

a contradiction. So u is

bounded from above.
To summarize, along the twisted K¨hler ricci flow [SzCo] obtained
a
Proposition 7 (Sz-Co). Along TKRF with g(0) = g0 there exist a constant C depending
continuously on the C 3 norm of g0 (and a uniform lower bound of g0 ) such that

|u| + | u|g(t) + |∆g(t) u| ≤ C

Now we are in a position to start bounding scalar curvature and also justify the first
inequality in (2.58). For this it suffices to show for the twisted K¨hler Ricci flow:
a
ωn

Lemma 19. Suppose ωt ≥ K0 for all t ∈ [0, ∞) where K0 is a constant independent of t.
n
Then there exist positive constants k0 , K independent of t > 0 such that for all t > 0 the
following estimates hold

0 <n + ∆ω φt < K

(2.83)

√
n

(2.84)

|∂∂φt |ω <k +

−1
k0 ω <ωt < Kω

(2.85)

Remark 26. (2.83) appears as the parabolic analogue of Yau’s C 2 estimate. But a little
modification is needed. In K¨hler-Ricci flow case [Pa] uses Perelman’s estimate |u| < C.
a
Recall in [ST] this depends on scalar curvature but thanks to [SzCo] this dependency can be
removed.
Proof. For (2.83) the ingredients remain the same as in [Pa]. Consider the quantity appearing

104

in K¨hler-Ricci flow
a
A := log(trω ωt ) − k(φt + ct ) M × [0, ∞)
Follow a computation similar to that in [BBEGZ]. Start with τ, τ K¨hler forms on a complex
a
manifold. There is a lower bound B > 0 for Riijj (τ ) so that

∆ log trτ τ ≥ −
τ

trτ Ric(τ )

− Btr τ
τ

trτ τ

Apply this when τ = ω and τ = ωt . So we obtain when

t log(trτ τ

)≥−
=−

∂t trτ τ
trτ τ

−

t

trτ Ric(τ )
trτ τ

= −(∂t − ∆t )

− Btr τ
τ

(∂t trτ τ + trτ Ric(τ ))

− Btr τ
τ

trτ τ

Note
˙
∂t ∆τ τ = ∆ω φ = trτ τ − trτ Ric(τ ) + trτ η
Since C ≥ trτ η ≥ 0 and n ≤ trτ τ tr τ we have
τ

t log(trτ τ

)≥−

trτ τ + trt η

− Btr τ
τ

trτ τ
C

− Btr τ
τ
trτ τ
C
≥ − 1 − ( + B)tr τ
τ
n

≥−1−

= − 1 − Ctr τ
τ

105

So choosing k > C and using tr τ = trt ω = n − ∆t φ = n − ∆ φ
τ

tA

τ

˙
≥ −1 + (k − C)tr τ + k(φt − n + at )
τ

(2.86)

Recall
τn
trτ1 τ2 ≤ 2 (trτ2 τ1 )n−1
n
τ1
ωn

˙ ˆ

so using ωt = eh+φ−φ
n

tr τ ≥(
τ

τn
(τ )n

1

trτ τ ) n−1

ˆ
˙
φ−h−φ
1
n−1 (trω ωt ) n−1
=e
ˆ
˙
A (k+1)φ−h−φ
n−1 e
n−1
=e

˙
Since estimates |φ|, |φt + ct | < C are available obtain
A

tr τ ≥ C0 e n−1
τ

where C > 0.
So (2.86) becomes
A

tA

≥ −C + C0 e n−1

It follows by applying the maximum principle that we have a uniform upper bounds on the
ˆ
maximum of eA < C n−1 ; since a maximum of A is a maximum of eA . Using boundedness φ

106

on [0, ∞) the upper bound of (2.83) follows:

n + ∆t φ = trω ωt < K

The upper bound in (2.85) follows directly from here since

1 + φii < trω ωt < K

so
ωt < Kω
To get the first inequality in (2.85) use the uniform estimate to get
ωn
K0 ≤ t = Πn (1 + φjj ) < K n−1 (1 + φii )
j=1
ωn
K

−1
0
Conclude ωt > k0 ω for k0 := n−1 > 0. In particular, it follows that Tt,η is uniformly
K

equivalent to ω.
Another pointwise calculation yields (2.84). At a point p

(1 + λii )2

ωt , ωt g =
i

(1 + λii ))2

<(
i

=(trω ωt )2
<K 2

107

i (1 + λii )

