STABILITY OF THE ALMOST HERMITIAN CURVATURE FLOW
By
Daniel J Smith

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

ABSTRACT
STABILITY OF THE ALMOST HERMITIAN CURVATURE FLOW
By
Daniel J Smith
The Almost Hermitian Curvature flow was introduced in [7] by Streets and Tian in order
to study almost hermitian structures, with a particular interest in symplectic structures.
This flow is given by a diffusion-reaction equation. Hence it is natural to ask the following:
which almost hermitian structures are dynamically stable? An almost hermitian structure
(ω, J) is dynamically stable if it is a fixed point of the flow and there exists a neighborhood
N of (ω, J) such that for any almost hermitian structure (ω(0), J(0)) ∈ N the solution of the
Almost Hermitian Curvature flow starting at (ω(0), J(0)) exists for all time and converges
to a fixed point of the flow. We prove that on a closed K¨hler-Einstein manifold (M, ω, J)
a
such that either c1 (J) < 0 or (M, ω, J) is a Calabi-Yau manifold, then the K¨hler-Einstein
a
structure (ω, J) is dynamically stable.

ACKNOWLEDGMENTS

The author would like to thank his advisor Jon Wolfson for all of his help and encouragement
on this project. The author would also like to thank his colleagues Andrew Cooper, Cheryl
Balm, Luke Williams, Chris Hays, Jeff Chapin, and Tom Jaeger for great conversations,
mathematical and otherwise, during graduate school.

iii

TABLE OF CONTENTS

Chapter 1 Preliminaries . . . . . . . . . . . . . . .
1.1 Introduction . . . . . . . . . . . . . . . . . . . .
1.2 Linear Stability and Parabolic Estimates . . . .
1.2.1 L2 bounds of ψ in terms of . . . . . . .
1.2.2 L1,2 bounds of ψ in terms of . . . . . .
1.2.3 L2,2 bounds of ψ in terms of . . . . . .
1.2.4 Lm+1,2 bounds on ψ given Lm,2 bounds

.
.
.
.
.
.
.

1
1
4
16
17
18
20

Chapter 2 Dynamic Stability when c1 (J) < 0 . . . . . . . . . . . . . . . . . . .

24

Chapter 3 Dynamic Stability in the Calabi-Yau Case . .
3.1 The Kernel of L and the Space of Calabi-Yau Structures
3.2 A New Calabi-Yau Structure . . . . . . . . . . . . . . . .
3.3 Finding the Limit Calabi-Yau Structure . . . . . . . . .

.
.
.
.

32
32
36
48

. . . . . . . . . . . . . . . . .

55

BIBLIOGRAPHY

. . . . . . . . . . . . .

iv

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.

.
.
.
.
.
.
.

.
.
.
.

.
.
.
.
.
.
.

.
.
.
.

.
.
.
.
.
.
.

.
.
.
.

.
.
.
.
.
.
.

.
.
.
.

.
.
.
.
.
.
.

.
.
.
.

.
.
.
.
.
.
.

.
.
.
.

.
.
.
.
.
.
.

.
.
.
.

.
.
.
.
.
.
.

.
.
.
.

.
.
.
.
.
.
.

.
.
.
.

Chapter 1
Preliminaries

1.1

Introduction

Let (M, J, g) be a closed almost complex manifold such that J is compatible with the Riemannian metric g, that is for any vector fields X and Y we have g(X, Y ) = g(J(X), J(Y )).
To the metric g we associate the 2-form ω defined by ω(X, Y ) = g(J(X), Y ). We call such
a pair (ω, J) an almost hermitian structure.
The Ricci flow has proven to be a successful tool in studying the Riemannian geometry
of manifolds. Therefore it is natural to attempt to use a parabolic flow to understand the
almost hermitian geometry of almost complex manifolds. However, the Ricci flow does not,
in general, preserve the set of almost hermitian structures. In [7], Streets and Tian introduce
the Almost Hermitian Curvature flow (AHCF), which is a weakly-parabolic flow on the space
of almost hermitian structures.
AHCF generalizes K¨hler Ricci flow in the sense that if the initial structure (ω0 , J0 ) is
a
K¨hler, then the evolution of (ω(t), J(t)) by AHCF coincides with K¨hler Ricci flow. In [8],
a
a
Streets and Tian construct a parabolic flow on the space of hermitian structures (ω, J) (here
J is integrable), called Hermitian Curvature flow (HCF). AHCF also generalizes HCF.
As we will see below AHCF is, in fact, a family of geometric flows. Streets and Tian have
a particular interest in one of these flows, called Symplectic Curvature flow (SCF). Given

1

an almost hermitian structure (ω0 , J0 ) such that dω0 = 0, under SCF ω(t) is a closed form
as long as the flow exists. Therefore, SCF is a tool which can be used to study symplectic
structures. Hence we see that AHCF is a very general family of geometric flows.
The Almost Hermitian Curvature flow is a coupled flow of metrics and almost complex
structures. It is written
∂
ω = −2S + H + Q
∂t
∂
J = −K + H.
∂t

(1.1)

S is a “Ricci-type” curvature. In particular, Sij = ω kl Ωklij and Ω is the curvature of the
almost-Chern connection

. That is,

is the unique connection satisfying

and T 1,1 = 0. T 1,1 is the (1, 1) component of the torsion of
quadratic in the torsion of

ω = 0,

J =0

. Q is any (1, 1) form that is

i
i
. Kj = ω kl k Nlj where N is the Nijenhuis tensor with respect

to J. H is any endomorphism of T M that is quadratic in N and skew commutes with J.
The term H(X, Y )= 2 ω((−K + H)(X), J(Y )) + ω(J(X), (−K + H)(Y )) is required in order
˙1
to maintain the compatibility of ωt with Jt . Streets and Tian prove short-time existence and
uniqueness (see Theorem 1.1 in [7]) of the flow starting at an almost hermitian structure
(ω(0), J(0)). Notice that the generality with which the tensors Q and H are defined implies
that (1.1) is in fact a family of geometric flows. This family of geometric flows includes
Hermitian Curvature flow, Symplectic Curvature flow and K¨hler Ricci flow. Associated
a
to AHCF is the volume-normalized version of the flow (VNAHCF), the volume-normalized
version is the one with which we will work.
One natural question to ask is: does M admit a K¨hler-Einstein structure? If so, is it
a
detected by VNAHCF? The main result of the paper is the following:

2

Theorem 1. Let (M 2n , ω, J) be a closed complex manifold with (ω, J) a K¨hler-Einstein
a
structure such that either c1 (J) < 0 or (M, ω, J) is a Calabi-Yau manifold. Then there
exists > 0 such that if (ω(0), J(0)) is an almost hermitian structure with (ω(0) − ω, J(0) −
J) C ∞ < , then the solution to the volume normalized AHCF starting at (ω(0), J(0)) exists
for all time and converges exponentially to a K¨hler-Einstein structure (ωKE , JKE ).
a
Remark 2. Theorem 1 gives evidence that the Almost Hermitian Curvature flow reflects
the underlying almost hermitian geometry of M .
Remark 3. In this paper we define a Calabi-Yau manifold (M, ω, J) to be a compact K¨hler
a
manifold with trivial canonical bundle such that ω is a K¨hler-Einstein metric with Ric(ω) =
a
0.
Remark 4. In the case when c1 (J) < 0, the K¨hler-Einstein structure that the flow starts
a
close to is the same one that the flow converges to, in other words (ω, J) = (ωKE , JKE ).
This is proved in Theorem 13.
In the Calabi-Yau case we cannot guarantee that (ω, J) and (ωKE , JKE ) are the same
Calabi-Yau structure.
The notion of stability in Theorem 1 is often referred to as dynamic stability. Dynamic
stability has also been studied in the case of the Hermitian Curvature flow by Streets and
Tian ([8]) and for the Ricci flow by Sesum ([6]) and by Guenther, Isenberg, and Knopf ([4]).
The first step in proving Theorem 1 is to show that K¨hler-Einstein structures behave
a
like sinks of the linear flow associated to VNAHCF, this is done in Section 1.2. Also in
Section 1.2, we derive parabolic estimates for the VNAHCF (see Theorem 12).
Next, in Chapter we prove Theorem 1 in the case when c1 (J) < 0. Finally, in the
last chapter we complete the proof of Theorem 1 by showing how to find a K¨hler-Einstein
a
3

structure (ωKE , JKE ) to which the flow exponentially converges in the Calabi-Yau case.

1.2

Linear Stability and Parabolic Estimates

To prove theorem 1 we first show that, on a linear level, any K¨hler-Einstein structure (ω, J)
a
is a “degenerate sink” with respect to the Almost Hermitian Curvature flow, meaning that the
linear operator associated to the non-linear flow is negative semi-definite at K¨hler-Einstein
a
structures. For notation sake write VNAHCF:
∂
ω=F
∂t
∂
J = G.
∂t

(1.2)

˙ ˙
To the operator (F, G), we have the associated linear operator (F, G). In particular, we
consider a one-parameter family of compatible, unit volume, almost hermitian structures
. ∂
˙ ˙ . ∂
(ω(a), J(a)) and (F, G) = ∂a a=0 (F, G)(ω(a), J(a)). Similarly we write (ω, J) = ∂a a=0 (ω(a), J(a)).
˙ ˙
Definition 5. An almost-hermitian structure (ω, J) is called static provided (F(ω, J), G(ω, J)) =
. ˙ ˙
0. Moreover, a static structure (ω, J) is linearly stable if the linearization L = (F, G)(ω,J) is
negative semi-definite, that is L·, · L2 (g) ≤ 0.

Notice that K¨hler-Einstein structures are static under VNAHCF. Next, we prove
a
Theorem 6. Let (M, J) be a closed complex manifold with c1 (J) ≤ 0, then any K¨hlera
Einstein structure (ω, J) on M is linearly stable.
Proof. Employing the DeTurck trick as in Proposition 5.4 and 5.5 of [7] the weak-ellipticity
of (F, G) follows. Furthermore, computing the linearization of (F, G) at a K¨hler-Einstein
a
4

structure, in complex coordinates with respect to J, the linearization is written

˙
˙
˙
Fij = −2 ∗ ωij + 2ω kl Rklij

(1.3)

˙
Fij = −2 ∗ ωij
˙

(1.4)

i
i
˙i
Gj = −2 ∗ J˙j + 2J˙pq Rjpq .

Here

∗

is the L2 (g) adjoint of

(1.5)

and R denotes the Riemannian curvature of g.

