LONG-TIME CONVERGENCE OF HARMONIC MAP HEAT FLOWS FROM
SURFACES INTO RIEMANNIAN MANIFOLDS
By
Kwangho Choi

A DISSERTATION
Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Mathematics
2011

ABSTRACT
LONG-TIME CONVERGENCE OF HARMONIC MAP HEAT FLOWS
FROM SURFACES INTO RIEMANNIAN MANIFOLDS
By
Kwangho Choi
We study the long-time convergence of harmonic map heat ﬂows from a closed Riemann
surface into a compact Riemannian manifold. P. Topping constructed an example of a ﬂow
that does not converge in the inﬁnite-time limit. Motivated by the observation that Topping’s ﬂow has accumulation points at which the Hessian of the energy function is degenerate,
we prove convergence under the assumptions that (a) the Hessian of the energy at an accumulation point is positive deﬁnite, and (b) no bubbling occurs at inﬁnite time. In addition,
we present examples of heat ﬂows for geodesics which show that the convexity of the energy
function and convergence as t → ∞ may not hold even for 1-dimensional harmonic map heat
ﬂows.

ACKNOWLEDGMENT

I sincerely thank my advisor, Professor Thomas Parker for his continued encouragement and
thoughtful guidance. This thesis would never have taken shape without his persistent help.
My thank goes to Professor Jon Wolfson for his invaluable suggestions and lectures on related
topics that helped me improve my knowledge in diﬀerential geometry. Thanks are also due
to former and current graduate students, Dr. Junho Lee, Dr. Jens Von Bergmann, Oh
Hoon Kwon, Harrison Cheuk-Man Hwang, and Manousos Maridakis for numerous helpful
discussions. Last but not least, I would like to acknowledge that none of this would have
been possible without love and patience of my family in Korea.

iii

TABLE OF CONTENTS

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 Introduction

v

1

1 Background
1.1 Energy . . . . . . . . . . . . . . . . . . . . .
1.2 Euler-Lagrange equation and harmonic maps
1.3 Existence of harmonic maps from a surface .
1.4 Harmonic map heat ﬂow . . . . . . . . . . .

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14

2 Degenerate Hessian of the energy
2.1 Failure of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Degenerate Hessian of the energy . . . . . . . . . . . . . . . . . . . . . . . .

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3 Hessian of the energy and W 2,2 convergence of harmonic map heat ﬂows 23
3.1 Preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 W 2,2 convergence of harmonic map heat ﬂows . . . . . . . . . . . . . . . . . 27
4 Continuity of Hessian
4.1 Sobolev spaces of maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Continuity of Hessian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31
31
33

5 Proof of the main theorem

40

6 Examples of geodesic heat ﬂows
6.1 Geodesic heat ﬂows on surfaces of revolution . . . . . . . . . . . . . . . . . .
6.2 The energy function E(t) needn’t be convex. . . . . . . . . . . . . . . . . . .
6.3 A non-converging geodesic heat ﬂow . . . . . . . . . . . . . . . . . . . . . . .

44
45
47
51

Bibliography

54

iv

LIST OF FIGURES

6.1

u : (S 1 , g) → (M, h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

6.2

u : (S 1 , g) → (S 2 , h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

6.3

E(t) is not convex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

v

Chapter 0

Introduction

A map u between two Riemannian manifolds (M, g) and (N, h) is called a harmonic map if it
1´
is a critical point of the energy function E(u) = 2 M |du|2 dvolg in C 2 (Σ, N). A harmonic
map heat ﬂow is a gradient ﬂow of the energy function on a space of maps. If one embeds
the target manifold N isometrically into Rr , then the harmonic map heat ﬂow is a weak
solution u : M × [0, ∞) → N to a system of nonlinear parabolic PDEs with initial condition:
u = ∆u + A(u)(du, du),
˙

(0.1)

u(·, 0) = u0 (·),

where we consider u : M → N ֒→ Rr to be a map into Rr , and where A denotes the
second fundamental form of N in Rr . Intrinsically, the right-hand side of (0.1) is minus the
gradient of E for the L2 Riemannian metric on the space of maps from M to N. Thus one
can consider a solution u = u(t) to be a family of maps that starts from u0 and ﬂows in the
energy-minimizing direction.
1

Standard results show the existence and smoothness of solutions to (0.1) for short time.
Eells and Sampson [6] used harmonic map heat ﬂows to prove that if the target manifold
N is a Riemannian manifold with non-positive sectional curvature then the solution to (0.1)
exists for all time and therefore any smooth map u0 : M → N is homotopic to a harmonic
map.

For dim M = 2, the energy E is invariant under the conformal change of metrics on
Σ and it is possible that the energy density concentrates at a ﬁnite set of points. Sacks
and Uhlenbeck [16] discovered a “bubbling phenomena”, in which each point of energy
concentration yields a harmonic 2-sphere S 2 , called a bubble. For sequences of harmonic
maps, Parker [13] proved a “bubble tree convergence theorem,” in which the maps converge
to a limit map together with bubbles so that energy is conserved in the limit and the image
of the limit is connected.
For harmonic map heat ﬂows, Struwe [18] showed that a global weak solution u(t) to
(0.1) exists and is smooth away from a ﬁnite set of “singular points” (x1 , T1 ), . . . , (xk , Tk )
in spacetime Σ × R. At each singular point, energy concentrates at xi and a bubble splits
oﬀ. Struwe also showed that there is a sequence tn → ∞ such that the maps u(tn ) converges weakly to a harmonic map u∞ together with possibly some bubbles (see Section 1.4).
Struwe’s theorem was reﬁned by Lin and Wang [9] and by Qing and Tian [15] to a bubble
tree convergence result. Speciﬁcally, they showed that, at both the ﬁnite and inﬁnite times,
enough bubbles appear to make the energy function continuous and the image of the limit
map u∞ (including bubbles) is connected. Notice that all of these results require passing to
subsequences.
In general, the limit of the solution u(tn ) as tn → ∞ is not unique and does depend on
2

the choice of the sequence tn of time. The uniformity of convergence of the heat ﬂow at
inﬁnity is proved for some interesting cases. For example, Topping [19] showed convergence
of the harmonic map heat ﬂow u(t) from S 2 to S 2 as t → ∞, assuming that u∞ and the
bubbles are all holomorphic or all anti-holomorphic maps as harmonic maps between two
S 2 ’s. In case of real analytic target manifolds, the uniformity of convergence of harmonic
map heat ﬂows follows by Simon [17].
In Chapter 2, we present details of the example of P. Topping that motivates our work
in this thesis. It is an example of a harmonic map heat ﬂow from a 2-sphere to a compact
Riemannian manifold that exists for all time but does not converge as t → ∞. Instead, it
exhibits ‘winding behavior’ — the ﬂow is asymptotic to a ﬂow that moves around a circle of
harmonic maps with constant speed.
Topping outlined his construction in Section 5 of [19]. He later gave a more sophisticated
example ([20]) for domains with boundary. Section 2.1 shows how the details in [20] yield
a complete proof of the construction of [19] (and ﬁx a small error in the original). For this,
we consider the manifold N = R2 × S 2 with the warped product Riemannian metric used
by Topping in [20], and then consider the harmonic map heat ﬂow u(t) from S 2 to N of the
form Topping uses in [19], namely u(q, t) = (p(t), q) where

p(t) = −∇f (p(t)).
˙

(0.2)

This gives the existence of a harmonic map heat ﬂow from S 2 into N having tension
uniformly bounded in L2 , but not converging anywhere as t → ∞, although there exist
sequences {tn } such that u(tn ) → u∞ in C ∞ as n → ∞. Roughly, one can think of u(t) as
S 2 ﬁbers “orbiting” the cylinder inﬁnitely many times with orbits asymptotic to the center
3

circle.
In fact, the accumulation points of the above ﬂow are maps u∞ that lie in the set M
of absolute minima of the energy function. In Section 2.2 we show that M is a manifold
diﬀeomorphic to S 1 × P SL(2, C) and we prove that the Hessian Hess(E) of the energy
function on the space of maps is degenerate as a bilinear form in the normal bundle along
M. The key observation of this thesis is that, in general, convergence is controlled by the
degeneracy or non-degeneracy of Hess(E) along the limiting set.