Since trω ωt > 0 it follows

(λ2 + λii ) <
ii
i

n
i=1 λii

> 0 and so −

< n. It follows

(1 + λii )2 < K 2
i
n

=⇒

|ddc φ|2
g

= (λii

)2

<

K2

λii < K 2 + n < (K +

−

√ 2
n)

i=1

2.8

Twisted Perelman entropy

We note, following [Tao], that this functional can be obtained by analyzing variations of
known functionals. Temporarily replacing the volume form by a static measure, a critical
quantity, which also happens to be a coercive quantity (in the sense of Tao) can be obtained
due to Perelman. It is also monotone with special type of critical points. Denote the volume
form by dµ. Consider the functionals

E(f ) =

1
| f |2 dµ
g
2 M

H(M, g) =

Rdµ
M

Provided g is static the E functional deforms like:

d
E=−
∆g f f˙dµ
dt
M

108

(2.87)
(2.88)

The first variation of the H functional is given by

d
d
˙ 1
H=
Rdµ =
(R + trg (g))dµ
˙
dt
dt M
2
M
=
M

=

(−Ricαβ gαβ − ∆trg g +
˙
˙

α

βg
˙ αβ

1
+ trg (g)αβ )dµ
˙
2

1
(−Ricαβ gαβ + R trg (g)αβ )dµ
˙
˙
2
M

The gradient flow for the negative functional is known to be not parabolic in general(g =
˙
Ric − Rg n ≥ 3). Replacing dµ by a static measure dm = e−f dµ (so the potential f deforms
1
like f˙ = 2 trg (g)) removes the contribution from R 1 trg gαβ causing this issue. In this way
˙
˙
2

obtain modified functionals H mod , E mod . H mod deforms like

d mod
˙
H
=
Rdm
dt
M
=
M

=
M

(−Ricαβ gαβ − ∆trg g +
˙
˙

α

β g )dm
˙ αβ

(−Ricαβ gαβ + (∆f − | f |2 )g αβ gαβ
˙
˙
g

+ gαβ ( α f β f ) − gαβ ( α β f ))dm
˙
˙

Note that in the deformation of H mod the second term following the third equality ∆f trg g
˙
above comes with an opposite sign to that of the variation of Dirichlet energy, which gives

109

another motivation for the choice of E mod . Its first variation is given by

d mod
d αβ
E
=
(g
α f β f )dm
dt
M dt
=
M

(−gγδ γ f δ f + g γδ f˙γ fδ + g γδ fγ f˙δ )dm
˙

1
1
(−gγδ γ f δ f + g γδ ( tr(g))γ fδ + g γδ fγ (trg g)δ )dm
˙
˙
˙
2
2
M

=
=

M

(−gγδ γ f δ f − (∆f − | f |2 )g γδ gγδ )dm
˙
˙
g

Define
Fm (M, g) := H mod + E mod =
M

(| f |2 + R)dm
g

So Fm deforms along g = −(2Ric + 2Hessf ) as
˙

∂t Fm (M, g, f ) = 2

|Ric(g) + Hess(f )|2 dm
M

It follows Fm is non decreasing along the flow. Along this gradient flow using the relation
1 ˙
f˙ = 2 trg we see that the potential deforms according to f˙ = −∆f − R. Using L f gαβ =

˙
2 α β f = 2Hessf and L f f = | f |2 and ∂t φ∗ ωt = φ∗ (LX ωt + ωt ) it follows the gradient
g
t
t
flow and the potential flow can be conjugated by a diffeomorphism to g = −2Ric(g) and
˙
f˙ = −∆f − R + | f |2 .
g
After conjugation f does not define a static measure but since Fm is invariant under
diffeomorphism, its variation remains the same whether modified by a diffeomorphism or
not. In particular under the modified flow induced by the diffeomorphism: g = −2Ric(g)
˙
and f˙ = −∆f − R + | f |2 , Fm is monotone non-decreasing.
g
From the monotonicity property it can be deduced the periodic solutions (ones for which

110

φ∗ g(t2 ) = g(t1 )) are critical points i.e satisfy Ric = −Hessf . Similarly we can consider
1
functionals with critical points solutions to Ric + Hessf − 2τ g = 0 (gradient shrinking

solitons), which when f = 0 has positively curved Einstein metrics as critical points.
Note that

|Ric + Hessf −

1
g 2
n
| = |Ric + Hessf |2 − (R + ∆f ) + 2
g
2 g
τ
2τ
4τ

(2.89)