To show that (ω, J) is linearly stable we have to deal with the fact that in (1.3) and (1.5)
the lower order terms do not have a sign. To see that the linearized operator is negative
semi-definite at K¨hler-Einstein structures we use a couple of Weitzenb¨ck-Bochner formulas
a
o
(cf. [2]).
Lemma 7. Let α and β be a (0,2) and (1,1) form respectively. If g is a K¨hler-Einstein
a
metric, then we have

s
(∆d α)ij = 2 ∗ αij + 2 αij
n

(1.6)

s
(∆d β)ij = 2 ∗ βij − 2β kl Rklij + 2 βij .
n

(1.7)

Where ∆d is the Hodge Laplacian with respect to g and s is the scalar curvature of g. In
addition we will use another Weitzenb¨ck-Bochner formula.
o
Lemma 8. Given a T 1,0 (M, ω, J) valued (0, 1)-form, φ and K¨hler-Einstein metric g we
a
have:

i
(∆∂ φ)j =

∗

i
φi − φpq Rjpq +
j

5

s i
φ .
n j

(1.8)

∗

∗

Where ∆∂ represents the complex laplacian ∂∂ g + ∂ ∂ g .
Therefore using equations (1.3), (1.4), (1.6) and (1.7), we have that

s
˙
˙
F = −∆d ω + 2 ω.
˙
n

(1.9)

Similarly, using equations (1.5) and (1.8), we have

s ˙
˙
G = −∆∂ J˙ + 2 J.
n

(1.10)

Combining (1.9) and (1.10) we see that

s
s
˙
L(ω, J) = −∆d ω + 2 ω, −∆∂ J˙ + 2 J˙ .
˙ ˙
˙
n
n

(1.11)

Finally since c1 (J) ≤ 0 implies that s ≤ 0; by integrating the theorem follows.
Notice that if c1 (J) < 0, then the scalar curvature of g is negative; that is s < 0. Hence
from (1.11) it follows that if c1 (J) < 0, then L is strictly negative definite with respect
to L2 (g). Let λ = min{|λi | : λi is an eigenvalue of L}. Further let C denote the space of
almost hermitian structures modulo diffeomorphism. Therefore we have proved the following
corollary.
Corollary 9. Let (M, ω, J) be a closed complex manifold such that (ω, J) is a K¨hlera
Einstein structure and moreover c1 (J) < 0. Let ψ ∈ T(ω,J) C, then

L(ω,J) ψ, ψ L2 (g) ≤ −λ|ψ|2 2 .
L (g)

6

Corollary 9 will be crucial to proving Theorem 1 in the case when c1 (J) < 0 (see Chapter
).
Fix a K¨hler-Einstein structure (ω, J) and let (ω(t), J(t)) be a solution of the coupled
a
system (1.2) starting at an initial almost hermitian structure (ω(0), J(0)). We will quantify
the amount by which the solution deviates from (ω, J) using

ρ(t) = (ω(t) − ω, J(t) − J).

Notice that ρ(t) ∈ Λ2 (M ) × End(T M ). Throughout the paper we use the operator norm on
End(T M ).
As noted in the proof of Theorem 1.1 in [7], C is a non-linear manifold. In the following
lemma we will see that ρ(t) ∈ T(ω,J) C, however ρ(t) can be estimated by an element of the
/
tangent space T(ω,J) C.
Lemma 10. Fix t and let (ω(t), J(t)) be an almost hermitian structure. Write ω(t) = ω+h(t)
and J(t) = J + K(t), in other words ρ(t) = (h(t), K(t)). If |ρ(t)|C 0 < 1, then there exists
ψ(t) ∈ T(ω,J) C so that

|ψ(t)|C k ≤ |ρ(t)|C k ,

(1.12)

|ρ(t)|L2 ≤ |ψ(t)|L2 + C1 |ψ(t)|2 2
L

7

(1.13)

and

|ρ(t)|C k ≤ |ψ(t)|C k + C2 |ψ(t)|2 k
C

(1.14)

where C1 and C2 depend on the L2 and C k norms of ρ(t) respectively.
Proof. We will begin by studying the tangent space T(ω,J) C. Let (ωs , Js ) denote a path of
.
∂
almost hermitian structures such that (ωs , Js )|s=0 = (ω, J) and let ∂s s=0 (ω(s), J(s)) =
(ω, J). Given vector fields X and Y , the compatibility condition is written:
˙ ˙

ωs (X, Y ) = ωs (Js (X), Js (Y ))

and the almost complex condition is written:

2
Js (X) = −X.

Hence the linearized compatibility and almost complex conditions are given by:

˙
˙
ω(X, Y ) = ω(J(X), J(Y )) + ω(J(X), J(Y )) + ω(J(X), J(Y ))
˙
˙
˙
0 = J˙ ◦ J(X) + J ◦ J(X).

(1.15)
(1.16)

From (1.16) we see that the tangent space to the space of almost complex structures is given
by endomorphisms that skew-commute with J. Equivalently, J˙ can be viewed as a section
of Λ0,1 ⊗ T 1,0 ⊕ Λ1,0 ⊗ T 0,1 . Here we use J to decompose T M = T 1,0 M ⊕ T 0,1 M .
First we will prove that the endomorphism K(t) can be estimated by an element of the
tangent space to the space of almost complex structures at J. For the sake of notation we
8

will often write K(t) = K.
1,0

0,1

1,0

0,1

1,0

Using J we decompose K = K0,1 + K0,1 + K1,0 + K1,0 where K0,1 : T 0,1 → T 0,1 ,
equivalently

1,0

K0,1 ∈ Λ0,1 ⊗ T 1,0 .

Take ψ(t) ∈ T(ω,J) C and write ψ(t) = (ψ1 (t), ψ2 (t)) ∈ Λ2 (M ) × End(T M ). We define
. 1,0
0,1
ψ2 (t) = K0,1 + K1,0 .

(1.17)

That is, ψ2 (t) is defined to be the projection of K onto Λ0,1 ⊗ T 1,0 ⊕ Λ1,0 ⊗ T 0,1 . Next
0,1

1,0

0,1

we will show that K0,1 and K1,0 are quadratic in ψ2 (t). We will only prove this for K0,1
1,0

since the same argument applies to K1,0 .
Using that J(t) is an almost complex structure we see that K(t) satisfies:

0 = K ◦ J(X) + J ◦ K(X) + K 2 (X).

0,1

(1.18)

1,0

Now for K acting on T 0,1 we will write K = K0,1 + K0,1 . Therefore using (1.18), on T 0,1
we have
√
0,1
0,1
0,1
1,0
0,1
1,0
1,0
0,1
1,0
0 = −2 −1K0,1 + K0,1 ◦ K0,1 + K0,1 ◦ K0,1 + K1,0 ◦ K0,1 + K1,0 ◦ K0,1

and so by type consideration,
√
0,1
K0,1

=−

−1
0,1
0,1
0,1
1,0
K0,1 ◦ K0,1 + K1,0 ◦ K0,1 .
2

9

(1.19)

0,1

1,0

0,1

Notice that on T 0,1 , K1,0 ◦ K0,1 = ψ2 (t)2 . Hence we are able to write K0,1 in terms of
0,1 2

K0,1

and a term that is quadratic in ψ2 (t).
0,1 2

Next, consider the first term on the right-hand side of (1.19), K0,1
0,1 2

to show that K0,1

0,1 4

can be expressed as K0,1
0,1 2

Plugging (1.19) into each factor of K0,1
0,1 2

K0,1

=−

1
4

0,1 4

K0,1

. We will use (1.19)

plus terms which are quadratic in ψ2 (t).

, we see that

0,1 2

+ K0,1

0,1

0,1

4
2
2
◦ ψ2 + ψ2 ◦ K0,1 + ψ2 ,

0,1

which can be substituted into the term K0,1 ◦ K0,1 in (1.19). Iterating this process by
0,1

0,1

successively plugging (1.19) into the highest power term in K0,1 , we see that K0,1 can be
0,1

expressed as a series. Notice that since |ρ|C 0 < 1 it follows that K0,1 C 0 < 1, and so this
series converges. Therefore

0,1

K0,1 = ψ2 (t)2 + [higher-power terms in ψ2 ◦ higher-power terms in K].

(1.20)

Next we will show that the two form h(t) can be estimated by an element of the tangent
space to the space of compatible metrics. Notice that for vector fields X and Y ,

ω(X, Y ) − ω(J(X), J(Y )) = 2ω (2,0)+(0,2) (X, Y ),
˙
˙
˙

and so by (1.15)

˙
˙
2ω (2,0)+(0,2) (X, Y ) = ω(J(X), J(Y )) + ω(J(X), J(Y )).
˙

Using the compatibility of ω(t) and J(t), we see that h(t) = ω(t) − ω and K(t) = J(t) − J

10

satisfy:

h(X, Y ) = h(J(X), J(Y )) + ω(K(X), J(Y )) + ω(J(X), K(Y ))
+ ω(K(X), K(Y )) + h(K(X), J(Y )) + h(J(X), K(Y ))

(1.21)

+ h(K(X), K(Y )).

We define ψ1 (t) as follows

(1,1)

ψ1
(2,0)+(0,2)

ψ1

.
= h(1,1)

(1.22)

.
(X, Y ) = ω(K(X), J(Y )) + ω(J(X), K(Y ))
= ω(ψ2 (X), J(Y )) + ω(J(X), ψ2 (Y )).

(1.23)
(1.24)

The last equality follows from the definition of ψ2 and the fact that ω is of type (1, 1).
(2,0)+(0,2)

Next we will show that h(2,0)+(0,2) − ψ1

can be expressed as terms that are

quadratic in ψ(t). Combining (1.21), (1.23) and (1.24) we have

(2,0)+(0,2)

2 h(2,0)+(0,2) (X, Y ) − ψ1

(X, Y ) = h(K(X), J(Y )) + h(J(X), K(Y ))
(1.25)
ω(K(X), K(Y )) + h(K(X), K(Y )).

As we proved above in (1.17) and (1.20), K can be written in terms of ψ2 and hence
the terms in the second line of (1.25) are higher-power in ψ2 . Next we consider the term
h(K(X), J(Y )). Since the left-hand side of (1.25) is a section of Λ(2,0)+(0,2) , let X, Y ∈
T 0,1 M . So for X, Y ∈ T 0,1 M we can write the components of h(K(X), J(Y )) as

0,1

1,0

K0,1 h(0,2) + K0,1 h(1,1) .

11

(1.26)

From (1.17) and (1.22) we see that the second term in (1.26) is quadratic in ψ. By
(1.20) the first term is quadratic in ψ plus higher-power terms in ψ composed with higherpower terms in ρ. Notice that the same argument can be applied to h(J(X), K(Y )). Abusing
notation we let ψ ∗ψ denote terms which are quadratic in ψ plus terms that are higher-power
in ψ composed with terms that are higher-power in ρ. Therefore we have

(2,0)+(0,2)

h(2,0)+(0,2) (X, Y ) − ψ1

(X, Y ) = ψ ∗ ψ.