Chapters 3 - 5 develop criteria that insure a harmonic map heat ﬂow will converge as
t → ∞. Our approach is to avoid the bad behavior of the above example by controlling
Hessian of the energy function E(u(t)) along the heat ﬂow. Consider a harmonic map heat
ﬂow u(t) from a compact Riemann surface Σ into a compact Riemannian manifold N. From
Struwe’s theorem, we know that supΣ |du(t)| is necessarily unbounded at each ﬁnite singular
point. We also know that as t → ∞ there is a subsequence u(tn ) that converges weakly
in W 1,2 to a harmonic map u∞ . This limit may not involve bubbles, and there may be a
uniform bound on sup |du(t)| as t → ∞. In fact, if u(t) is very close to a stable harmonic
map u∞ for some large t then one expects that u(t) will ﬂow to u∞ without bubbles or
energy concentration points. For such ﬂows, we can translate time so that t = 0 corresponds
to a time, beyond the ﬁnite singular times, such that we have a uniform sup bound on |du|
valid for all t ≥ 0.
Thus we will suppose that supΣ×[0,∞) |du(t)| < C for some positive constant C. Further, if the Hessian of the energy E is positive deﬁnite at u∞ , one expects the exponential convergence of the solution, that is, there should be constants C and λ such that
dist(u(t), u∞) ≤ C eλt . Notice that this is an assumption about the Riemannian geom4

etry of the target space N. With these two assumptions, we prove the following result:
Theorem A. Let u : Σ2 ×[0, ∞) → N be a harmonic map heat ﬂow such that u(t) converges
to a harmonic map u∞ weakly in W 1,2 . Suppose
(a) the Hessian of the energy E is positive deﬁnite at u∞ .
(b) supΣ |du(t)| < C for all large t > T .
Then u(t) converges to u∞ exponentially fast in W 2,2 and hence in C 1 .

Consequently, under the hypothesis of Theorem A, u(t) converges uniquely to a harmonic
map u∞ , independent of any choice of subsequence. The proof of Theorem A is given in
Chapter 5.

The organization of this thesis is as follows.
In Chapter 2, we review Topping’s construction of a harmonic map heat ﬂow from a twodimensional domain that fails to converge and that behaves problematically as described
above.
In Chapter 3 and thereafter, we assume that the energy density of the harmonic map
heat ﬂow is uniformly bounded in t < ∞ for technical reasons. We consider the second
variation of the energy along the harmonic map heat ﬂow u(t) to justify the deﬁnition of a
symmetric bilinear tensor B along u(t). Then we prove that if the symmetric bilinear tensor
B is positive deﬁnite along a harmonic map heat ﬂow u(t) then u(t) converges exponentially
to a harmonic map in W 2,2 norm where a parabolic estimate for harmonic map heat ﬂows
plays an important role.
5

In Chapter 4, we discuss topologies on the space of maps to work with and then prove
that the symmetric bilinear tensor B is continuous on the space of maps in the W 1,2 ∩ C 0
topology. Roughly, this implies that the symmetric bilinear tensor B is positive deﬁnite at
u nearby the accumulation point u∞ under the assumptions (a) and (b) of Theorem A.
Chapter 5 contains the proof of Theorem A. For the proof, we take a sequence u(tn )
converging to u∞ in W 1,2 ∩ C 0 and show by contradiction that, once u(tn ) enters in a
δ
W 1,2 ∩ C 0 4 -neighborhood, it stays in a W 1,2 ∩ C 0 δ-neighborhood for all n. Hence our
main theorem follows from the exponential convergence in Chapter 3.
Finally, in Chapter 6, we present examples of geodesic heat ﬂows. This case is technically
simpler, but still displays interesting non-convergence behavior, and it provided motivation
for our work in Chapters 3 - 5. For the 1-dimensional domain, the harmonic map heat ﬂows
from S 1 are called geodesic heat ﬂows. Here one can use some facts from Morse theory,
working on the (inﬁnite-dimensional) manifold of W 1,2 loops in N. It is known that PalaisSmale compactness condition is satisﬁed on this loop space with its W 1,2 Riemannian metric,
but this does not insure convergence of downward W 1,2 gradient ﬂows (see [1]). The same
is true for geodesic heat ﬂows: they may not converge as t → ∞.
We illustrate such behavior by constructing explicit examples which show that the convexity of the energy function and convergence as t → ∞ may not hold. For the ﬁrst two
examples, we consider geodesic heat ﬂows from S 1 into a surface of revolution in R3 that are
equivariant under an S 1 action and then derive a general solution of the geodesic heat ﬂow.
As an application, we analyze the solution of the equivariant geodesic heat ﬂow from S 1 into
S 2 and show that the energy function is not convex. For the third example, we construct
the geodesic heat ﬂow u(t) : S 1 → T 3 , showing that the convergence fails as t → ∞.
6

Further studies on the geodesic heat ﬂows, including a C ∞ convergence theorem, are in
progress [1].

7

Chapter 1
Background
We deﬁne the basic concepts and notations of harmonic maps and give a short introduction
to harmonic map heat ﬂows in the light of the bubble tree convergence. In this thesis we
denote by Σ a closed Riemann surface, and by N a compact Riemannian manifold unless
speciﬁed otherwise.

1.1

Energy

Let (M, g) be a compact Riemannian manifold without boundary and (N, h) a compact
Riemannian manifold. If u : M → N is a smooth map, we consider du to be a u∗ T N1
valued 1-form of M. The energy density e(u) of u is deﬁned to be e(u) = 2 |du|2. In local
coordinates (xα ) and (ui ) around x and u(x), we have

e(u) =

∂ui ∂uj
1 αβ
g hij (u) α
.
2
∂x ∂xβ
8

The energy E(u) of u is
E(u) =

ˆ

M

e(u) dvolg

where dvolg is the volume form on (M, g). We note that, as a rule, Greek indices are used
for tensors on the domain manifold and Latin ones for tensors on the target manifold and
we use the summation convention.

1.2

Euler-Lagrange equation and harmonic maps

In this section, we derive the Euler-Lagrange equation of the energy E.