With respect to Ricci flow scaling τ has dimension 2. So the derivative of the scale invariant
quantity must have dimension −2. But each of the three terms have dimension −4. So we
must consider a quantity like 2τ M |Ric + Hessf − g2 |2 dm. Recalling the Nash entropy
2τ g
functional Nm := M log dm dm = − M f dm deforms like (provided dm is static)
dµ
d
Nm = −
f˙dm =
(∆F + R)dm =
(| f |2 + R)dm
g
dt
M
M
M
we may integrate to obtain Wm (M, g, f, τ ).
Similarly
Proposition 26. The twisted entropy functional given by

α
Wm (M, g, τ, f ) = (4πτ )−n

M

(τ (R − T rg α + | f |2 ) + (f − 2n))e−f dm
g

deforms like

|Ric − α + Hessf −

2τ
M

111

g 2
| dm
2τ g

It is monotone provided (g(t), f (t), τ (t)) solves the coupled system

∂g g = − 2(Ric − α)
˙
∂t f = − ∆g f + | f |2 − R + T rg α +
g

n
τ

∂t τ = − 1

on some interval [0, T ].
Proof. Following the same heuristics one obtains the twisted entropy functional. For the
same reasons it will be both monotone and a critical quantity (in the sense of Tao).
Along g = −2(Ric − α + Hessf ) with dm static so f˙ = −∆g f − R + trg α, it follows
˙

d
Ric − α + Hessf, α dm
−g sα g βt gαβ αst dm = 2
˙
trg αdm =
dt M
M
M
Just as in (2.89) we have

|Ric − α + Hessf −

g 2
1
n
|g = |Ric − α + Hessf |2 − (R + ∆f − trg α) + 2
g
2τ
τ
4τ

α
Similarly we have that for Fm := M (R + | f |2 − trg α)dm
g

d α
Fm = −
(Ricαβ +
dt
M

112

α

βf

− ααβ )gαβ dm
˙

(2.90)

α
Likewise for Nm := M −f dm. So if dm is static along g = −2(Ric − α + Hessf ) then
˙

d α
−f˙dm =
(R + ∆g f − trg α)dm
N =
dt m
M
M
=
M

(R + | f |2 − trg α)dm
g

It follows that

g
n
d
α
α
|Ric − α + Hessf − |2 dm
(τ Fm − Nm − log τ ) = 2τ
dt
2
2τ g
M
normailized so that dm is a probability measure.
n
˜
˜
Write e−f dµ = dm = (4πτ )− 2 e−f dµ so f = f + n log(4πτ ). Since dm is a probability
2

measure up to an arbitrary constant we may write the functional as

M

˜
˜
(τ (R + | f |2 − T rg α) + f − cnst)(4πτ )−n dm
g

Normalize the arbitrary constant to n so that in the euclidean setting, when also α = 0, dm
is gaussian measure and the expression vanishes.
n
˜
Note that f˙ = −∆g f − R + trg α + 2τ so conjugating by a diffeomorphism induced by

the vector field

f obtains the coupled system

g = − 2(Ric − α)
˙
n
˜
˜
˜
f˙ = (−∆g f − R + T rg α +
+ | f |2 )
g
2τ

113

2.9

Extracting canonical metric

In this section we show how the canonical metric g∞ , solving (2.1), can be extracted. This
follows along the same lines as [Pa] for the K¨hler-Einstein case. Recall we have in hand the
a
following (with the exception of the fourth bullet point)
ωn

˙
• φ = log ωφ + φ + f + ct
n
η

• νω is bounded below.
η

• νω is decreasing along the twisted K¨hler-Ricci flow.
a
ˆ
• |φt |C 0 (X) + |∂∂φt |C 0 (X) + | ∂∂φt |C 0 (X) < C
provided η ≥ 0 and the non-collapsing estimate along the flow holds.
η

A consequence of the second and third bullet points is limt→∞ νω < ∞. So for sequences
{tk }

∞ we have
n
˙
| φtk |2 ωt = 0
tk k
k→∞ X

lim

Since otherwise we may integrate and contradict boundedness along the flow.
From the uniform C 2 , C 3 estimates we obtain that the (1, 1) forms ∂∂φt are uniformly
bounded in C 0,α (X) topology.
By differentiating the first bullet by ζ = ∂zk , ∂zk we have from [Pa]

t (ζφt ) + ζφt

The laplacian term in

t

= (T rω − T rt )(Lζ ω) + ζhω,η

−1
and T rt contain gt so are bounded in C 0,α norm. By schauder

regularity theory for parabolic equations we obtain ζφt is uniformly bounded in C 2,α .