(1.27)

Notice that by the definition of ψ(t), given in (1.17) (1.22) and (1.23), the inequality

|ψ(t)|C k ≤ |ρ(t)|C k

follows immediately. Again using the definition of ψ(t) along with (1.20) and (1.27) we see
that

|ρ(t)|C k ≤ |ψ(t)|C k + C|ψ(t)|2 k
C

where C depends on the C k norm of ρ(t). Notice that (1.13) follows analogously.
In Theorem 6 we proved that the linearization of (F, G), denoted L, is negative semidefinite on T(ω,J) C. The goal is to use the sign on L to prove exponential decay of ρ(t).
However as we observed in the previous lemma, ρ(t) ∈ T(ω,J) C. To deal with this we will
/
prove exponential decay of ψ(t) ∈ T(ω,J) C which, by (1.14), will prove exponential decay of
ρ(t).
Next we show that ψ(t) evolves by a parabolic flow equation and moreover that we have
estimates on the non-linear part of the flow.
12

Lemma 11. Let L be the differential operator defined by (1.11). Then for ψ(t) ∈ T(ω,J) C
defined by (1.17) (1.22) and (1.24) we have
∂
1. ∂t ψ(t) = L(ψ(t)) + A((ω, J), ψ(t))

2. |A((ω, J), ψ(t))|C k ≤ C(|ψ(t)|C k | 2 ψ(t)|C k + | ψ(t)|2 k )
C
where C depends on the C k norm of ρ(t).
∂
Proof. To prove ∂t ψ(t) = L(ψ(t)) + A((ω, J), ψ(t)) we first study the evolution of ρ(t).

Notice that since (ω, J) is independent of t,

∂
∂
ρ(t) = (ω(t), J(t)) = F(ω(t), J(t)), G(ω(t), J(t)) .
∂t
∂t

(1.28)

Furthermore since (ω, J) is a static structure, when we linearize (F, G) at (ω, J) in the
direction ψ(t), we have

(F, G) = L(ψ(t)) + A((ω, J), ρ(t)).

(1.29)

Hence from (1.28) and (1.29) it follows that

∂
ρ(t) = L(ψ(t)) + A((ω, J), ρ(t)),
∂t

(1.30)

where A represents the error in approximating (F, G) by the linearization L. As in [8] and
[6] we have the following C k bounds on A:

|A|C k ≤ C(|ρ|C k | 2 ρ|C k + | ρ|2 k ).
C

13

Therefore by Lemma 10 we have the following bounds on A:

|A|C k ≤ C(|ψ|C k | 2 ψ|C k + | ψ|2 k ).
C

(1.31)

Next we will use the definition of ψ(t) and the evolution of ρ(t) to derive an evolution
equation for ψ(t). By the definition of ψ2 (t) and the (1, 1) part of ψ1 (t) (see (1.17) and
(1.22) respectively) we have

∂
∂
1,0
0,1
ψ2 (t) =
K0,1 + K1,0
∂t
∂t
∂
∂ (1,1)
ψ1 (t) = h(1,1) (t).
∂t
∂t

(1.32)
(1.33)

∂ (2,0)+(0,2) (t), it follows from (1.25) that
For ∂t ψ1

∂
∂ (2,0)+(0,2)
ψ1
(t) = h(2,0)+(0,2) (t) +
∂t
∂t

∂
ρ ∗ ρ,
∂t

(1.34)

since the (2, 0) + (0, 2) components of ψ1 (t) and h(t) differ by terms that are quadratic in
∂
ρ(t) = (h(t), K(t)). Notice that by (1.30) and (1.31) we have ∂t ρ is second order in ψ and

hence the final term in (1.34) may be absorbed in the error estimate A. Therefore from
(1.32), (1.33), (1.34) and (1.30) it follows that

∂
ψ(t) = L(ψ(t)) + A((ω, J), ψ(t)),
∂t
where A is a different tensor than in (1.30), but we still have |A|C k ≤ C(|ψ|C k | 2 ψ|C k +
| ψ|2 k ).
C
Roughly speaking, the following theorem says that given any finite time T > 0, by
14

starting the flow very close to (ω, J), the solution (ω(t), J(t)) remains close to (ω, J) on the
interval [0, T ).
Theorem 12. Given T > 0,
such that if |ρ(0)|C ∞ <

> 0 and an integer k ≥ 0, there exists

= (T, , k) > 0

then, (ω(t), J(t)) exists on [0, T ) and moreover |ρ(t)|C k <

on

[0, T ).
Proof. First, for sufficiently small, work of Streets and Tian (see Theorem 1.1 in [7]) shows
that there exists T > 0 such that the solution (ω(t), J(t)) exists on [0, T ) and moreover
|ρ(t)|C k <

on [0, T ). Suppose by way of contradiction that there exists a maximal T so

that for all

> 0 the solution exists and |ρ(t)|C k <

on [0, T ) with T < T . Fix T < T .

To derive a contradiction we will produce bounds on the C k norm of ρ(t) on [0, T ] in terms
of , independent of T .
Recall from Lemma 10 that associated to ρ(t) we have ψ(t) ∈ T(ω,J) C. In order to obtain
C k estimates on ρ(t) in terms of

we will produce C k bounds on ψ(t) and employ (1.14).

To this end, we study the evolution of ψ(t). Recall from Lemma 11 part (1) that

∂
ψ(t) = L(ψ(t)) + A((ω, J), ψ(t))
∂t

(1.35)

where L is negative semi-definite and A represents the error in approximating (F, G) by L.
From part (2) of Lemma 11 we have

|A|C k ≤ C(|ψ|C k | 2 ψ|C k + | ψ|2 k ).
C

Notice that C depends on the C k norm of ρ(t) which we are assuming is bounded by
t ∈ [0, T ) ⊃ [0, T ].
15

(1.36)

for

In the estimates that follow Rm will denote the curvature of the fixed metric g and
will denote the Levi-Civita connection of g. Moreover we will use the fact that M is compact
and hence there exists a constant C such that | Rm |C ∞ < C.

1.2.1

L2 bounds of ψ in terms of

The linear stability of K¨hler-Einstein structures will allow us to produce L2 bounds on ψ(t)
a
in terms of

which are independent of T . Indeed, for t ∈ [0, T ] by (1.35) and using that

L(ω,J) ·, · L2 (g) ≤ 0, we have
1∂
|ψ|2 dvolg =
g
2 ∂t M
M

∂
ψ, ψ
∂t

≤

A ∗ ψ.

(1.37)

M

Now, using the bound on A given in (1.36), we see that

A∗ψ =

ψ ∗2 ∗

2ψ

+

ψ ∗2 ∗ ψ.

Using integration by parts on the second term yields

ψ ∗2 ∗

A∗ψ ≤
M

2 ψ.

(1.38)

M

For t ∈ [0, T ], by assumption and (1.12), |ψ(t)|C k <

therefore M ψ ∗2 ∗ 2 ψ ≤ C

M

|ψ|2 .

Hence combining (1.37) and (1.38) we have

1∂
|ψ|2 dvolg ≤ C1
g
2 ∂t M

16

M

|ψ|2 dvolg .
g

(1.39)

Therefore for any t ∈ [0, T ],

|ψ(t)|2 2 ≤ eC1 T |ψ0 |2 2 ≤ eC1 T .
L
L

(1.40)

L1,2 bounds of ψ in terms of

1.2.2

Given the L2 bounds above, we bootstrap to obtain higher-order bounds. Notice that linear
stability was only used to start the bootstrapping process. Using (1.3), (1.4), (1.5) and (1.35)
we have

1∂
|ψ|2 dvolg =
g
2 ∂t M
M

∂
ψ, ψ
∂t

−

=

∗

ψ + Rm ∗ψ + A, ψ .

(1.41)

M

Since M is compact there exists a constant C2 such that | Rm(g)|C ∞ < C2 . Moreover,
we can bound the term associated with A as we did in (1.38) and (1.39) to get

1∂
|ψ|2 dvolg ≤ −
g
2 ∂t M

| ψ|2 dvolg + C3

|ψ|2 dvolg .

(1.42)

Integrating from 0 to T , we see that
T
0

| ψ|2 +
M

T
1
1
|ψ(T )|2 ≤
|ψ0 |2 + C3
|ψ|2 .
2 M
2 M
0
M

Now the L2 bounds from (1.40) imply
T
0

| ψ|2 ≤ C4 T eC1 T |ψ0 |2 2 ≤ C4 T eC1 T .
L
M

17

(1.43)

1.2.3

L2,2 bounds of ψ in terms of

Next, we use the L1,2 bounds above to produce L2,2 bounds. Similar to (1.41),

1∂
| ψ|2 dvolg =
g
2 ∂t M
M

(− ∗ ψ + Rm ∗ψ + A), ψ .

(1.44)

(− ∗ ψ), ψ above. Commuting covariant derivatives and

First consider the term

using integration by parts we get

(− ∗ ψ), ψ = −
M

| 2 ψ|2 +
M

Rm ∗ ψ ∗

(Rm ∗ψ), ψ =

Next we obtain estimates on the term

ψ.

(1.45)

M

Rm ∗ψ + Rm ∗ ψ, ψ

from equation (1.44). Since | Rm |C ∞ < C2 , we can use Young’s Inequality, to show

| ψ|2 + C

(Rm ∗ψ), ψ ≤ C
M

|ψ|2 .

M

A∗

Finally, we consider the final term in (1.44),

(1.46)

M

ψ. Using the estimates on A from

(1.36), we have

A∗

ψ=

ψ ∗2 ∗

2ψ

A∗

Integration by parts on the last term yields

+

ψ∗

ψ=

ψ∗

ψ ∗2 ∗

3 ψ.

2ψ +

ψ∗

2 ψ ∗2

and

hence

A,
M

ψ ≤C
M

| ψ|2 + C7

18

| 2 ψ|2
M

(1.47)

since |ψ|C k <

for t ∈ [0, T ].

Combining (1.44), (1.45), (1.46), and (1.47) we see that

1∂
| 2 ψ|2 + C5
|ψ|2 + C6
| ψ|2 dvolg ≤ −
| ψ|2 + C7
g
2 ∂t M
M
M
M

| 2 ψ|2 .
M

(1.48)

Hence, we choose
T
0

1
small enough so that C7 < 2 . Integrating (1.48) from 0 to T we have

| 2 ψ|2 +

| ψ(T )|2 ≤

M

M

M

| ψ0 |2 + 2C5

T
0

M

|ψ|2 + 2C6

T
0

| ψ|2 .
M

(1.49)
Therefore, using the L2 estimate from (1.40) and the L1,2 estimate from (1.43) we have
T
0

| 2 ψ|2 ≤ C7 T eC1 T |ψ0 |2 1,2 ≤ C7 T eC1 T .
L
M

(1.50)

Notice that (1.49) also gives bounds on | ψ(T )|2 2 . Moreover by integrating (1.48) from
L

0 to t for t ∈ [0, T ] these bounds hold not just at T but for any t ∈ [0, T ]. Hence we also
have
sup | ψ|2 2 ≤ C7 T eC1 T .
L

[0,T ]

T
∂
∂
Now since ∂t ψ is second order in ψ, estimate (1.50) also gives 0 M | ∂t ψ|2 ≤ CT eC1 T .