Let V ∈ Γ(u∗ T N) be a vector ﬁeld along the image of u : M → N and consider a family
of maps u(t) : M → N such that u(0) = u and

∂u(t)
= V . We consider the family u as
∂t t=0

a map u : M × R → N and let u denote ∂u . Then we have
˙
∂t
ˆ
d
d
e(u(t)) dvolg t=0
E(u(t)) t=0 =
dt
dt M
ˆ
1
∂
=
du(t), du(t) dvolg t=0
2 M ∂t
ˆ
=
∇ ∂ du(t), du(t) dvolg t=0
M ∂t
where ∇ is the covariant derivative in T ∗ (M × R) ⊗ u∗ T N. For X ∈ T M, we have
˙
˙
∇ ∂ du X = ∇X u + du([ ∂ , X]) = ∇X u,
∂t
∂t
9

since [ ∂ , X] = 0. Then
∂t
d
E(u(t)) t=0 =
dt

ˆ

M
ˆ

∇u(t), du(t) dvolg t=0
˙

∇V, du dvolg
M
ˆ
= −
V, τ (u) dvolg
M
=

where τ (u) = trace ∇du is called the tension ﬁeld of u. Hence the tension ﬁeld τ (u) of
u ∈ C 2 (M, N) is the negative of the gradient of the energy function with respect to the L2
Riemannian metric on the space of maps. In local coordinates (xα ) and (ui), the tension
ﬁeld τ (u) is given by

τ (u) =

∂
where ∆ui = √ 1
det g ∂xα

∂uj ∂uk
∆ui + g αβ Γi (u) α
jk ∂x ∂xβ
√

i
det g g αβ ∂u
β
∂x

∂
∂ui

=: τ i (u)

∂
∂ui

,

(1.1)

and Γi is the Christoﬀel symbol of N.
jk

Deﬁnition 1.1. A smooth map u : M → N is harmonic if it is a critical point of the energy
E.
Note that u is a harmonic map if and only if τ i (u) = 0 for all i. For systematic developments of the theory of harmonic maps we refer to [2], [3] and [4].

1.3

Existence of harmonic maps from a surface

As a generalization of the problem of ﬁnding harmonic functions, one can consider the
problem of the existence of harmonic maps. In many ways, the most interesting case is
10

when the domain is a 2-dimensional Riemannian manifold (Σ, g). Then, given a Riemannian
manifold (N, h), the existence problem takes this form:

Given ϕ : Σ → N, ﬁnd a harmonic map u : Σ → N with u homotopic to ϕ.

In general, the existence of nontrivial harmonic maps does not hold. For example, it is
known that there is no harmonic map of degree one from a smooth torus T 2 to the round
sphere S 2 (see Eells and Wood [5]).
A classical result on the existence of harmonic maps from S 2 was given by Sacks and
Uhlenbeck [16]. Unlike the 1-dimensional case, the Palais-Smale compactness condition for
the energy E in general fails on the space of maps of Σ with any reasonable topology. Hence
the direct minimization method does not apply in the 2-dimensional case. To overcome this,
Sacks and Uhlenbeck considered a perturbed α-energy Eα : for α > 1

Eα (u) =

ˆ

Σ

(1 + |du|2)α dvolg

for u in the separable Banach manifold W 1,2α (Σ, N) (See (1.3) below). Note that, for
α = 1, the critical points of E1 are harmonic maps. For α > 1, the α-energy Eα (u) satisﬁes
Palais-Smale compactness condition and hence (smooth) critical maps uα exist. Thus one
can consider a sequence uk of αk harmonic maps with αk → 1 and with Eα (uk ) uniformly
k
bounded. Sacks and Uhlenbeck showed that there exists a subsequence which converges
weakly in W 1,2 (S 2 , Rr ) to a limit u∞ and a further subsequence ul converges to u∞ in C 1
away from a ﬁnite set of points x1 , . . . , xK in Σ. It follows that (after using a removable
singularity theorem) u∞ is a smooth harmonic map. Furthermore, after renormalizing and
11

passing to a subsequence, they obtained a nontrivial harmonic 2-sphere at each singular point
xi to (partially) capture the energy loss as ul → u∞ . As an application, they answered the
above problem when the fundamental group π2 (N) is trivial: if π2 (N) = 0, for ϕ : Σ → N,
there exists a smooth harmonic map u : Σ → N homotopic to ϕ.
A nontrivial harmonic map of the 2-sphere S 2 → N is called a bubble when it arises by
a renormalization process like the one used by Sacks and Uhlenbeck.
For technical reasons, it is convenient to ﬁx an isometric embedding N ⊂ Rr and write
the harmonic map equation extrinsically. Then τ (u) is the tangential component of ∆u and
satisﬁes
τ (u) = ∆u + A(u) (∇u, ∇u) ,

(1.2)

where ∆u is the Laplacian of Rr (our sign convention is ∆ = g αβ ∂α ∂β + lower order terms)
and A is the second fundamental form of N ⊂ Rr and we abbreviate A(u) (∇u, ∇u) =
g αβ A(u)

∂u , ∂u .
∂xα ∂xβ

To proceed with the extrinsic setting, a natural choice for admissible family of maps is
the Sobolev space W 1,2 (Σ, N):

W 1,2 (Σ, N) = {u ∈ W 1,2 (Σ, Rr ) | u(x) ∈ N for a.e. x ∈ Σ},

(1.3)

where u ∈ W 1,2 (Σ, Rr ) if and only if ui ∈ W 1,2 (Σ) for each i = 1, . . . , r. It is well-known
that W 1,2 (Σ, N) is not embedded into C 0 (Σ, N) though the space C ∞ (Σ, N) is dense in
W 1,2 (Σ, N). In particular, dim Σ = 2 is the borderline case of the Sobolev embedding
theorem W 1,2 ֒→ C 0 . The “bubbling phenomena” of Sacks and Uhlenbeck is related to the
borderline failure of this Sobolev embedding and also to the invariance of the energy E(u)
12

under the conformal change of metrics on Σ.
Now we consider a critical point of the energy E among variations of a map in W 1,2 (Σ, Rr )
composed with the nearest point projection onto the compact Riemannian manifold N ⊂ Rr .
Deﬁnition 1.2. u ∈ W 1,2 (M, N) is a weakly harmonic map if
∆u + A(u) (∇u, ∇u) = 0 weakly

(1.4)

that is,
ˆ

M

− ∇u, ∇ϕ + A(u) (∇u, ∇u) , ϕ dvolg = 0

(1.5)

∞
for any ϕ ∈ Cc (M, Rr ).
A weakly harmonic map is called stationary if it is also a critical point of the energy
with respect to domain variations. In general, stationary weakly harmonic maps are not
necessarily smooth. We state results on the regularity of stationary weakly harmonic maps
without proof below. Proofs and references can be found in Moser [11], and Lin and Wang
[10].
Theorem 1.3. A stationary weakly harmonic map u : M → N is smooth away from a
singular set of vanishing (m − 2)-dimensional Hausdorﬀ measure where dim M = m.
Theorem 1.4. A continuous weakly harmonic map is smooth.
For the 2-dimensional case we have by H´lein [7]
e
Theorem 1.5. If dim Σ = 2, every weakly harmonic map of Σ is smooth.