114

We recall that C k,α → C k is a compact embedding. So a bounded sequence in C k,α lies
in a compact set in C k ( k ≥ 0, 0 < α < 1) (see [Jo]).
Since ζφt ∈ C 2,α is uniformly bounded, φt ∈ C 3,α is uniformly bounded. Further, thanks
ˆ
ˆ
to the C 0 uniform estimate |φt | < C we have φt lies in a bounded set in C 3,α . So by the compact embedding we may arrange for a subsequence of φtk so that (φˆk , dφsk , ∂∂φsk , ∂∂φsk )
s
converges uniformly to (φ∞ , dφ∞ , ∂∂φ∞ , ∂∂φ∞ ).
The uniform estimate

n
ωt
ωn

≥ K > 0 gives that

n
ωφ
∞
ωn

> K. By (2.85) we have ωφ∞ > 0.

The equation in the limit on this sequence then reads
n

ω
˙
ψ := lim φsk = log ∞ + φ∞ − hω,η
ωn
k→∞
In particular ψ is C 1 . Since φsk ∈ C 3,α converges in C 3

˙
lim ∂ φsk = lim (∆sk ∂φsk + ∂φsk − ∂hω,η ) = ∆∞ ∂φ∞ + ∂φ∞ − ∂hω,η = ∂ψ

k→∞

k→∞

The convergence being uniform gives that

0 = lim

k→∞ X

n
n
˙
| φsk |2 ωs =
|∂ψ|2 ∞ ωφ∞
sk k
φ
X
˙

n
So ψ is a constant and from the normalization X e−φt ωt = V we obtain that ψ = 0. So

φ∞ satisfies
0 = log

n
ωφ

∞
n
ω

+ φ∞ − hω,η =: F (φ∞ )

115

Recall that hω corresponds to the twisted ricci potential. Since ω∞ > 0 ellipticity follows
from computing the linearization:

dFφ∞ (v) = ∆φ∞ v + v

By Schauder regularity theory for elliptic equations we can conclude that φ∞ is smooth. So
φ∞ is a desired solution to the equation Ric(ω∞ ) = ω∞ + η.

2.10

C 3 Estimate

g
Following [PSS] work with the quantity hα := g αk gkβ (ˆ−1 g is an endomorphism). We follow
ˆ
β
the notation gkβ since most quantities appearing in this formulation come as endomorphisms.
Let g denote the initial metric. Note that T rh = T rg g. Similarly h−1 = g −1 g . Recall also
ˆ
ˆ
ˆ
that in the K¨hler setting the connection looks like g −1 ∂g, is of pure type i.e. Γk = g ks ∂j gis
a
ij
(old notation) and the torsion free condition gives symmetry in permutation of i, j. Similarly
the curvatures look like R·

·k·

= ∂k Γ· .
··

The change in connection with respect to the initial metric can be written in terms of
the quantity ( h)h−1 .
ˆ
( m h)k hλ = (Γ − Γ)k
λ l
ml

116

so as not to distract from our main goal we only outline the calculation schematically:

( h)h−1 = (∂h − hΓ + hΓ)(h−1 )
= (∂h − g −1 gg −1 ∂g + hg −1 ∂g)(h−1 )
ˆ
ˆ
= (∂h − g −1 ∂g)(h−1 ) + Γ
= (∂(ˆ−1 g) − g −1 ∂g)h−1 + Γ
g
ˆ
= (∂ˆ−1 gh−1 + Γ)
g
= (−ˆ−1 g −1 (∂ˆ)ˆ + Γ)
g ˆ
g g
= (−ˆ−1 ∂ˆ + Γ)
g
g
ˆ
= (−Γ + Γ)

Here in the first line the minus sign comes from the connection extended to forms ( m in
the first line is the covariant derivative induced on endomorphisms). In passing from line 4
to 5 the torsion free property is used to cancel out −ˆ−1 ∂g appearing in line 4.
g
On forms and vector fields the Levi-Cevita connection differs by a sign we have the
following expression for the change in connection acting on forms and vector fields:

( m − ˆ m )Vl = −Vα ( m hh−1 )α
l
( m − ˆ m )V l = ( m hh−1 )l V α
α

Similarly for curvature
ˆ
(R − R)α = ∂k ( j hh−1 )α
β
jkβ

117

(2.91)

Also
φjkm = ˆ m φjk = −gkα ( m hh−1 )α
j

(2.92)

The minus sign here is attributed to following the convention that gkα = gkα + φkα .
ˆ
To establish the C 3 estimate one considers the quantity