Next we use induction to show that for any p we have both:
T
0

| p ψ|2 ≤ C(p)T eC1 T
M

19

and
sup | p−1 ψ|2 2 ≤ C(p)T eC1 T .
L

[0,T ]

1.2.4

Lm+1,2 bounds on ψ given Lm,2 bounds

To produce Lm+1,2 bounds on ψ given Ls,2 estimates for s = 1, 2, . . . , m we compute the
evolution of the L2 norm of

m ψ.

1∂
| m ψ|2 dvolg =
g
2 ∂t M
M

m (− ∗

m (− ∗

First we consider the term

ψ + Rm ∗ψ + A), m ψ .

(1.51)

ψ), m ψ above. Similar to, (1.45) commuting

the covariant derivatives and using integration by parts yields
m−1
m (− ∗

ψ),

mψ

=−

M

|

m+1 ψ|2

j

+

M

Rm ∗ m−j ψ ∗

m ψ.

M j=0

Furthermore, using that | Rm |C ∞ < C2 and employing Young’s Inequality on each of the
final m − 1 terms on the right-hand side we have that there exists a constant C such that:
m
m (−

∗

ψ),

mψ

≤−

M

m+1 ψ|2

|
M

Next we study the term

m (Rm ∗ψ),

mψ

| j ψ|2 2 .
L

+C
j=0

from equation (1.51). Again using that

| Rm |C ∞ < C2 , by Young’s Inequality we see that

m (Rm ∗ψ),
M

mψ

mψ

=
M

20

(1.52)

∗

mψ

+ ··· +

ψ∗ψ
M

and hence there exists a constant C such that
m
m (Rm ∗ψ),

mψ

| j ψ|2 2 .
L

≤C

M

(1.53)

j=0
m A,

Finally consider the term

mψ

from (1.51). Again we use the estimates on A

from (1.36). Here we have
m
m A,

mψ

M

m
j+2 ψ

≤

∗

m−j ψ

∗

mψ

j+1 ψ

+

M j=0

∗

m+1−j ψ

∗

m ψ.

M j=0

We will now show how to estimate the highest order terms in the right-hand side of the
m+2 ψ

above inequality. First we rewrite the right-hand side as
mψ

∗

∗

mψ

∗ψ+

m+1 ψ

∗

ψ + lower order terms. Integration by parts on the first term yields

m A,

mψ

m+1 ψ

≤

∗

m+1 ψ

∗ψ+

m+1 ψ

∗

mψ

∗

ψ + lower order terms.
(1.54)

Next we use Young’s Inequality on the second term on the right-hand side. In particular,
Young’s Inequality is written ab ≤ ηa2 + C(η)b2 where η > 0 can be taken arbitrarily small
at the expense of making C(η) large. Hence by Young’s Inequality,

m+1 ψ

∗

mψ

∗

ψ≤η

| m+1 ψ|2 + C(η)

| m ψ|2 | ψ|2 .

Therefore combining (1.54) and (1.55) and using that |ψ|C k <
M

m A,

mψ

(1.55)

for t ∈ [0, T ] we get

≤ (C8 + η) M | m+1 ψ|2 + C9 M | m ψ|2 + lower order terms. Hence

21

1
we choose η = 4 and

sufficiently small so that

1
C8 < .
4

(1.56)

And so,

m A,

mψ

≤

M

1
| m+1 ψ|2 + C9
| m ψ|2 + lower order terms.
2 M
M

We now have estimates for each term in the evolution of the L2 norm of

mψ

(1.57)

given in

(1.51). In particular combining (1.51) (1.52), (1.53), and (1.57) we have
m

1
1∂
| m ψ|2 dvolg ≤ −
| m+1 ψ|2 + C10
| j ψ|2 2 .
g
L
2 ∂t M
2 M
j=0

(1.58)

Integrating from 0 to T we get
T

|
0

m+1 ψ|2

|

+

M

M

m ψ(T )|2

≤

|

m ψ(0)|2

M

T m

+ 2C10

0

| j ψ|2 2 .
L

(1.59)

j=0

Now we can employ the Ls,2 estimates for s = 1, . . . , m to get Lm+1,2 bounds. In
particular,
T
0

M

| m+1 ψ|2 ≤ C11 T eC1 T |ψ0 |2 m,2 ≤ C11 T eC1 T .
L

Notice that (1.59) also gives bounds on | m ψ(T )|2 2 . Moreover by integrating (1.58) from
L

0 to t for any t ∈ [0, T ], the bound on | m ψ|2 2 holds not just at T but for any t ∈ [0, T ].
L

22

Hence we also have

sup | m ψ|2 2 ≤ C11 T eC1 T |ψ0 |2 m,2 ≤ C11 T eC1 T .
L
L

[0,T ]

This proves that for any p
T
0

| p ψ|2 ≤ C(p)T eC1 T |ψ0 |2 p−1,2 ≤ C(p)T eC1 T
L
M

(1.60)

and

sup | p−1 ψ|2 2 ≤ C(p)T eC1 T |ψ0 |2 p−1,2 ≤ C(p)T eC1 T .
L
L

(1.61)

[0,T ]

T
∂q
∂
Furthermore, since ∂t ψ is second order in ψ, (1.60) also implies that 0 M ∂tq

rψ 2

≤C

for any q, r > 0, where C is independent of T .
Now use the Sobolev Embedding Theorem, with respect to g, to obtain C k bounds on ψ
in terms of . And hence by (1.14) we have C k bounds on ρ in terms of . In [7], Theorem
1.9, Streets and Tian prove that if there is a finite time singularity τ of the flow, then
limt→τ sup{| Rm |C 0 , |DT |C 0 , |T |2 0 } = ∞. Here D denotes the Levi-Citia connection and
C

Rm is the curvature of D. Therefore, the fact that the estimates above are independent of
T implies that the solution exists on [0, T ]. Again, using the short-time existence result of
Streets and Tian ([7]), the solution can be extended past time T . Moreover, for sufficiently
small, we maintain the C k estimates on ρ(t) past time T . This contradicts the maximality
of T .

23

Chapter 2
Dynamic Stability when c1(J) < 0
In this chapter we prove that when c1 (J) < 0, VNAHCF converges exponentially to the
K¨hler-Einstein structure (ω, J). As above we let ρ(t) = (ω(t) − ω, J(t) − J).
a
Theorem 13. Let (M 2n , ω, J) be a closed complex manifold where (ω, J) is a K¨hlera
Einstein structure such that c1 (J) < 0. Given a positive integer k, there exists

= (k) > 0

such that if (ω(0), J(0)) is an almost hermitian structure with ρ(0) C ∞ < , then the solution to the volume-normalized AHCF starting at (ω(0), J(0)) exists for all time and converges
exponentially in C k to (ω, J).
Proof. We prove Theorem 13 using two lemmas. As in Chapter 2 we have to deal with the
non-linearity of the space of almost hermitian structures. To prove Theorem 13 we will show
that there exists

so that if ρ(0) C ∞ < , then ψ(t) exponentially decays in C k . Finally

employing (1.14) exponential C k decay of ρ(t) will follow from exponential C k decay of ψ(t).
Lemma 14. Given δ > 0 and an integer k ≥ 0, there exists 1 = 1 (δ, k) > 0 such that if
ρ(0) C ∞ < 1 then |ψ(t)|C k < δ for all t ≥ 0 and moreover |ψ(t)|2 2 ≤ Ce−λt for all t ≥ 0.
L
Proof. As in Section 1.2, let λ = min{|λi | : λi is an eigenvalue of L}. Further let ψ(t) ∈
T(ω,J) C be the element of the tangent space, from Lemma 10, associated to ρ(t). Recall that

24

from Corollary 9 that c1 (J) < 0 implies that

M

L(ω,J) ψ, ψ g ≤ −λ|ψ|2 2 .
L (g)

(2.1)

And by (1.35),

1∂
|ψ|2 dvolg =
g
2 ∂t M
M

∂
ψ, ψ
∂t

Lψ + A, ψ .

=

(2.2)

M

Then for any t for which |ψ(t)|C 2 < δ, we can employ the bound

A, ψ ≤ C1 δ|ψ|2 2
L

(2.3)

derived in (1.38) and (1.39). Combining (2.1), (2.2) and (2.3) yields

1∂
|ψ|2 dvolg ≤ −λ|ψ|2 2 + C1 δ|ψ|2 2 .
g
L
L
2 ∂t M
Here we choose δ so that

1
C1 δ < λ.
2

(2.4)

Integrating from 0 to t yields L2 exponential decay of ψ(t). In particular,

|ψ(t)|2 2 ≤ e−λt |ψ(0)|2 2
L

L

(2.5)

for any t for which |ψ(t)|C 2 < δ. Therefore to complete the lemma, we will show that
25

given k ≥ 2, there exists 1 (k, δ) > 0 such that if |ρ(0)|C ∞ < 1 , then |ψ(t)|C k < δ
for all t ∈ [0, ∞). Notice again that from (1.12) it follows that |ρ(0)|C ∞ < 1 implies
|ψ(0)|C ∞ < 1 .
By Theorem 12 we know that given T > 0, k ≥ 0 and
|ρ(0)|C ∞ <

implies that |ψ(t)|C k <

> 0, there exists > 0 such that

on [0, T ). We apply Theorem 12 with

= δ and δ

sufficiently small so that (2.4) holds. Let 2 denote a constant that is small enough so that
if |ψ(0)|C ∞ < 2 then |ψ(t)|C k < δ on [0, T ) and assume that

|ψ(0)|C ∞ < 2 .

(2.6)

Given T , let t0 < t < T , then integrating (1.42) from t0 to t we have
t

t
1
| ψ|2 2 ≤ |ψ(t0 )|2 2 + C3
|ψ(s)|2 2 .
L
L
L
2
t0
t0

(2.7)

Furthermore since (2.5) holds on [0, T ),
t
t0

|ψ(s)|2 2 ≤
L

t
t0

e−λs |ψ(0)|2 2 =
L

1 −λt
e 0 |ψ(0)|2 2 .
L
λ

(2.8)

Therefore combining (2.7) and (2.8) yields
t

1
C
| ψ|2 2 ≤ |ψ(t0 )|2 2 + 3 e−λt0 |ψ(0)|2 2 ≤
L
L
L
2
λ
t0

1 C3
+
2
λ

e−λt0 |ψ(0)|2 2 .
L

(2.9)

The last inequality is again by (2.5). The key observation here is that the L1,2 estimate in
(2.9) is independent of t.