13

1.4

Harmonic map heat ﬂow

Another approach on the existence of harmonic maps is the heat ﬂow method. This approach
is the main topic of this thesis.
Let M, N be Riemannian manifolds of any dimension. A harmonic map heat ﬂow is the
downward L2 gradient ﬂow of the energy E(u):

u(t) = τ (u(t))
˙
u(·, 0) = u0 ,

(1.6)
(1.7)

where u0 ∈ C ∞ (M, N) and we abbreviate u(t) = u(·, t) for t > 0.
In a seminal paper [6], Eells and Sampson showed that the nonlinear parabolic equation
(1.6) with the initial condition (1.7) has a short-time solution which is unique, and that if the
target manifold N is a compact Riemannian manifold with non-positive sectional curvature
then the solution to (1.6) exists for all time and therefore any smooth map u0 : M → N is
homotopic to a harmonic map.
When the domain is a compact Riemann surface Σ, Struwe [18] showed the following
existence theorem:
1,2
Theorem 1.6. Given u0 ∈ C ∞ (Σ, N), there exists a weak solution u ∈ W (Σ×[0, ∞), N)
loc
to the harmonic map heat ﬂow equations

u = ∆u + A(u) (∇u, ∇u)
˙
u(·, 0) = u0 .
14

(1.8)

satisfying
(a) u : Σ×[0, ∞) → N is smooth on Σ×[0, ∞) away from a ﬁnite set of points in Σ×(0, ∞),
(b) At each bubble point xT , a bubble ‘separates’, so that the energy function E(u(t)) is
decreasing in t,
(c) Near t = ∞ there exist a sequence of time tn → ∞ and a smooth harmonic map
u∞ : Σ → N such that un := u(tn ) converges to u∞ weakly in W 1,2 (Σ, N), and
strongly in W 2,2 away from a ﬁnite set of bubble points x∞ of Σ at each of which a
bubble ‘separates.’
We note that there are a ﬁnite number of bubble points at each singularity time and bubbles
separate in the sense that, for each bubble point xT with possibly T = ∞, there are sequences
xn → xT , tn → T , 0 < Rn → 0 as n → ∞ such that, after precomposing exponential map
expxn at xn , rescaling un ◦ expxn with scale Rn , it converges to a nonconstant harmonic
map of R2 → N with ﬁnite energy and hence induces, by the removable singularity theorem
[16], a nonconstant harmonic map via a ﬁxed stereographic projection.
By renormalizing sequences of harmonic maps to obtain “bubbles on bubbles”, Parker
[13] proved a “bubble tree convergence” theorem in which the limit map and bubbles are
connected for a sequence of harmonic maps from a compact Riemann surface Σ.
Qing and Tian [15], also considering maps from surfaces, proved that any Palais-Smale
sequence of the energy functional with uniformly L2 bounded tension ﬁelds τ (un ) converges,
after passing to a subsequence and relabeling, pointwise to the image of the limit map, called
the bubble tree map. Building on the “bubble tree convergence” theorem of Parker and
Wolfson [14] and Parker [13], Lin and Wang [9] gave another proof of the Qing and Tian’s
15

result.
In this thesis we examine the convergence of harmonic map heat ﬂows u(t) from Σ to N
as t → ∞. It is natural to study the asymptotic behavior of the harmonic map heat ﬂow as
t → ∞. More precisely, one can ask if the weak convergence is actually convergence in C 0 or
a stronger norm, and whether the convergence is independent of the choice of subsequences
as tn → ∞. The remainder of this thesis is devoted to answering these questions.

16

Chapter 2
Degenerate Hessian of the energy
In this chapter, we review an example of P. Topping of a harmonic map heat ﬂow that
exists for all time but does not converge as t → ∞. We then observe that Topping’s ﬂow is
asymptotic to a manifold M of limit maps u∞ , and that a key feature of the ﬂow is that
the Hessian of the energy is degenerate in the normal bundle to M.

2.1

Failure of convergence

Topping’s construction is outlined in [19], but for technical reasons it is necessary to replace
the ﬂat metric on the torus T 2 used in [19] with the modiﬁed metric that Topping uses in
[20] for a more sophisticated example. To start, consider the strip [-1, 1]×R with coordinates
(w, z) and with the metric

dw 2 + 2w 2 dw dz + (1 + w 4 ) dz 2 .
17

Let C be the cylinder obtained by taking the quotient of [-1, 1] × R by the group of isometries
Γ = {(w, z) → (w, z+n) | n ∈ Z}, and let N be the warped product C ×S 2 with the “warped
product” metric

h = dw 2 + 2w 2 dw dz + (1 + w 4 ) dz 2 + f (w, z) dα2 + sin2 α dθ2

where f is a smooth function of C deﬁned by the equation

f (w, z) =



− 2π

 1 + e |w|


√

2 + sin 2π 1 − z − 1
8
|w|




1

if w = 0
if w = 0.

Using coordinates x = 1 − z and y = z, the metric is
|w|
h = (x + y)−4 dx2 + dy 2 + f (x, y) dα2 + sin2 α dθ2

where
√
2 + sin 2π(x − 1 )
f (x, y) = 1 + e−2π(x+y)
8
and satisﬁes

for w = 0

∂f
(0, ·) ≡ 0. We consider a path of maps u(t) : S 2 → N in (x, y, α, θ) coordi∂x

nates by
u(t)(r, θ) =

0, y(t), arccos

−1 + r 2
, θ
1 + r2

where we parametrize S 2 by stereographic projection, so our domain is R2 with polar coordinates and with the conformally euclidean metric

g =

4
(1 + r 2 )2
18

dr 2 + r 2 dθ2 .

√
1
Theorem 2.1. y(t) = 2π ln 2 2π 2 (t + t0 ), for t0 > 0 suﬃciently large, yields a solution to
(1.6) for u(t):

y(t) = −
˙
=

√

∂f
(0, y)
∂y
2πe−2πy(t) .

(2.1)

Proof. Each u(t), as a map of (S 2 , g) → (S 2 , g), is the identity with respect to the metric g
on S 2 and thus harmonic, that is, τ α (u) ≡ 0 ≡ τ θ (u). We also have ∆x = 0 = ∆y, grθ = 0,
and all partial derivatives are zero except for ∂α = −2 2 , ∂θ = 1. Using the formula (1.1),
∂r
1+r ∂θ
we have
∂θ 2
∂α 2
+ g θθ Γx (u)
τ x (u) = g rr Γx (u)
αα
θθ
∂r
∂θ
(1 + r 2 )2 (−1)
∂f
(−2) 2
=
(x + y)4 (0, y)
4
2
∂x
1 + r2
(1 + r 2 )2 (−1)
∂f
4r 2
+
12
(x + y)4 (0, y)
2
∂x
4r 2
(1 + r 2 )2
∂f
= −(x + y)4 (0, y)
∂x
∂f
(0, ·) ≡ 0
= 0, since
∂x
∂θ 2
∂α 2
y
y
+ g θθ Γ (u)
τ y (u) = g rr Γαα (u)
θθ
∂r
∂θ
(1 + r 2 )2 (−1) ∂f
(−2) 2
=
(0, y)
4
2 ∂y
1 + r2
(1 + r 2 )2 (−1) ∂f
4r 2
+
12
(0, y)
2
2 )2
2 ∂y
4r
(1 + r
∂f
= − (0, y)
∂y
√
=
2πe−2πy .
19

1
for t0 > 0.
2π(t+t0 )
√
−C0
1
1
Write y(t) = 2π (C0 + ln (t + t0 )). Using (2.1), we set y =
˙
= 2πe
for t ≥ 0
t+t0
2π(t+t0 )
√
and then get C0 = ln 2 2π 2 .
Suppose that y(t) solves (2.1). Then y = −2π y 2 and so y(t) =
¨
˙
˙

The target manifold N is given [20] where Topping considered the harmonic map heat
ﬂow from the 2-disc D into his manifold N with a suitable initial boundary condition and
proved a ﬁnite time singularity is developed at center and is “winding” in the sense of [20].
Given t0 > 0 suﬃciently large, the tension ﬁeld τ (u(t)) satisﬁes
ˆ

|τ (u(t))|2 dvol 2 =
S
2
S

Vol(S 2 )
1
< C
4π 2 (t + t0 )2

for all t ≥ 0. Hence we have
Corollary 2.2. There exists a harmonic map heat ﬂow from S 2 into N having tension uniformly bounded in L2 , but not converging anywhere as t → ∞ although there exist sequences
{tn } such that u(tn ) → u∞ in C ∞ as n → ∞.
Proof. For t0 > 0, tn = t0 e2πn yields such a sequence u(tn ) → u∞ as n → ∞.