S = g jr g sk g mt φjkm φrst

This can be written as

β

µ

S = g mγ gµβ g lα ( m hh−1 )l ( γ hh−1 )α = | hh−1 |2

using (2.92).
We summarize the process involved in establishing C 3 estimates. The following obtains

∆S = g −1 gg −1 (∆( hh−1 ) hh−1 + ( hh−1 )∆ hh−1 )
+| ( hh−1 )|2 + | ( hh−1 )|2

(2.93)

by direct calculation. The term with ∆ = g pq q p can be written in terms of ∆ by
γ

γ

commuting derivatives and introducing curvatures. That is, with (Tα )j = ( j hh−1 )α obtain

γ

γ

γ

µ

µ

γ

µ

γ

∆(Tα )j = ∆(Tα )j − Rµ (Tα )j + Rα (Tµ )j + Rj (Tα )µ

Replacing the ∆ expression in (2.93) by this gives another expression for ∆S with three
extra terms involving curvatures. See [PSS] for the formula. Note that ∆S is a fifth order

118

term and the expression in [PSS] (2.43) contains terms with ∆( hh−1 ) which are also fifth
order. In order to obtain an expression of the form ∆S ≥ −C1 S − C2 to apply a maximum
principle we need to reduce the order of the terms appearing. At worst to fourth and third
order terms, and of course the fourth order terms must come with favorable sign so they can
be dropped.
An application of Bianchi identity gives

·

·
ˆ
∆( hh−1 ) = − · R· + g −1 · R·· ·

(2.94)

The first term in (2.94) and curvature terms Rβα appearing in [PSS] (2.43) need to cancel
out since these are not controlled.
˙
˙
Next S can be expressed in terms of h−1 h. Noting
˙
• g = g h = g(h−1 h)
˙ ˆ˙
˙
˙
• g −1 = −g −1 = −(h−1 h)g −1
˙
•

h = gg −1 ∂g −1 gh = g −1 ∂(ghg −1 )g

The appearance of g after the second equality in the first bullet corresponds to lowering the
˙
endomorphism (h−1 h) to a (0,2) tensor. Similarly in the second bullet g −1 corresponds to
raising it to a (2,0) tensor.
Some application of these bullet points give
1.

˙h = −h−1 h + ( h)h−1 h +
˙
˙

˙
2. ( hh−1 ) =

˙
(h−1 hh) =

˙
(h−1 h)

119

˙
(h−1 h)h +

˙
h(h−1 h)

Here the first equality in (1.) follows from time differentiating bullet three. With this the
˙
expression S can be computed. This gives rise to terms

β

µ

g mγ gµβ g lα (∂t ( m hh−1 )l ( γ hh−1 )α )

and time differentiating the g lα term in S using the second bullet and likewise gµβ using the
˙
˙
first bullet boils down to replacing g lα by −(h−1 h)lα and gµβ by (h−1 h)µβ . That is we get
terms like:

β

µ

β

µ

˙
−g mγ gµβ (h−1 h)lα ( m hh−1 )l ( γ hh−1 )α
˙
g mγ (h−1 h)µβ g lα ( m hh−1 )l ( γ hh−1 )α

˙
˙
The formula for S appears in [PSS] as equation 2.47. Note that in 2.47 h−1 h is raised or
lowered with the metric. So under the action of ∆ − ∂t , S deforms as

β

µ

(∆ − ∂t )S =| ( hh−1 )|2 + | ( hh−1 )|2 + g mγ gµβ g lα ((∆ − ∂t )( m hh−1 )l ( γ hh−1 )α )
β

µ

˙
+ g mγ gµβ g lα (( m hh−1 )l (∆ − ∂t )( γ hh−1 )α ) + ((h−1 h + R)mγ gµβ g lα
β

µ

˙
˙
− g mγ (h−1 h + R)µβ g lα + g mγ gµβ (h−1 h + R)lα )( m hh−1 )l ( γ hh−1 )α
(2.95)
Now finally we may begin the verification of the C 3 estimates for our problem.
Proposition 27. Along the Tkrf the uniform estimate | ∂∂φt |C 0 (X) < C holds.
Restrict to twisted K¨hler-Ricci flow. Then
a

β

β

˙
(h−1 h)l = g βα (g + η − R)lα = (δ + η − R)l
120

It follows that

β

β

β

˙
(h−1 h + R)l = (δ + η)l = (Tη )l

With
cωt ≥ Tωt ,η ≥ ωt
β

β

β

˙
(δ)l ≤ (h−1 h + R)l ≤ c(δ)l

(2.96)

together with (2.94), 2. and (2.91) obtain

(∆ − ∂t )( j hh−1 )l = −
m

l
j Rm

+

p Rl
ˆ
pjm

−

−1 ˙ l
j (h h)m

= −

l
j Rm

+

p Rl
ˆ
pjm

−

j (δ

=

p Rl
ˆ
pjm

−

+ η − R)l
m

l
j (η)m

Using (2.96)