26

Next, to obtain a similar L2,2 estimate we integrate (1.48) from t0 to t.
t
t0

| 2 ψ|2 2 + | ψ(t)|2 2 ≤ | ψ(t0 )|2 2 + C5
L
L
L

t
t0

|ψ(t)|2 2 + C6
L

t
t0

| ψ(t)|2 2 .
L

(2.10)

Bounding the last two terms of (2.10) using (2.8) and (2.9) yields
t
t0

| 2 ψ|2 2 + | ψ(t)|2 2 ≤ | ψ(t0 )|2 2 + Ce−λt0 |ψ(0)|2 2 .
L
L
L
L

Again the key observation is that the estimate above is independent of t.
Using the same inductive argument as in the proof of Theorem 12 shows that for any p,
t
t0

| p ψ|2 2 + | p−1 ψ(t)|2 2 ≤ C1 (p)|ψ(t0 )|2 p−1,2 + C2 (p)e−λt0
L
L
L

(2.11)

where C1 (p) and C2 (p) are independent of t. Notice that there exists a constant C, such that
|ψ(t0 )|2 p−1,2 ≤ C|ψ(t0 )|C p−1 . Hence by (2.11) we have | p−1 ψ(t)|2 2 ≤ C1 (p)|ψ(t0 )|C p−1 +
L
L
C2 (p)e−λt0 . Therefore applying the Sobolev Embedding Theorem we have

|ψ(t)|C k ≤ C1 (k)|ψ(t0 )|C p−1 + C2 (k)e−λt0

(2.12)

where C1 (k) and C2 (k) are independent of t.
Since (2.12) is independent of t, to prove that |ψ(t)|C k < δ for all t ∈ [0, ∞), it suffices to
show that there exists a constant 1 with 0 < 1 ≤ 2 and such that |ρ(0)|C ∞ < 1 implies
that the right-hand side of (2.12) is bounded above by δ.
First we bound the second term on the right-hand side of (2.12). Notice that, given
δ > 0 small enough so that we have (2.4), the argument above which led to inequality (2.12)

27

holds under the assumption (2.6). Furthermore, notice that the estimate in (2.12) holds for
t0 < T , independent of T . Therefore we take T to be sufficiently large so that T > t0 and
1
C2 (k)e−λt0 < δ.
2

To bound the first term in (2.12) we again use Theorem 12 with

(2.13)

= 2C 1(k) δ and T > t0 .
1

Hence, by Theorem 12 there exists 3 > 0 such that |ρ(0)|C ∞ < 3 implies that

|ψ|C p−1 <

=

1
δ
2C1 (k)

(2.14)

for t ∈ [0, T ) ⊃ [0, t0 ].
Finally, choose 1 = min{ 2 , 3 }. Hence combining (2.12), (2.13) and (2.14) proves that
if |ψ(0)|C ∞ < 1 , then (2.12) holds independent of t. Therefore it follows that |ψ(t)|C k < δ
for all t ≥ 0 and moreover the L2 decay estimate in (2.5) holds for all t ≥ 0.
To finish the proof of Theorem 13, we show that the L2 decay estimate above and
parabolic theory can be used to prove C k decay of ψ(t).
Lemma 15. Given an integer k ≥ 2, there exists δ = δ(k) > 0 such that if both |ψ(t)|C k < δ
for all t ∈ [0, ∞) and |ψ(t)|2 2 ≤ Ce−λt then |ψ(t)|C k ≤ C(k)e−λt .
L
Proof. We begin the proof by deriving an L1,2 exponential decay estimate. The same argument that was used to derive (1.41) shows that there exists a constant C1 such that
∂
|ψ|2 ≤ −| ψ|2 2 + C1 |ψ|2 2 ,
L
L
∂t L2

where C1 depends on both (ω, J) and |ψ(t)|C 2 ; but by assumption |ψ|C 2 < δ for all t ≥ 0.
28

Integrating from t to ∞ yields,
∞
t

| ψ|2 2 ≤ |ψ(t)|2 2 + C1
L
L

∞
t

|ψ|2 2 ≤ Ce−λt .
L

(2.15)

The last inequality follows from the assumed L2 exponential decay estimate.
Next, for a fixed t, let θ(s) be a smooth function which is 0 for s ∈

t − 1, t − 1 ,
2

monotonically increasing from 0 to 1 for s ∈ t − 1 , t and 1 for s ≥ t. As we shall see
2
below, θ(s) will be used to deal with boundary terms which arise in the parabolic estimates
that follow. The same argument that was used to produce (1.48) shows that there exist
constants such that

1∂
| ψ|2 dvolg ≤ −
| 2 ψ|2 + C2
|ψ|2 + C3
| ψ|2 + C4 δ
| 2 ψ|2 ,
g
2 ∂t M
M
M
M
M
(2.16)

again these constants depend on both (ω, J) and |ψ(t)|C 2 . Now we choose δ sufficiently small
1
so that C4 δ < 2 . Hence using (2.16) and that both θ(s) and its derivative are uniformly

bounded,

∂
θ(s)| ψ(s)|2 2 ≤ C5 | ψ(s)|2 2 + C6 |ψ(s)|2 2 .
L
L
L
∂s

(2.17)

1
We integrate (2.17) in s from t − 2 to t for t ≥ 1. Using that θ t − 1 = 0 and θ(t) = 1 we
2

29

get
t

t

| ψ(s)|2 2 + C6
|ψ(s)|2 2
L
L
t− 1
t− 1
2
2

| ψ(t)|2 2 ≤ C5
L

∞

≤ C5

1
t− 2

|

ψ(s)|2 2
L

∞

(2.18)

+ C6
|ψ(s)|2 2 .
L
1
t− 2

Hence using the L2 decay assumption and (2.15) it follows from (2.18) that

| ψ(t)|2 2 ≤ Ce−λt .
L

(2.19)

This proves exponential L1,2 decay.
Next we prove L2,2 decay. By (2.16) with δ small enough so that C4 δ < 1 ,
2
∂
| ψ|2 2 ≤ −| 2 ψ|2 2 + 2C3 | ψ|2 2 + 2C2 |ψ|2 2 .
L
L
L
L
∂t

Integrating from t to ∞ yields
∞
t

| 2 ψ|2 2 ≤ | ψ(t)|2 2 + 2C3
L
L

∞
t

| ψ|2 2 + 2C2
L

∞
t

|ψ|2 2 ≤ Ce−λt .
L

(2.20)

Where (2.19), (2.15), and the L2 decay assumption were used in the first, second, and third
term on the right-hand side respectively.
Now, as in (1.56) we choose δ small enough so that (1.58) holds for m = 2. Therefore
there exists a constant C6 such that
2

1
1∂
| 2 ψ|2 dvolg ≤ −
| 3 ψ|2 + C6
| j ψ|2 2 .
g
L
2 ∂t M
2 M
j=0

30

Hence, there exists a constant C7 so that
∂
θ(s)| 2 ψ(s)|2 2 ≤ C7 | 2 ψ(s)|2 2 + C7 | ψ(s)|2 2 + C7 |ψ(s)|2 2 .
L
L
L
L
∂s

(2.21)

1
We integrate (2.21) in s from t − 2 to t for t ≥ 1. Using that θ t − 1 = 0 and θ(t) = 1 we
2

get

|

2 ψ(t)|2
L2

t

≤ C7

2

1
t− 2 j=0

|

j ψ(s)|2
L2

∞

≤ C7

2

t− 1 j=0
2

| j ψ(s)|2 2 .
L

Hence using (2.20), (2.15) and the assumed L2 decay we get

| 2 ψ(t)|2 2 ≤ Ce−λt .
L

This gives exponential L2,2 decay.
Continuing in this way we get

|ψ(t)|2 p,2 ≤ C(p)e−λt .
L

Furthermore by the Sobolev Embedding Theorem we get |ψ(t)|C k ≤ C(k)e−λt .
By Lemma 15 we know that given k ≥ 2, there exists δ > 0 so that if both |ψ(t)|C k < δ
and |ψ(t)|2 2 ≤ Ce−λt hold for all t ≥ 0, then |ψ(t)|C k ≤ C(k)e−λt for all t ≥ 0. Furthermore
L

by Lemma 14 we know that there exists 1 > 0 such that if |ψ(0)|C ∞ < 1 , then both
|ψ(t)|C k < δ and |ψ(t)|2 2 ≤ Ce−λt hold for all t ≥ 0. Hence let δ be determined by Lemma
L
15. To finish the proof of Theorem 13, we apply Lemma 14 with

= 1 and note that by

(1.14) exponential decay of ρ(t) follows from exponential decay of ψ(t).
31

Chapter 3
Dynamic Stability in the Calabi-Yau
Case

3.1

The Kernel of L and the Space of Calabi-Yau Structures

In Chapter we proved Theorem 1 when c1 (J) < 0 by using that, in this case, the linearization
L is negative definite. However in the Calabi-Yau case, the kernel of L is non-trivial and so
the non-linear part of the flow is no longer controlled by the linear part. In this chapter we
will show that in the Calabi-Yau case we can find a Calabi-Yau structure to which the flow
exponentially converges.
In order to find a Calabi-Yau structure to which the flow exponentially converges we will
construct a sequence {(ωj , Jj )} of successively better Calabi-Yau structures; in the sense
that the solution (ω(t), J(t)) to the VNAHCF converges exponentially on larger and larger
intervals. Moreover we will prove that each of these Calabi-Yau structures is contained in
a fixed neighborhood of the original Calabi-Yau structure (see Theorem 20 part (2)). This
will allow us to extract a limit (ωKE , JKE ) to which {(ωj , Jj )} subconverges. One could
imagine that if this sequence failed to converge that we would only be able to conclude that
the solution becomes asymptotic to the space of Calabi-Yau structures.
32

In order to choose a new Calabi-Yau structure we will use Koiso’s Theorem. Before
stating Koiso’s Theorem we need a definition.
Definition 16. Let AC denote the space of almost complex structures on M modulo diffeo˙ ˙
morphism. A complex structure J is unobstructed if for any J˙ ∈ TJ AC such that N (J) = 0,
∂
˙
there exists a path of complex structures J(a) such that J(0) = J and ∂a a=0 J(a) = J.

Again, N denotes the Nijenhuis tensor.
Theorem 17. (Koiso [5]) Let (ω, J) be a K¨hler-Einstein structure on M . Assume that:
a
1. the first Chern class of J is zero;
2. J is unobstructed.
Then the space of K¨hler-Einstein structures, modulo diffeomorphism, around (ω, J) is a
a
manifold.
In order to make use of Koiso’s Theorem we employ a theorem of Tian and Todorov.
Theorem 18. (Tian [9] and Todorov [10]) Let (M, J) be a closed Calabi-Yau manifold.
Then J is unobstructed.
Next we will describe the tangent space of Calabi-Yau structures at (ω, J). Using the
above two theorems we prove that the kernel of L is isomorphic to the tangent space of
Calabi-Yau structures at (ω, J). Let U denote the space of Calabi-Yau structures near (ω, J)
modulo diffeomorphism.
Lemma 19. Let (M, ω, J) be a closed Calabi-Yau manifold, then T(ω,J) U ∼ Ker L.
=
Proof. Let (ω(a), J(a)) be a one-parameter family of unit volume almost hermitian structures
∂
˙ ˙
and write ∂a a=0 (ω(a), J(a)) = (ω, J). First since Calabi-Yau structures are static under

33

the system (1.2), we have T(ω,J) U ⊆ Ker L. To prove Ker L ⊆ T(ω,J) U, let (ω, J) ∈ Ker L.
˙ ˙
˙
˙
We make the following claim. If J˙ ∈ ker G then J˙ ∈ ker N . To see this, first notice that
˙
from (1.10) and using that the scalar curvature sg = 0, if J˙ ∈ ker G, then ∆∂ J˙ = 0. By
integrating we see that ∂ J˙ = 0. On the other hand, in coordinates, the Nijenhuis tensor is
written

p

p

p

p

i
i
i
i
i
Njk = Jj ∂p Jk − Jk ∂p Jj − Jp ∂j Jk + Jp ∂k Jj .