2.2

Degenerate Hessian of the energy

Deﬁne a family of smooth variations u(s, t) : S 2 → N in (w, z, α, θ) coordinates by
u(s, t)(r, θ) =

s, t, arccos
20

−1 + r 2
, θ .
1 + r2

(2.2)

1,2
Let W1 (S 2 , N) denote the space of W 1,2 maps from S 2 to N whose restriction to the
second component of N = C × S 2 is a degree 1 map S 2 → S 2 . Note that the group
P SL(2, C) of complex automorphisms of P1 = S 2 acts conformally on S 2 and acts on
N = C × S 2 by acting trivially on the ﬁrst factor. Composition then gives an action of
1,2
1,2
P SL(2, C) on W1 (S 2 , N): for γ ∈ P SL(2, C) and φ ∈ W1 (S 2 , N) the map γ · φ is the
composition γ ◦ φ.
1,2
Lemma 2.3. The absolute minima for the energy function E(u) on W1 (S 2 , N) is the set

M = { γ ◦ u(0, t) | γ ∈ P SL(2, C)},
which is diﬀeomorphic to S 1 × P SL(2, C).
Proof. Any map u that minimizes E(u) is harmonic, and hence smooth. Writing u =
(u1 , u2 ) : S 2 → C × S 2 , we have
ˆ
1
|du1 |2 + f |du2 |2 2 dvol
E(u) =
C
S
2 S2
ˆ
1
f |du2 |2 2 dvol
≥
2
S
2 S
ˆ
1
|du2 |2 2 dvol
≥
S
2 S2

where the ﬁrst inequality is strict unless u1 is a map to a point p ∈ C and the second
1
inequality is strict unless f (p) = 1, which occurs only if p lies on the circle S0 = {w = 0} in
1,2
1
C. Thus any u ∈ W1 (S 2 , N) with minimal energy has the form u = (p, u2 ) where p ∈ S0
and u2 is a degree 1 map S 2 → S 2 with minimal energy. But minimal energy degree 1 maps
21

P1 → P1 are holomorphic, and hence are elements of P SL(2, C).
1,2
Proposition 2.4. Along the minimum set M of the energy function E on W1 (S 2 , N),
the Hessian of E is degenerate.
Proof. We will show that, for each t, the maps u(s) = u(s, t) deﬁned by (2.2) are a smooth
is not
1-parameter family of maps with u0 in the absolute minimum set M, d u(s)
ds
s=0
tangent to M, and
d
= 0
E(u(s))
ds
s=0

and

d2
E(u(s))
ds2

= 0.

(2.3)

s=0

1
The energy density for u(s) is given by e(u(s)) = 2 e(s):

e(s) = f (s, 0) =



 − 2π
 e s ϕ(s) + 1 for s > 0,



1


for s ≤ 0

√
1
where ϕ(s) = sin 2π( 1 − 8 ) + 2 for s > 0 . Direct calculation shows that the derivatives of
s
ϕ(s) satisfy ϕ(n) = O(s−2n ) as s → 0, and that
2π
e′ (s) = e− s 2πs−2 ϕ(s) + ϕ′ (s)
2π
e′′ (s) = e− s 4π 2 s−4 − 2πs−3 + 2πs−2 ϕ′ (s) + ϕ′′ (s) .
Thus as s → 0 we have e′ (s) → 0 and e′′ (s) → 0 (and in fact e(n) (s) → 0 for all n). This
gives the variations (2.3). Also note that the description of M = S 1 × P SL(2, C) given in
the proof of Lemma 2.3 shows that d u(s)
= (1, 0, 0, 0) is not tangent to M.
ds
s=0

22

Chapter 3

Hessian of the energy and W 2,2
convergence of harmonic map heat
ﬂows

In this chapter, we prove that the harmonic map heat ﬂow u(t) converges exponentially to a
harmonic map in W 2,2 norm if the second variation of E(u(t)) is positive deﬁnite along u(t)
and the energy density e(u(t)) is uniformly bounded in t. Key ingredients of the proof are
the ﬁrst and second variations formulas (3.1) and (3.9) for the energy along the harmonic
map heat ﬂows, and W 2,2 estimates (3.4) on harmonic map heat ﬂows. For completeness
we record at the end of this chapter the proof of regularity of weakly harmonic maps from
Σ under the assumption that e(u(t)) is uniformly bounded in t.

23

3.1

Preliminary lemmas

We start with the following formulas for the harmonic map heat ﬂow.

Lemma 3.1. For a smooth harmonic map heat ﬂow u(t) : M × [0, T ) → N, we have
ˆ
d
E(u(t)) = −
|u(t)|2 dvolg
˙
dt
M

(3.1)

and
E(u(t)) +

ˆ tˆ
0

M

|u(t)|2 dvol dt = E(u0 ) ,
˙

∀t ∈ [0, T ).

(3.2)

Thus E(u(t)) is a nonincreasing function of t.

Proof. Multiplying (1.8) by u and integrating over M yields
˙

ˆ
d
E(u(t)) +
|u(t)|2 dvolg = 0 ∀t ∈ [0, T ) .
˙
dt
M

(3.3)

Then (3.2) follows by integrating in t.

For simplicity, we note that sup |du| are used interchangeably with |du|∞ and numbered
Σ
constants below depend only on the geometry of Σ and N and |du|∞ .
Lemma 3.2. Let u(t) : M × [0, ∞) → N be a harmonic map heat ﬂow. If sup |du(t)| < C
Σ
∀t > T for some T ≥ 0, we have
u(t) − u(s) 2,2 ≤ c1

u(t) − u(s) 2 + u(t) − u(s) 1,2
˙
˙
24

∀s, t > T.

(3.4)

Proof. Regarding u(t) as maps into Rr , let v := u(t) − u(s). Then v is a weak solution of
∆v = (u(t) − u(s)) − Φ(u(t), u(s))
˙
˙

(3.5)

where Φ(u(t), u(s)) := A(u(t))(du(t), du(t)) − A(u(s))(du(s), du(s)). By the elliptic estimate
for the Laplacian, we have

v 2,2 ≤ c2 ( ∆v 2 + v 2 )
≤ c2 ( u(t) − u(s) 2 + Φ(u(t), u(s)) 2 + u(t) − u(s) 2 ) .
˙
˙

(3.6)

Since A is a smooth symmetric bilinear tensor,

Φ(u(t), u(s)) 2 ≤

(A(u(t)) − A(u(s))) (du(t), du(t)) 2
A(u(s))(du(t) − du(s), du(t) + du(s)) 2

+

≤ c3 |∇A|∞ |du(t)|2 u(t) − u(s) 2
∞
+ c4 |A|∞ |du(t) + du(s)|∞ u(t) − u(s) 1,2
≤ c5 u(t) − u(s) 1,2

∀t, s > T.