(∆ − ∂t )S ≥ | ( hh−1 )|2 + | ( hh−1 )|2
ˆβ
+ g mγ ( p Rpml −

β
−1 l
m (η)l )( γ hh )β )

ˆµ
+ g mγ ( m hh−1 )α ( p Rpγα −
µ

µ
γ (η)α ))
β

µ

+ (2 − c)g mγ gµβ (g)lα ( m hh−1 )l ( γ hh−1 )α

(2.97)

√
Lemma 20. The second and third terms in (2.97) are O(S + c1 S)
ˆβ
Proof. We check this for g mγ p Rpml ( γ hh−1 )l , and terms involving the conjugate expresβ

121

sion are similar.

p Rβ
ˆ
pml

ˆβ
ˆβ
= g jp ( ˆj Rpml − ( j hh−1 )α Rpαl
m
β ˆα
ˆβ
− ( j hh−1 )α Rqmα + ( j hh−1 )α Rqml )
l

ˆ
ˆβ
= g jq ( ˆj Rqml + O( hh−1 R))

Claim 16. Along Tkrf the following estimates are available:
√
ˆβ
ˆ
g jq ˆj Rqml ( γ hh−1 )l ≥ − | ˆ R|| hh−1 | ≥ −C S
β
β

ˆ
−g mγ g jq ( j hh−1 )α Rqαl ( γ hh−1 )l ≥ − CS
m
β
β

ˆ
g jq g mγ ( j hh−1 )α Rqmα ( γ hh−1 )l ≥ − CS
l
β
β

ˆα
g jq g mγ ( j hh−1 )α Rqml ( γ hh−1 )l ≥ − CS
β
β
ˆ
ˆβ
This follows by using R is bounded, so Rqαl ≤ Cδα gql (by the C 2 estimate g·,· is equivalent

to g ) and cauchy schwartz.
ˆ
β

Similarly, but with less effort, −g mγ m ηl ( γ hh−1 )l can be handled. Since η is fixed
β
β

β

β

there is a C > 0 so that −Cδα ≤ ηα ≤ Cδα . So

β

β

β

−g mγ m ηl ( γ hh−1 )l = − g mγ ˆ m ηl ( γ hh−1 )l + g mγ ( m hh−1 )α ηα ( γ hh−1 )l
l
β
β
β
β

α
− g mγ ( m hh−1 )α ηl ( γ hh−1 )l
β
β

≥ − η, hh−1 − 2Cg mγ ( m hh−1 )l ( γ hh−1 )l
β
√
≥ − |η|| hh−1 | − 2C| hh−1 |2 ≥ C1 S − 2CS

122

proof of proposition. Applying the lemma, (2.97) becomes
√
˜
˜
(∆ − ∂t )S ≥ | ( hh−1 )|2 + | ( hh−1 )|2 − CS − C1 S ≥ −C1 S − C2

To conclude, exactly the same argument as in [PSS] applies. That is, with A sufficiently
large the following expression

ˆ
(∆ − ∂t )(S + A∆φ) ≥ C3 S − C4

(2.98)

C3 > 0 is available. An application of the maximum principles allows to conclude that S is
bounded by a positive number.

123

Chapter 3
Future direction

3.1

Coupled system

Assume H 0 (X, KX ⊗ L) is endowed with the natural L2 inner product induced from the
hermitian metric on the adjoint bundle KX ⊗ L. Consider the coupled system

si ∧ si e−φ )e−u =

(ωu )n = (
i

X

hK ⊗L (si , sj )e−u =
X

i

n
|si |2 ω0 e−u := µs e−u
h,ω0

2

X

in si ∧ sj e−(φ+u) = si , sj u = Cδij

(3.1)
(3.2)

We note that for the coupled system above solutions are balanced metrics solving the mean
field equation (3.1).
Equation (3.1) is equivalent to the density of states condition

1 = in

2

i si

2
∧ si e−(φ+u)
= in
n
ωu

i

ωn
|si |2 e−u 0
ω,h
n
ωu

(3.3)

very much in the spirit of Donaldson’s double quotient in [Do]. However the hermitian metric
is defined on KX ⊗ L and is a coupling of a hermitian metric on KX and one on L with