Hence,

p
p
p
p
p
p
p
p
i
i
i
i
i
i
i
i
˙i
Njk = J˙j ∂p Jk + Jj ∂p J˙k − J˙k ∂p Jj − Jk ∂p J˙j − J˙p ∂j Jk − Jp ∂j J˙k + J˙p ∂k Jj + Jp ∂k J˙j .

˙
Now each of the terms above of the form J˙ ∗ ∂J can be written as J˙ ∗ ∂J = J( J + Γ ∗ J).
So using normal, complex coordinates (with respect to the Calabi-Yau structure (g, J)), at
a point p ∈ M , we have that each of these terms vanish. Here we also made use of the fact
that when (g, J) is K¨hler the Chern connection coincides with the Levi-Civita connection
a
and so J is parallel with respect to the connection. Next, since J˙ ∈ Λ0,1 ⊗ T 1,0 in these
normal, complex coordinates at p ∈ M , we have,

p
p
p
p
i
i
i
i
˙i
Njk = J ∂p J˙k − J ∂p J˙j − Jp ∂j J˙ + Jp ∂k J˙ = 0.
j

k

k

j

This proves the claim.
By Theorem 18 there exists a path of complex structures J(a) where J(0) = J and
d
da J a=0

˙
˙
= J. Next, using (1.9) we have that ω ∈ ker F implies that ∆d ω = 0, that is ω is
˙
˙
˙

harmonic. Therefore, by the Calabi-Yau Theorem ([1], [11], [12], also see Theorem 2.29 in

34

[3]) ω(a) is a variation through K¨hler metrics such that [ω(a)] = [ωKE (a)], where ωKE (a)
a
is Ricci-flat. Moreover by the Hodge Decomposition Theorem there is a unique harmonic
representative in each cohomology class. Hence ω(a) = ωKE (a) and we have that ω arises
˙
as a variation through Calabi-Yau metrics.
Notice that Λ2 (M ) × End(T M ) is an affine space which can be viewed as a vector space
by taking (ω, J) to be the origin. Throughout this chapter we will view Λ2 (M ) × End(T M )
as a vector space. Let π0 : Λ2 (M ) × End(T M ) → Ker L be the projection onto the kernel of
L.
Let (ω, J) denote the Calabi-Yau structure from Theorem 1. Roughly speaking, we will
next prove that there exists a better Calabi-Yau structure (ωI , JI ); in the sense that the
solution (ω(t), J(t)) to VNAHCF exponentially converges to (ωI , JI ) on an interval I (see
.
Theorem 20 and Lemma 21). Throughout this chapter ρI (t) = (ω(t) − ωI , J(t) − JI ) will
quantify the distance the solution is from this new Calabi-Yau structure. As above let
ρ(t) = (ω(t) − ω, J(t) − J). Notice that we may view both ρ(t) and ρI (t) as elements of
Λ2 (M ) × End(T M ).
As in Section 1.2 we have to deal with the non-linearity of the space of almost hermitian
structures modulo diffeomorphism denoted C. Notice that we may write ρI (t) = ρ(t) − ρI
.
where ρI = (ωI − ω, JI − J). From Lemma 10 associated to ρ(t) we have ψ(t) ∈ T(ω,J) C
and analogously associated to ρI we have ψI ∈ T(ω,J) C. Hence associated to ρI (t) we have
ψI (t) ∈ T(ω,J) C defined by
.
ψI (t) = ψ(t) − ψI .

35

Moreover, employing the same argument as in the proof of Lemma 10 we have

|ψI (t)|C k ≤ |ρI (t)|C k ,

(3.1)

|ρI (t)|L2 ≤ |ψI (t)|L2 + C|ψI (t)|2 2

(3.2)

|ρI (t)|C k ≤ |ψI (t)|C k + C|ψI (t)|2 k .

(3.3)

L

and

C

Similarly by the proof of Lemma 11 we have

∂
ψ (t) = L(ψI (t)) + A((ω, J), ψI (t))
∂t I

(3.4)

where

|A((ω, J), ψI (t))|C k ≤ C(|ψI (t)|C k | 2 ψI (t)|C k + | ψI (t)|2 k ).
C

3.2

(3.5)

A New Calabi-Yau Structure

Next we will use the identification of the kernel of L and the tangent space of Calabi-Yau
structures at (ω, J), from Lemma 19, to find a new Calabi-Yau structure denoted (ωI , JI )
such that |π0 (ψI (t))|L2 is small relative to |ψI (t)|L2 . Furthermore we will show that the
new Calabi-Yau structure is contained in a fixed neighborhood of the original Calabi-Yau

36

structure (ω, J).
Theorem 20. Given t0 and T > 0, let I = [t0 , t0 + T ]. There exists δ(T, g) such that if
supI |ψ(t)|C k < δ with k ≥ 2, then there exists a Calabi-Yau structure (ωI , JI ) with the
following properties:
1. |π0 (ψI )|2 2

L (g)

≤ 1 |ψI |2 2
4

L (g)

on I

2. |(ωI − ω, JI − J)|C k ≤ C supI |ψ|C k .
Proof. First by Theorem 17 we know that U has a manifold structure near (ω, J) and moreover by Lemma 19 we have Ker L ∼ T(ω,J) U.
=
By identifying Ker L and T(ω,J) U we will view (ω, J) as the origin of Ker L. Let Φ =
Φ(ω,J) : Ker L → U denote the exponential map at (ω, J). Now since D(ω,J) Φ is the identity
map, the inverse function theorem may be applied to Φ. By the inverse function theorem
there exists a neighborhood V ⊂ Ker L of (ω, J) on which the exponential map is invertible.
Let δ1 be small enough so that

|π0 (ψ(t0 ))|C k < δ1 implies that π0 (ψ(t0 )) ∈ V.

(3.6)

Notice that if supI |ψ|C k < δ1 then since t0 ∈ I, it is clear that |π0 (ψ(t0 ))|C k < δ1 . Hence
by the inverse function theorem there is a Calabi-Yau structure (ωI , JI ) ∈ U such that

(Φ|V )−1 (ωI , JI ) = π0 (ψ(t0 )).

(3.7)

Applying Φ to each side of (3.7), it follows from the inverse function theorem that there

37

exists a constant C so that

|(ωI − ω, JI − J)|C k ≤ C|π0 (ψ(t0 ))|C k
≤ C sup |ψ|C k .
I

This proves (2).
Next, using that (Φ|V )−1 (ωI , JI ) = π0 (ωI − ω, JI − J) , from (3.7) we have

π0 (ψI (t0 )) = 0.

(3.8)

In other words, there exists a Calabi-Yau structure (ωI , JI ) such that at time t0 , ψI (t) is
orthogonal, with respect to L2 (g), to Ker L.
To prove (1) we will carefully study the evolution of ψI (t) starting at t = t0 in order
to get L2 estimates on π0 (ψI ). First let ||ψI ||M ×I = I |ψI |L2 (g) denote the L2 norm on
˙
M × I. Let {Bi } be an orthonormal basis, with respect to L2 (g), of T(ω,J) C determined by
the eigenspace decomposition of L. Then there exist constants ci so that {ci Bi eλi t } is an
∂
orthonormal basis, with respect to ||·||M ×I , of Ker ∂t − L

M ×I

where λi is the eigenvalue

associated to Bi .
∂
We let π I ψI (t) denote the projection of ψI (t) onto Ker ∂t − L

∂ I
π (ψI (t)) = L(π I (ψI (t))).
∂t

38

M ×I .

In other words,

(3.9)

From (3.8), we have π I (ψI (t0 )) =

λi =0 ki Bi .

π I (ψI (t)) =

It then follows that

ki Bi eλi (t−t0 ) .

(3.10)

λi =0

We write

ψI (t) = π I (ψI (t)) + ξI (t).

(3.11)

Since π I ψI (t) is orthogonal to Ker L on I, it follows that for t ∈ I,

|π0 (ψI (t))| ≤ |ξI (t)|.

(3.12)

Therefore to obtain estimates on π0 ψI (t) we compute the evolution of ξI (t). Moreover,
from (3.10) and (3.11), since λi < 0 is bounded away from 0 for all i, we have that ξI (t)
converges exponentially to ψI (t). Therefore there is a uniform constant C so that on I,

|ξI (t)| ≤ C|ψI (t)|.

(3.13)

To compute the evolution of ξI (t) we compare two evolution equations for ψI (t). From
∂
(3.4), ψI (t) satisfies ∂t ψI (t) = L(ψI (t)) + A((ω, J), ψI ) and hence,

∂
ψ (t) = L(π I (ψI (t))) + L(ξI (t)) + A((ω, J), ψI (t)).
∂t I

39

(3.14)

Furthermore, π I (ψI (t)) satisfies (3.9) and so
∂
∂ I
∂
ψI (t) =
π (ψI (t)) + ξI (t) = L(π I (ψI (t))) + ξI (t).
∂t
∂t
∂t

(3.15)

Combining equations (3.14) and (3.15) we have that ξI (t) evolves by
∂
ξ (t) = L(ξI (t)) + A((ω, J), ψI (t)).
∂t I

(3.16)

Recall that L is negative semi-definite; and so by (3.16), on I we have

∂
|ξ (t)|2
=2
∂t I L2 (g)
M

∂
A (ω, J), ψI (t) ∗ ξI (t).
ξ (t), ξI (t) dvolg ≤ 2
∂t I
M
g

Now using the bounds on A from (3.5), the same computation as (1.37) shows that

∂
|ξ (t)|2
≤ C1
| 2 ψI ||ψI ||ξI |.
∂t I L2 (g)
M
Hence by (3.13),

∂
|ξ (t)|2 2 ≤ C2
| 2 ψI ||ψI |2 .
L (g)
∂t I
M

(3.17)

Next we assume supI |ψ(t)|C k < δ with k ≥ 2 and δ ≤ δ1 where δ1 is from (3.6). Using
part (2) of Theorem 20 and the triangle inequality, from (3.17) it follows that on I

∂
|ξ (t)|2 ≤ C3 δ|ψI (t)|2 2 .
L
∂t I L2

40

Now since ξI (t0 ) = 0,

|ξI (t)|2 2

L (g)

t
∂
2
ξI (s) L2 (g) ds ≤ C3 δ
|ψI (s)|2 2 ds.
L (g)
∂s
t0
t0
t

=

(3.18)

∂
∂
Notice that since ∂t ψI (t) = ∂t ψ(t) is second order in ψ(t) and supI |ψ(t)|C k < δ with k ≥ 2,
∂
∂t ψI (t)

is uniformly bounded in terms of δ and hence each ψI (s) for s ∈ I is uniformly

equivalent. Therefore

|π0 (ψI (t))|2 2

L (g)

≤ |ξI (t)|2 2

L (g)

t

≤ C4 δ

t0

|ψI (t)|2 2

L (g)

ds = C5 δ(t − t0 )|ψI (t)|2 2

L (g)

,

where the first inequality follows from (3.12) and the second is from (3.18). To finish the
1
proof we choose δ small enough so that both C5 T δ < 4 and δ ≤ δ1 hold.