(3.7)

Thus (3.4) follows from (3.6) and (3.7).

Lemma 3.3. If u(x, t) satisﬁes the harmonic map heat ﬂow equation then

1
˙
− ∂t |u|2 =
2

∇∗ ∇u, u + tr Ru (u, du)u, du ,
˙ ˙
˙
˙
25

(3.8)

and hence
∆|u|2 = ∂t |u|2 + 2 |∇u|2 + tr Ru (u, du)u, du
˙
˙
˙
˙
˙

.

(3.9)

Proof. By assumption, u(t) satisﬁes u = tr∇du. Pick (x0 , t0 ) ∈ M × R and an orthonormal
˙
frame {eα } on a neighborhood U of x0 such that [eα , et ] = 0, ∇et eα = 0 and ∇et (∇eα eβ ) =
0 at each point in U and we may assume (∇eα )x = 0 and hence [eα , eβ ]x0 = 0, ∀ α, β.
0
Then at the point (x0 , t0 ),

∇t u = ∇t (tr ∇du)
˙
= ∇t (∇α ∇α u)
= ∇α (∇t ∇α u) + Ru (u∗ et , u∗ eα )u∗ eα
= ∇α (∇α ∇t u) + Ru (u, u∗ eα )u∗ eα
˙
= −∇∗ ∇u + tr Ru (u, du)du,
˙
˙

(3.10)

˙ ˙
where we abbreviate ∇α = ∇eα and ∇t = ∇et . Noting that 1 ∂t |u|2 = ∇t u, u gives (3.8)
2 ˙
and (3.8) together with
∆|u|2 = −2 ∇∗ ∇u, u + 2|∇u|2
˙
˙ ˙
˙

(3.11)

yields (3.9).
Deﬁnition 3.4. For u ∈ C ∞ (M, N) a symmetric bilinear tensor B on u∗ T N is deﬁned as
Bu (V, V ) =

ˆ

M

|∇V |2 + trg Ru (V, du)V, du dvolg

∀V ∈ u∗ T N

(3.12)

Remark 3.5. If u is a harmonic map then Bu is the Hessian of the energy at u, i.e. the
26

usual second variation of the energy at u.

W 2,2 convergence of harmonic map heat ﬂows

3.2

The following theorem holds for any closed Riemannian manifold M of any dimension.

Theorem 3.6. Let u(t) : M × [0, ∞) → N be a smooth harmonic map heat ﬂow and suppose
(a) there exist T ≥ 0 and λ > 0 such that
Bu(t) (u(t), u(t)) ≥ λ u(t) 2
˙
˙
˙
2

∀t > T.

(3.13)

Then u(t) converges in L2 exponentially fast in t to a map u∞ ∈ L2 (M, N).
(b) If further sup |du(t)| < C ∀t > T , the convergence u(t) → u∞ is in W 2,2 and u∞ is a
Σ
weakly harmonic map. If dim M ≤ 3 then the convergence is in C 0 and if dim M = 2
then u∞ is a smooth harmonic map.
Proof.

(a) Let E(t) denote the energy E(u(t)) of the solution u(t). Then

E ′ (t) = −

ˆ

M

τ (u(t)), u(t) dvolg = − u(t) 2 ≤ 0.
˙
˙
2

(3.14)

Integrating both sides of (3.9) (noting that ∂M = ∅), and then using (3.13), we have
E ′′ (t) = 2Bu(t) (u(t), u(t)) ≥ 2λ u(t) 2 = −2λE ′ (t)
˙
˙
˙
2
27

(3.15)

for all t. Integrating in t,

E ′ (t) ≥ E ′ (T ) e−2λt = − τ (u(T )) 2 e−2λt ,
2

(3.16)

and so
τ (u(T )) 2 e−λt

u(t) 2 ≤
˙

∀t > T.

(3.17)

Since the map of t ∈ [0, ∞) → u(t) ∈ L2 (M, Rr ) is C 1 , we have
u(t) − u(s) 2 ≤

ˆ s
t

u(τ ) 2 dτ ≤
˙

τ (u(T )) 2 −λt
e
λ

(3.18)

for all T < t ≤ s (the ﬁrst inequality in (3.18) is proved in Palais [12]). Hence {u(t)} is
Cauchy in L2 and so limt→∞ u(t) = u∞ exists and is unique in L2 (M, Rr ). Moreover,
u∞ (x) ∈ N almost everywhere x, since u(t)(x) ∈ N ∀(x, t).
(b) Now suppose that |du(t)|∞ < C ∀t > T . By interpolation, let C = C(ǫ) > 0 be a
positive constant such that

u(t) − u(s) 1,2 ≤ ǫ u(t) − u(s) 2,2 + C(ǫ) u(t) − u(s) 2 .

(3.19)

1
Taking ǫ = 2c and using (3.4), we have
1

u(t) − u(s) 2,2 ≤ c6 ( u(t) − u(s) 2 + u(t) − u(s) 2 )
˙
˙
≤ c7 e−λt

∀t ≤ s,

(3.20)

where c7 = c6 τ (u(T )) 2 2 + 1 . Thus u(t) is also Cauchy in W 2,2 and so converges
λ
28

to u∞ in W 2,2 as t → ∞. Moreover, (3.17) implies
∆u∞ + A(du∞ , du∞ ) 2 = lim ∆u(t) + A(du(t), du(t)) 2 = lim u(t) 2 = 0.
˙
t→∞
t→∞
If dim M ≤ 3 then C 0 convergence follows from the Sobolev embedding theorem. If
dim M = 2 then the well-known theorem of H´lein [7], or Lemma 3.8 below, implies
e
that u∞ is a smooth harmonic map.

Remark 3.7. Under the hypotheses of Theorem 3.6 with M = Σ2 , u∞ is the unique harmonic map in the weak W 1,2 closure of the ﬂow {u(t) | t > T }. In particular, the weakly
convergent subsequences {un } of Struwe’s Theorem 1.6 all converge to u∞ .
Instead of using H´lein’s theorem, we can use the following bootstrap argument in the
e
last sentence of the proof of Theorem 3.6.
Lemma 3.8. If dim Σ = 2, any weakly harmonic W 1,2 map u : Σ → N with sup |du| < C
Σ
is smooth.
Proof. Using sup |du| < C and H¨lder inequality, we have u ∈ W 2,p (Σ, N) ∀p ∈ (2, ∞),
o
Σ
since

u 2,p ≤ c8 ( ∆u p + u p )
≤ c9 ( A(u)(du, du) p + u p )
≤ c10 (1 + u 2 ),

(3.21)

where the constant c10 depends on p > 2 and Vol(Σ). By the Sobolev embedding theorem
29

W 2,p ֒→ C 1 , u ∈ C 1 (Σ, N).
Using the Bochner formula for functions,

|∆∇α u| = |∇α ∆u + Rαβ ∇β u|
≤ ∇α A(u)(du, du) + Rαβ ∇β u
≤ |∇A|∞ |du| + 2|A|∞ |∇2 u| |du| + |R|∞ |du|
≤ c11 |∇2 u|2 + |du|2 + |du|2 ,
where the last inequality holds by Young’s inequality. By (3.21), we have

u 3,p ≤ c12 u 2
2,2p < ∞
for all p < ∞. Hence u ∈ C 2 (M, N). By induction, we have
∆∇α1 · · · ∇αk+1 u = ∇α1 · · · ∇αk+1 ∆u + lower order terms.
Hence u k+1,p ≤ c13

k
k
l=0 u l,p ∀p and thus u ∈ C (M, N) for each k.