124

ωu ∈ c1 (L). The same effect is obtained by choosing a hermitian metric only from L:

n2

1=i

i

ωn
2
|si |2 e−u 0 = in
ω,h
n
ωu

|si |2
ω,h

ωn
−u+log 0
n
ωu
e

i
ωn

So the hermitian metric on L giving (3.3) is determined by the weight φ + u − log ω0 .
n
u

The orthogonality condition (3.2) on a basis of H 0 (X, KX ⊗ L) with respect to ·, · φ+u
can now be written as:

Cδij = in

2
X

si ∧ sj e−(φ+u) =

ωn
−(φ+u+log u ) ω n
n u
2
ω0
si ∧ sj e
in
n
ω0
X

(3.4)

Note that for a solution of the mean field equation that (3.4) can also be written as

Cδij =

si ∧ sj e−φ
X

−φ
k sk ∧ sk e

n
ωu

(3.5)

Definition 3. Given an embedding into CPN , for some N > 0, induced by an L2 orthonormal basis (si ) of H 0 (X, KX ⊗ Lk ) (with respect to ·, · kφ+ku ) we say that embedding is
balanced if (3.5) holds.
Fix notation
Bk(φ+u) e−k(φ+u) :=

sj ∧ sj e−k(φ+u)

then from [Bo09] the following asymptotics are available

a
n
Bk(φ+u) e−k(φ+u) = k n (a0 + 1 S + O(k −2 ))ωu
k
n
= k n ωu (1 + O(k −1 ))

125

Applying this to right hand side of (3.5) obtain
si ∧ sj e−φ

X

si ∧ sj e−kφ
n
n
k n ωu
ωku = C
n
k n ωu (1 + O(k −1 ))
sγ ∧ sγ e−kφ e−ku
X
γ
=C
X

si ∧ sj e−k(φ+u) (1 + O(k −1 ))

(3.6)

where C is the constant appearing in (3.4).
From [BBEGZ] the mabuchi energy is introduced in a more general setting by defining
it as
M abµs (φ) :=

log(
X

n
ωφ

µs

)ω n + J(φ) − I(φ)

(3.7)

On Hω this restricts to the usual mabuchi functional νω (φ). Also if M abµ is proper then the
n
corresponding mean field equation ωu = µs e−u can be solved, see [BBEGZ]. Next we show

properness is independent of the choice of s.
Proposition 8. If M abµs is proper on Hω then so is M abµ .
s

Proof. Recall that the mabuchi functional satisfies the cocyle property [Tian00]

νω (φ) − ν (φ ) = νω (ψ)
ω

where ω = ω + ∂∂ψ and φ = φ − ψ. This follows from computing the differential of the left
hand side. Since it vanishes we obtain the left hand side above is constant. Setting φ = ψ
determines the constant.
In the smooth case using the cocyle condition and properness as defined by Tian [Tian00]

126

we get

ν

≥ µ(Jω (φ)) − C

ω

Since the function µ is increasing, it suffices to show

J (φ ) ≤ αJω (φ) + β
ω

where α, β > 0 are constants. We have

µ (·) = µ(

·−β
)
α

is increasing and then

ν

ω

≥ µ (J (φ )) − C
ω

Begin by recalling

Jω (φ) =

1
V

n−1

i+1
n−i−1
∂φ ∧ ∂φ ∧ ω i ∧ ωφ
n+1 X
i=0

(3.8)

where V = X ω n . Throughout we assume that φ ∈ Hω . This gives, through a simultaneous
diagonalization argument that the integrands are non-negative so Jω (φ) ≥ 0.
Also recall that

n+1
Jω (φ) ≤ Iω (φ) ≤ (n + 1)Jω (φ)
n2

127

(3.9)

where

Iω (φ) =

1
n
φ(ω n − ωφ )
V X

(3.10)

Lemma 21. With the notations above the following identity holds:

Iω (φ) =

1
n
φ(ω n − (ω )n ) +
φ((ω )n − ωφ )
V X
X
∂φ ∧ ∂ψ ∧ (ω n−1 + ω n−2 ∧ ω . . . + (ω )n ) +

=

X

X

+
X

n
φ ((ω )n − ωφ )

n
ψ((ω )n − ωφ )

≥I (φ ) +
ω

X

n
ψ((ω )n − ωφ ) + O(1)

(3.11)

To obtain inequality (3.11) first
Claim 17. X ∂φ ∧ ∂ψ ∧ (ω n−1 + ω n−2 ∧ ω . . . + (ω )n−1 ) > −2CnV
Rewrite the expression as