We will now use part (1) of Theorem 20 to prove L2 exponential decay of ψI (t) on I.
Notice that by (3.3) this implies exponential decay of ρI (t) = (ω(t) − ωI , J(t) − JI ) on I.
Lemma 21. Let I and (ωI , JI ) be as in Theorem 20. There exists β > 0 such that if
|ψ|C 2 < β on I, then
Tλ

sup
t0 + 1 T,t0 +T
2

M

|ψI |2 dvolg ≤ e− 2

sup
1
t0 ,t0 + 2 T

M

|ψI |2 dvolg

where λ = min{|λi | : λi is a non-zero eigenvalue of L } > 0.
Proof. We compute the evolution of |ψI |2 2 and as in (3.4) we have
L

∂
|ψ |2 dvolg = 2
L(ψI ), ψI dvolg +
A (ω, J), ψI (t) ∗ ψI dvolg .
∂t M I
M
M

41

(3.19)

Recall that by the definition of π0 , ψI − π0 (ψI ) is the component of ψI orthogonal to the
kernel of L. Hence by the definition of λ,

2
M

L(ψI ), ψI dvolg ≤ −2λ|ψI − π0 (ψI )|2 2

(3.20)

L

≤ −2λ |ψI |2 2 − |π0 (ψI )|2 2 .
L

L

(3.21)

Let δ be the constant from Theorem 20. By Theorem 20 part (1), if supI |ψ(t)|C 2 < β1
with β1 ≤ δ, then from (3.20) and (3.21) it follows that

2

(3.22)

A ∗ ψI from (3.19). We use the estimate on A from (3.5) to

Next consider the term
bound

3
L(ψI ), ψI dvolg ≤ − λ|ψI |2 2 .
L
2
M

A ∗ ψ. Notice that if |ψI |C 2 < β3 , then as in (1.38),

M

A (ω, J), ψI (t) ∗ ψI dvolg ≤ Cβ3 |ψI |2 2 .
L

(3.23)

Now we choose β3 small enough so that

Cβ3 <

λ
.
2

(3.24)

Let β2 be sufficiently small so that |ψ|C 2 < β2 on I implies that |ψI |C 2 < β3 on I. Notice
that this can be done using the triangle inequality and part (2) of Theorem 20.
Finally we choose β = min{β1 , β2 }. Combining (3.19), (3.22), (3.23) and (3.24) it follows
that if |ψ|C 2 < β on I, then

42

∂
|ψ |2 dvolg ≤ −λ
|ψI |2 dvolg .
∂t M I
M

(3.25)

Integrating from t0 to t gives

|ψI (t)|2 2 ≤ e−λ(t−t0 ) |ψI (t0 )|2 2 .

(3.26)

L

L

Finally since (3.25) implies that |ψI (t)|2 2 is decreasing, plugging t0 + 1 T into (3.26) proves
2
L

the lemma.
This gives exponential L2 decay of ψI (t) on I. Next we prove a general result about
parabolic flows (cf. Lemma 8.8 in [8]). The following lemma says roughly that exponential
decay at a later time implies exponential decay at an earlier time.
∂
Lemma 22. There exists ν > 0 so that if κ solves the parabolic flow equation ∂t κ = L(κ) +

A(κ) and |κ(t)|C k < ν for all t ∈ [0, t0 + T ], then
Tλ

|κ|2 ≤ e− 2

sup
1
t0 + 2 T,t0 +T

M

|κ|2

sup
t0 ,t0 + 1 T
2

(3.27)

M

implies that

Tλ

|κ|2 ≤ e− 2

sup
1
t0 ,t0 + 2 T

M

|κ|2 .

sup
t0 − 1 T,t0
2

(3.28)

M

Proof. Suppose, by way of contradiction, that the lemma fails to hold. Then there is a
∂
sequence νi → 0 with κi (t) solving ∂t κi = L(κi ) + A(κi ) and |κi |C k < νi on [0, t0 + T ] and

43

moreover (3.27) holds but (3.28) does not. Parabolically rescale the solution κi ; that is let
−1
−1
κ(t)i = νi κi (νi t). Now for all i, |κi |C k < 1 on [0, νi (t0 + T )] ⊃ [0, t0 + T ] and so by

compactness we get a convergent subsequence κ(t)i → κ(t)∞ on [0, t0 + T ] as i → ∞. Now
∂
since κi solves ∂t κi = L(νi κi ) + A(νi κi ) and A(κ) is quadratic in κ this implies that κ∞ (t)

solves the linear equation

∂
κ∞ = L(κ∞ ).
∂t

(3.29)

Furthermore since (3.27) and (3.28) are scale invariant it follows that for κ∞ (3.27) holds
but (3.28) does not. This is a contradiction.
To see the contradiction, first notice that (3.29) implies that

∂
|κ∞ |2 2 ≤ 0
L
∂t

(3.30)

as L is negative semi-definite. It then follows that

|κ∞ (t0 + 1 T )|2 2 =
2
L

sup
t0 + 1 T,t0 +T
2
Tλ

≤ e− 2

|κ∞ |2 2

(3.31)

L

sup
1
t0 ,t0 + 2 T

Tλ

|κ∞ |2 2

= e− 2 |κ∞ (t0 )|2 2 ,
L

L

(3.32)

(3.33)

where the inequality follows from (3.27). As above, let {Bi } be an orthonormal basis, with
respect to L2 (g), of T(ω,J) C determined by the eigenspace decomposition of L. We can now

44

write κ∞ (t) with respect to this basis,

Bi eλi (t−t0 ) .

κ∞ (t) = κ∞ (t0 )

(3.34)

i
1
Notice that at time t = t0 + 2 T we have

eT λi

|κ∞ (t0 + 1 T )|2 2 = |κ∞ (t0 )|2 2
2
L

L

.

(3.35)

i

By combining (3.31), (3.32), (3.33) and (3.35), it follows that

Tλ

eT λi ≤ e− 2 .

(3.36)

i

From (3.34) it follows that

e−T λi

|κ∞ (t0 − 1 T )|2 2 = |κ∞ (t0 )|2 2
2
L

L

.

(3.37)

i
1
Finally using (3.37) and the concavity of f (x) = x we see that

|κ∞ (t0 )|2 2 = |κ∞ (t0 − 1 T )|2 2
2
L

L

1
−T λi
ie

≤ |κ∞ (t0 − 1 T )|2 2
2

eT λi

L

(3.38)

i
Tλ
1
≤ e− 2 |κ∞ (t0 − 2 T )|2 2 .
L

The last inequality in (3.38) follows from (3.36). Notice that the above inequality along with
(3.30) imply that (3.28) holds. This is a contradiction.

45

Corollary 23. Given T > 0 and j ≥ 1 there exists α = α(T, j) > 0 such that if |ψ(t)|C 2 < α
on [0, (j + 1)T ] then there exists a Calabi-Yau structure (ωj , Jj ) so that ρj (t) = (ω(t) −
ωj , J(t) − Jj ) satisfies the following exponential decay estimate:
λt

|ρj (t)|2 2 ≤ Ce− 2
L

for t ∈ [0, (j + 1)T ] and a constant C independent of j.
Proof. First notice that by (3.2) it suffices to prove exponential decay of ψj (t), the tangent
vector associated to ρj (t). We will prove exponential decay of ψj (t) using the previous two
lemmas and Theorem 20.
Let δ, β and ν be the small constants from Theorem 20, Lemma 21 and Lemma 22
respectively. In order to apply the above lemmas and Theorem 20 we let α = min{δ, β, ν}
and assume that |ψ|C 2 < α on [0, (J + 1)T ]. Employing Theorem 20 (with t0 = jT ) and
Lemma 21, there exists a Calabi-Yau structure, denoted (ωj , Jj ), such that
Tλ

sup
1
j+ 2 T,(j+1)T

|ψj |2 2 ≤ e− 2
L

sup
jT, j+ 1 T
2

|ψj |2 2 .
L

(3.39)

Now, Lemma 22 says that exponential decay at a later time implies exponential decay
at an earlier time. In particular, from Lemma 22 and (3.39) it follows that for any k ∈ { n :
2
n ∈ Z and 1 ≤ n ≤ 2j + 1},

sup
kT, k+ 1 T
2

Tλ

|ψj |2 2 ≤ e− 2
L

46

sup
k− 1 T,kT
2

|ψj |2 2 .
L

(3.40)

1
1
Applying (3.40) with k = 2 implies that for any t ∈ 2 T, T ,

Tλ

|ψj (t)|2 2 ≤ e− 2
L

λt

sup |ψj |2 2 ≤ e− 2
L

0, 1 T
2

sup |ψj |2 2 .

1
0, 2 T

L

(3.41)

1
Next, using (3.40) with k = 2 and k = 1 yields

Tλ

sup |ψj |2 2 ≤ e− 2
L

3
T, 2 T

sup |ψj |2 2 ≤ e−T λ sup |ψj |2 2 .
L

1 T,T
2

1
0, 2 T

L

3
Therefore it follows that for t ∈ T, 2 T ,

λt

|ψj (t)|2 2 ≤ e−T λ sup |ψj |2 2 ≤ e− 2
L

L

1
0, 2 T

sup |ψj |2 2 .
L
1

(3.42)

0, 2 T

3
Combining (3.41) and (3.42) yields L2 exponential decay of ψj (t) on 1 T, 2 T . Iterating
2
1
this argument, we see that for t ∈ 2 T, (j + 1)T ,

λt

|ψj (t)|2 2 ≤ e− 2
L

λt

sup |ψj |2 2 ≤ Ce− 2 .

1
0, 2 T

L

Notice that C is independent of j. Indeed by assumption |ψ(t)|C 2 < α on [0, (j +1)T ]. Hence
part (2) of Theorem 20 and the triangle inequality imply that |ψj (t)|C 2 < C on [0, (j + 1)T ],
where C is independent of j.