30

(3.22)

Chapter 4
Continuity of Hessian
In this chapter we discuss topologies on the space of maps to work with and then prove
that the symmetric bilinear tensor B is continuous on the space of maps in the W 1,2 ∩ C 0
topology. Roughly, this implies that the symmetric bilinear tensor B is positive deﬁnite at
u nearby the accumulation point u∞ under the assumptions (a) and (b) of Theorem A.

4.1

Sobolev spaces of maps

For a closed Riemann surface Σ let M denote the completion of C ∞ (Σ, N) with respect to
the W 1,p norm (p > 2) where N is isometrically embedded in Rr as above. For dim Σ = 2,
2
1− p
(Σ, N) and
the Sobolev embedding theorem implies that M embeds continuously in C
hence M is a C 2 separable Banach manifold. Then Tu M is the set of W 1,p vector ﬁelds
along image:
Tu M =

p
V ∈ Γ u∗ T N | V 1,p < ∞ ,
31

(4.1)

´
p
where V 1,p = M |∇V |p + |V |p dvolg . Hence M carries a weak L2 Riemannian metric
given by
V, W 2 =

ˆ

Σ

V, W dvolg

and a weak W 1,2 Riemannian metric given by

V, W 1,2 =

ˆ

Σ

∇V, ∇W + V, W dvolg

(4.2)

for V, W ∈ Tu M. Note that Tu M is not complete with respect to either L2 or W 1,2
topology.
Lemma 4.1. Suppose that the symmetric bilinear form B is positive deﬁnite at u ∈ M with
respect to

·, · 2 : there exists λ > 0 such that
∀V ∈ u∗ (T N).

Bu (V, V ) ≥ λ V 2
2
Then B is positive deﬁnite at u with respect to

(4.3)

·, · 1,2 : there exists µ > 0 depending on

Σ, N, λ and |du|∞ such that
Bu (V, V ) ≥ µ V 2
1,2

∀V ∈ u∗ (T N).

(4.4)

The constant µ > 0 is uniform on any set of maps with a uniform bound |du|∞ < C.
Proof. Using the H¨lder inequality and the interpolation inequality,
o

ˆ

Σ

ˆ
2 |du|2 dvol +
|V |
|V |2 dvolg
g
Σ
Σ
ˆ
≤ Bu (V, V ) + 2|R|∞ V 2 du 2 +
|V |2 dvolg
4
4
Σ

|∇V |2 + |V |2 dvolg ≤ Bu (V, V ) + 2|R|∞

ˆ

≤ Bu (V, V ) + c14 ǫ ∇V 2 + (1 + C(ǫ)) V 2 ,
2
2
32

where c14 = c14 (|R|∞ , |du|∞, Vol(Σ)). Taking ǫ = 2c1 and absorbing the second term
14
to the left hand side,
V 2 ≤ 2Bu (V, V ) + c15 V 2
1,2
2
c
≤ 2 + 15 Bu (V, V ).
λ

4.2

Continuity of Hessian

Fix an isometric embedding N ֒→ Rr with a second fundamental form A. For small ǫ > 0 and
let Nǫ be an ǫ-neighborhood of N in Rr such that the nearest projection map proj : Nǫ →
N is well-deﬁned. Let u, v ∈ C ∞ (Σ, N). For V ∈ Γ(u∗ T N), there is a corresponding
ˆ
ˆ
V ∈ v ∗ T N such that V = πv V via the composition
πv
u∗ T N ֒→ u∗ T Rr ⋍ Σ × Rr ⋍ v ∗ T Rr → v ∗ T N
where πv : v ∗ Rr → v ∗ T N is the orthogonal projection to the image of v. Let D denote
the Levi-Civita connection on Rr . For X ∈ T Σ, A(u)(u∗ X, V ) is the normal component of
DX V such that

πu = dproju ,
Dπu (X, V ) = A(u)(u∗ X, V ),
∇X V = DX V − A(u)(u∗ X, V ).
33

Note the sign convention.
Now extend A smoothly to a tensor in a neighborhood Nǫ of N in Rr . For Theorem 4.3
we estimate the diﬀerence between the second fundamental forms at two points of N.
Lemma 4.2. There exists a constant c16 > 0 depending on the geometry of N such that

ˆ ˆ
|A(u)(V, W )| − |A(v)(V , W )|
ˆ
ˆ
ˆ
≤ c16 |u − v| |V | |W | + |V − V | |W | + |V | |W − W | ,

(4.5)

where c16 > 0 depends on the geometry of N.
Proof. Because N is compact, both |A| and |∇A| are bounded. We can then write
ˆ ˆ
A(u)(V, W ) − A(v)(V , W )
ˆ
ˆ
ˆ
= (A(u) − A(v))(V, W ) + A(v)(V − V , W ) + A(v)(V , W − W ),
and note that the Mean Value Theorem implies that |A(u) − A(v)| ≤ C dist(u, v) for some
C > 0. Lemma 4.2 follows because the Riemannian distance dist(u, v) in N is uniformly
equivalent to the euclidean distance |u − v|.
Deﬁne the W 1,2 ∩ C 0 topology by the norm
|||u − v||| :=

u − v 1,2 + sup |u(x) − v(x)|
x∈Σ

for maps u, v : Σ → N ֒→ Rr .

34

(4.6)

Theorem 4.3. Suppose there exists λ > 0 such that
Bv (V, V ) ≥ λ V 2
2

∀V ∈ Γ(v ∗ T N).

For C > 0 given, there exist µ = µ(Σ, N, λ, C) > 0 and a W 1,2 ∩ C 0 neighborhood U of v
such that
Bu (V, V ) ≥ µ V 2
2

∀V ∈ Γ(u∗ T N)

for all u ∈ UC := U ∩ {sup |du| < C}.
Σ

Proof. Fix v : Σ → N and let δ1 < 1 inj(N) be a small positive number to be chosen later
2
and set δ := sup dist(u(x), v(x)) < δ1 . Write
x∈Σ
ˆ ˆ
Bu (V, V ) − Bv (V , V )
ˆ
ˆ
2 − |∇V |2 dvol +
ˆ
ˆ
ˆ
=
|∇V |
trg Ru (V, du)V, du − trg Rv (V , dv)V , dv dvolg
g
Σ
Σ
= I + II.

(4.7)

(I) Let {eα } be a local orthonormal frame. For V ∈ Γ(u∗ T N), we have
ˆ
ˆ
ˆ
∇α V = Dα V − A(v)(v∗ eα , V )
ˆ
= Dα (πv V ) − A(v)(v∗ eα , V )
ˆ
= Dπv (v∗ eα , V ) + πv (Dα V ) − A(v)(v∗ eα , V )
ˆ
= A(v)(v∗ eα , V − V ) + πv (Dα V ),
where
35

πv (Dα V ) = (πv − πu )(Dα V ) + πu (Dα V )
= (πv − πu )(Dα V ) + ∇α V.
Applying the Mean Value Theorem with π = dproj, we have the bounds

|πv − πu |

≤

c17 δ,

ˆ
|V − V | = |(πv − πu )V |

≤

c18 δ |V |.