∂φ ∧ ∂ψ ∧ (ω n−1 + ω n−2 ∧ ω . . . + (ω )n−1 )
X

ψ(∂∂φ) ∧ (ω n−1 + ω n−2 ∧ ω . . . + (ω )n−1 )

=−
X

=−
X

ψωφ ∧ (ω n−1 + ω n−2 ∧ ω . . . + (ω )n−1 )
ψω ∧ (ω n−1 + ω n−2 ∧ ω . . . + (ω )n−1 )

+
X

Note that |ψ| < C. Then

128

(3.12)

Claim 18. On Hω with notations as above

ωφ ∧ ((ω n−1 + ω n−2 ∧ ω . . . + (ω )n−1 )) ≥ 0

At an arbitrary point p ∈ X

ωp = dzi ∧ dzi ,

ωp = (1 + λi )dzi ∧ dzi

where |λi | < 1

set (1 + λ)J = Πk∈J (1 + λ)k . Then

ωφ ∧ ω n−1−i ∧ (ω )i =ωφ ∧
p

(n − 1 − i)! i! (1 + λ)J dz I ∧ dz J
|I|+|J|=n,I∩J=Ø

|I|! |J|! (1 + λ)J

(1 + φkk )

=
k

I⊂{1...n}\{k}

ωn
n!

Since each term in the inner sum is nonnegative there is an > 0 satisfying

|I|! |J|! (1 + λ)I >
I⊂{1...n}\{k}

n

Therefore

ωφ ∧ ω n−1−i ∧ (ω )i >
p

(1 + φkk )
k

and the claim follows.

129

ωn
ωn
= trω ωφ
>0
n!
n!

So the first term in the last equality in (3.12) becomes

−
X

ψωφ ∧ (ω n−1 + ω n−2 ∧ ω . . . + (ω )n−1 )

≥−C
X

ωφ ∧ (ω n−1 + ω n−2 ∧ ω . . . + (ω )n−1 )
ω ∧ (ω n−1 + ω n−2 ∧ ω . . . + (ω )n−1 )

=−C
X

∂∂φ ∧ (ω n−1 + ω n−2 ∧ ω . . . + (ω )n−1 )

−C
X

ω ∧ (ω n−1 + ω n−2 ∧ ω . . . + (ω )n−1 )

=−C
X

This a consequence of

∂∂φ ∧ (ω n−1 + ω n−2 ∧ ω . . . + (ω )n−1 ) = 0
X

obtained using integration by parts and stokes theorem. The claim follows from

∂φ ∧ ∂ψ ∧ (ω n−1 + ω n−2 ∧ ω . . . + (ω )n−1 )
X

ω ∧ (ω n−1 + ω n−2 ∧ ω . . . + (ω )n−1 )

> − 2C
X

> − 2CnV

Next expand binomially

ω n−i ∧ (ω )i = ω n−i ∧ (ω i + cn,1 ω n−1 ∧ ∂∂φ + . . . (∂∂φ)i )

130

Integrating and combining with an application of by parts and stokes theorem obtain that

ω n−i ∧ (ω )i =

ωn = V

X

X

Now we can conclude

Iω (φ) > I (φ ) +
ω

X

n
ψ((ω )n − ωφ ) − 2CnV

Since ω , ωφ represent the same cohomology class

X

n
c((ω )n − ωφ ) = 0

So we may choose c > 0 so that ψ + c < −1. Then we have (suppressing the −2CnV term
since it is O(1))

Iω (φ) ≥I (φ ) +
ω

X

(ψ + c)(ω )n −

=I (φ ) +
ω

n
(ψ + c)((ω )n − ωφ )

X

X

n
(ψ + c)ωφ

Because

−
X

n
(ψ + c)ωφ > V > 0,

(ψ + c)(ω )n = O(1)
X

we finally obtain
Iω (φ) ≥ I (φ ) + C
ω

131

for some constant C. From (3.9) it follows

Jω (φ) ≥

1
C
1
1
C
Iω (φ) ≥
I (φ) +
≥ 2 J (φ) +
n+1
n+1 ω
n+1
n+1
n ω
2

So α = n2 and β = − Cn and we obtain properness of ν
n+1

ω

on Hω given that νω is.

We have seen from (3.6) that metrics in c1 (Lk ) solving the coupled system are approximately balanced for sufficiently large k. In view of the proposition we finally conclude by
posing the following question:
Question 1. Provided M abµs is proper, do solutions ωu ∈ c1 (Lk ) to the coupled system for
sufficiently large k exist?

132

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