47

3.3

Finding the Limit Calabi-Yau Structure

In this section we will show that there is a sequence of Calabi-Yau structures {(ωk , Jk )},
indexed by k, which subconverges as k → ∞ to a limit Calabi-Yau structure, which we will
denote (ωKE , JKE ). By the choice of the Calabi-Yau structures in {(ωk , Jk )}, it will follow
that the solution (ω(t), J(t)) of the system (1.2) exponentially converges to (ωKE , JKE ) as
t → ∞.
Notice that the decay estimate from Corollary 23 may fail to hold for intervals beyond
Ij . In order to maintain exponential decay we want to choose another Calabi-Yau structure
(ωj+1 , Jj+1 ) to which the solution exponentially converges. To ensure that we can continue
this process we need to prove that |ψ(t)|C 2 is small for all time so that Corollary 23 may
be applied on any interval. This is the purpose of the following theorem. As a corollary we
will prove the existence of a Calabi-Yau structure, denoted (ωKE , JKE ), to which the flow
exponentially converges.
Theorem 24. Let (M, ω, J) be a closed complex manifold with (ω, J) a Calabi-Yau structure.
Given

> 0 and k ≥ 0 there exists

> 0 so that |ρ(0)|C ∞ <

implies that |ψ(t)|C k <

[0, ∞).
Proof. We will employ Theorem 12. To do this we make explicit T , , and .
1. Let T be large enough so that
1
1
T 2 C3 (k + 2)
+ Tλ < .
Tλ − 1
2
e
e

Where C3 (k + 2) is a constant depending only on k and (ω, J).

48

on

2. Choose

= α.

Where α is the constant from Corollary 23.
3. Choose

sufficiently small so that (ω(t), J(t)) exists on [0, 3T ] and

sup |ψ(t)|C k < T λ .
e

[0,2T ]

We want to prove that |ψ(t)|C k <

on [0, ∞). Suppose by way of contradiction there

is a finite maximal time T such that |ψ(t)|C k <

on [0, T ) with k ≥ 2. Write [0, T ) =

[0, T ] ∪ [T, 2T ] ∪ · · · ∪ [N T, T ) and let [jT, (j + 1)T ] = Ij . Also let (ωj , Jj ) denote the
Calabi-Yau structure, from Corollary 23, to which (ω(t), J(t)) exponentially converges on
Ij . Using (3.39) and applying Lemma 22 iteratively we have

sup
Ij−1 ∪Ij

|ψj |2 2 ≤ e−λT (j−1) sup |ψj |2 2 .
L

[0, T ]
2

L

(3.43)

We again use a parabolic regularity argument to prove that from (3.43) we can obtain a
C k+2 decay estimate.
Lemma 25. There exists a constant α > 0 so that if both |ψ|C 2 < α on Ij and supIj−1 ∪Ij |ψj |2 2 ≤
L

e−λT (j−1) sup T |ψj |2 2 , then there exists a constant C(k + 2) such that
[0, 2 ]
L
sup |ψj |C k+2 < C(k + 2)T e−T λ(j−1) sup |ψj |2 2 .
L
I
1
j

0, 2 T

49

Proof. The proof of Lemma 25 uses essentially the same argument as the proof of Lemma
15 hence we omit some of the details.
From (1.42) we have

∂
|ψ |2 ≤ −| ψj |2 2 + C1 |ψj |2 2 .
L
L
∂t j L2

Fix t ∈ Ij and integrate from (j − 1)T to t;
t
(j−1)T

| ψj |2 2 ≤ |ψj ((j − 1)T )|2 2 + C1
L
L

t
(j−1)T

|ψj |2 2
L

≤ CT e−T λ(j−1) sup |ψj |2 2 ,
1
0, 2 T

(3.44)

L

where the second inequality follows from the L2 exponential decay assumption.
Next let θ(s) be a smooth function which is 0 for s ≤ (j − 1)T , monotonically increasing
from 0 to 1 for s ∈ [(j − 1)T, jT ] and equal to 1 for s ≥ jT . As in Lemma 25, θ(s) will be
used to deal with the boundary terms that arise in the estimates below. Now from (1.48)
we have

∂
| ψj |2 2 ≤ C5 |ψ|2 2 + C6 | ψ|2 2
L
L
L
∂t

and since θ(s) and its derivative are uniformly bounded, it follows that

∂
θ(s)| ψj (s)|2 2 ≤ C7 |ψj |2 2 + C8 | ψj |2 2 .
L
L
L
∂s

50

We now integrate from (j − 1)T to t ∈ Ij and use that θ((j − 1)T ) = 0 and θ(t) = 1,

| ψj (t)|2 2 ≤ C7
L

t
(j−1)T

|ψj |2 2 + C8
L

t
(j−1)T

| ψj |2 2
L

≤ CT e−T λ(j−1) sup |ψj |2 2 ,
0, 1 T
2

(3.45)
(3.46)

L

where the first and second terms on the right-hand side of (3.45) were bounded using the L2
exponential decay assumption and (3.44) respectively. Notice that (3.45) and (3.46) yield
the desired L1,2 exponential decay estimate.
Continuing in this way, on Ij we get

|ψj (t)|2 p,2 ≤ C(p)T e−T λ(j−1) sup |ψj |2 2 ,
L

1
0, 2 T

L

moreover by the Sobolev Embedding Theorem, for any t ∈ Ij

|ψj (t)|C k+2 ≤ C(k + 2)T e−T λ(j−1) sup |ψj |2 2 .
1
0, 2 T

L

From Lemma 25 it follows that

sup
Ij

∂
ψ
≤ C2 (k + 2)T e−T λ(j−1)
∂t C k

∂
∂
since ∂t ψ C k = ∂t ψj C k ≤ C supIj |ψj |C k+2 . Hence, for j ≥ 2 and any t ∈ Ij , by

51

integrating we see that

|ψ(t)|C k ≤ T sup
Ij
j

≤T

∂
ψ
+ sup |ψ|C k
∂t C k I
j−1
sup

l=2 Il

∂
ψ
+ sup |ψ|
∂t C k I0 ∪I1 C k

< C3 (k + 2)T 2

1

1
1
+ 2λT + · · · +
eλT
e
e(j−1)λT

≤

I0 ∪I1

C3 (k + 2)T 2
+ sup |ψ|C k
eλT − 1
I0 ∪I1

≤

+ sup |ψ|C k

C3 (k + 2)T 2
+ Tλ
eλT − 1
e

<

2

.

Where the final inequality is from our choice of T and . The key observation here is that
the above inequality is independent of both j ≥ 2 and t ∈ Ij . Hence the above inequality
holds for j = N which contradicts the maximality of T . Therefore T = ∞.
The important thing to notice about Theorem 24 is that it allows us to find a Calabi-Yau
structure (ωKE , JKE ) to which the flow converges.
Corollary 26. Under the assumptions of Theorem 1 with (M, ω, J) a Calabi-Yau manifold,
there exists a Calabi-Yau structure (ωKE , JKE ) to which the flow exponentially converges.
Proof. By Theorem 24 we know that given
if |ρ(0)|C ∞ < , then |ψ(t)|C k <

> 0 and k ≥ 0, there exists

for all t ≥ 0. Let

> 0 such that

= α, where α is the small constant

from Corollary 23. Recall that ρj (t) = (ω(t) − ωj , J(t) − Jj ). Now since |ψ(t)|C k < α for all
t ≥ 0, by Corollary 23 there exists a sequence of Calabi-Yau structures {(ωj , Jj )} such that

52

ρj (t) exponentially decays in L2 for all t ∈ [0, (j + 1)T ] and for each j. Specifically,
λt

|ρj (t)|L2 ≤ Ce− 2

(3.47)

for t ∈ [0, (j + 1)T ] and for each j.
Next, by part (2) of Theorem 20, each of these Calabi-Yau structures (ωj , Jj ) is contained
in a fixed neighborhood of (ω, J), in particular |(ωj − ω, Jj − J)|C k ≤ C . Therefore as
j → ∞ we have a convergent subsequence (ωj , Jj ) → (ωKE , JKE ). And hence by (3.47) we
have L2 exponential convergence of (ω(t), J(t)) to (ωKE , JKE ) for all t ≥ 0. Finally applying
the parabolic regularity argument of Lemma 15 it follows that the exponential convergence
of (ω(t), J(t)) to (ωKE , JKE ) is C k convergence, that is
λt

|(ω(t) − ωKE , J(t) − JKE )|C k ≤ Ce− 2 .

(3.48)

λt

In other words, (ω(t), J(t)) is contained in a ball of radius Ce− 2 of the limit Calabi-Yau
structure (ωKE , JKE ) for all t ≥ 0. This gives us the desired exponential decay estimate.

53

BIBLIOGRAPHY

54

BIBLIOGRAPHY
´
[1] Thierry Aubin. Equations du type Monge-Amp`re sur les vari´t´s k¨hleriennes come
ee a
pactes. C. R. Acad. Sci. Paris S´r. A-B, 283(3):Aiii, A119–A121, 1976.
e
[2] Arthur L. Besse. Einstein manifolds, volume 10 of Ergebnisse der Mathematik und
ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag,
Berlin, 1987.
[3] Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg,
Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni. The Ricci flow: techniques and
applications. Part I, volume 135 of Mathematical Surveys and Monographs. American
Mathematical Society, Providence, RI, 2007. Geometric aspects.
[4] Christine Guenther, James Isenberg, and Dan Knopf. Stability of the Ricci flow at
Ricci-flat metrics. Comm. Anal. Geom., 10(4):741–777, 2002.
[5] Norihito Koiso. Rigidity and stability of Einstein metrics—the case of compact symmetric spaces. Osaka J. Math., 17(1):51–73, 1980.
[6] Natasa Sesum. Linear and dynamical stability of Ricci-flat metrics. Duke Math. J.,
133(1):1–26, 2006.
[7] J. Streets and G. Tian. Symplectic curvature flow. ArXiv e-prints, December 2010.
[8] Jeffrey Streets and Gang Tian. Hermitian curvature flow. J. Eur. Math. Soc. (JEMS),
13(3):601–634, 2011.
[9] Gang Tian. Smoothness of the universal deformation space of compact Calabi-Yau
manifolds and its Petersson-Weil metric. In Mathematical aspects of string theory (San
Diego, Calif., 1986), volume 1 of Adv. Ser. Math. Phys., pages 629–646. World Sci.
Publishing, Singapore, 1987.
[10] Andrey N. Todorov. The Weil-Petersson geometry of the moduli space of SU(n ≥ 3)
(Calabi-Yau) manifolds. I. Comm. Math. Phys., 126(2):325–346, 1989.
[11] Shing Tung Yau. Calabi’s conjecture and some new results in algebraic geometry. Proc.
Nat. Acad. Sci. U.S.A., 74(5):1798–1799, 1977.
55

[12] Shing Tung Yau. On the Ricci curvature of a compact K¨hler manifold and the complex
a
Monge-Amp`re equation. I. Comm. Pure Appl. Math., 31(3):339–411, 1978.
e

56