(4.8)

Noting

|Dα V | ≤ |∇α V | + |A|∞ |du|∞ |V |

≤

c19 (|∇α V | + |V |),

we have
ˆ
|∇α V | − |∇α V |

ˆ
≤ |A(v)(v∗ eα , V − V )| + |(πv − πu )(Dα V )|
≤ c20 δ(|∇α V | + |V |),

(4.9)

where c20 = c20 (|A|∞ , |du|, |dv|). And also,
ˆ
|∇α V | + |∇α V | ≤ 2 |∇α V | + c21 δ(|∇α V | + |V |)
≤ c22 (|∇α V | + |V |),
where c22 = c22 (|A|∞ , |du|, |dv|). Then (4.9) and (4.10) yield

36

(4.10)

I

≤
≤
≤

ˆ

ˆ
ˆ
(|∇V | + |∇V |) |∇V | − |∇V | dvolg
ˆ
c23 δ
(|∇V | + |V |)2 dvolg
Σ
c24 δ V 2 .
1,2
Σ

(4.11)

(II) For N ֒→ Rr the Gauss-Codazzi equation becomes

R(X, Y )Z, W

= A(X, W ) · A(Y, Z) − A(X, Z) · A(Y, W )

for X, Y, Z, W ∈ T N. Hence we have
ˆ
ˆ
R(u)(V, u∗eα )V, u∗ eα − R(v)(V , v∗ eα )V , v∗ eα

≤ A1 + A2 ,

where

A1 :=

ˆ
|A(u)(V, u∗eα )|2 − |A(v)(V , v∗ eα )|2 ,

ˆ ˆ
A2 := A(u)(V, V ) · A(u)(u∗ eα , u∗ eα ) − A(v)(V , V ) · A(v)(v∗ eα , v∗ eα ) .
ˆ
(A1 ) Noting (4.8), |V | ≤ |V | and Lemma 4.2, we have
A1 ≤

ˆ
|A(u)(V, u∗ eα )| + |A(v)(V , v∗ eα )|

ˆ
|A(u)(V, u∗eα )| − |A(v)(V , v∗ eα )|

≤ c25 |V | ( δ |V | + |du − dv| |V |)
= c25 ( δ |V |2 + |du − dv| |V |2 ),
where c25 = c25 (|∇A|∞ , |A|∞ , |du|, |dv|).
37

(4.12)

(A2 ) Using Lemma 4.2,

A2

ˆ ˆ
A(u)(V, V ) − A(v)(V , V ) A(u)(u∗ eα , u∗ eα )

≤

ˆ ˆ
+ A(v)(V , V ) A(u)(u∗ eα , u∗ eα ) − A(v)(v∗ eα , v∗ eα )
≤

c26 δ |V |2 + c27 |V |2 (δ + |du − dv|)

≤

c28 (δ|V |2 + |du − dv| |V |2 ),

(4.13)

where c28 = c28 (|∇A|∞ , |A|∞, |du|, |dv|). But the H¨lder inequality and the Sobolev
o
embedding W 1,2 ֒→ L4 give

ˆ

Σ

|du − dv| |V |2 dvolg ≤

du − dv 2 V 2
4

≤ c29 u − v 1,2 V 2 .
1,2

(4.14)

Combining (4.12), (4.13) and (4.14), we have

II ≤ c30 δ + u − v 1,2

V 2 .
1,2

(4.15)

Hence, from (4.11) and (4.15),

ˆ ˆ
Bu (V, V ) ≥ Bv (V , V ) − c31 δ + u − v 1,2

V 2 ,
1,2

(4.16)

where c31 = c31 (Σ, N, |dv|∞ , |du|∞).

Now we prove Theorem 4.3. Using Lemma 4.1 and (4.8), one can choose a positive
38

number µ1 > 0, depending on |du|∞ , such that
ˆ ˆ
Bv (V , V ) ≥ µ1 V 2
1,2

∀V ∈ Γ(u∗ T N).

(4.17)

µ
Let δ1 > 0 be a positive number with δ1 < 2 c1 . If δ + u − v 1,2 < δ1 , we have, from
31
(4.16) and (4.17),

Bu (V, V ) ≥ µ1 V 1,2 − c31 δ1 V 2
1,2
µ
≥ 1 V 2 .
1,2
2

39

Chapter 5
Proof of the main theorem
In this chapter we complete the proof of our main result Theorem A. To that end, we consider
a harmonic map heat ﬂow u(t) : Σ → N whose energy density is uniformly bounded:
sup |du(t)| ≤ C
Σ

for all t > T0 .

(5.1)

Lemma 5.1 below shows that this bound implies that the ﬂow is adherent to a smooth
harmonic map u∞ at t → ∞. We then add the hypothesis that this limit u∞ is stable, that
is, the Hessian of the energy function is positive deﬁnite (in the sense of Deﬁnition 3.4) at
u∞ . With these hypotheses, Proposition 5.2 implies that at some large time, the ﬂow enters
and remains in a neighborhood of u∞ in which the symmetric bilinear tensor B is positive
deﬁnite. Theorem A then follows from the exponential convergence proved in Chapter 3.

To start, consider a harmonic map heat ﬂow u(t) : Σ → N satisfying (5.1). This assumption implies that the maps u(t) are uniformly bounded in W 1,p for any p, so by the
compactness of the Sobolev embedding W 1,p → C 0 for p > 2, there is a sequence tn → ∞
40

such that the maps u(tn ), which we denote as un , converge in C 0 . In fact, using the heat
ﬂow, we can also assume that the un converges in W 1,2 as follows.
Lemma 5.1. For any harmonic map heat ﬂow u(t) : Σ → N satisfying (5.1) , there is
a sequence tn → ∞ such that the maps un = u(tn ) converge in W 1,2 ∩ C 0 to a smooth
harmonic map u∞ .
Proof. As above let tn → ∞ be a sequence such that un converges in C 0 and so does in L2 .
By Lemma 3.1 we have
ˆ t+1 ˆ
t

Σ

|u(t)|2 dvolg → 0 as t → ∞.
˙

(5.2)

Hence, after passing to a subsequence and relabeling, there exists a sequence tn → ∞ such
that un 2 → 0 as n → ∞. Under the assumption (5.1), the elliptic estimate for ∆ shows
˙
un 2,2 ≤ c32

un 2 + un 1,2 .
˙

Interpolating the second term and absorbing to the left side, we have

un 2,2 ≤ c33 ( un 2 + un 2 ) .
˙
Hence un converges to u∞ in W 2,2 , and so in W 1,2 . Moreover, u∞ is a harmonic map,
since un 2 → 0 as n → ∞.
˙
The W 1,2 ∩ C 0 norm |||u||| = u 1,2 + sup |u(x)| used in (4.6) deﬁnes neighborhoods of
x∈Σ
u∞
Uδ := {u | |||u − u∞ ||| < δ}.
41

Proposition 5.2. Let u(t) : Σ × [0, ∞) → N be a harmonic map heat ﬂow and let δ1 > 0
be the constant in the proof of Theorem 4.3. Suppose that
(a) B is positive deﬁnite at u∞
(b) sup |du(t)| < C
Σ

∀t.

For each δ < δ1 there exist a T > 0 such that if t > T , u(t) lies in the neighborhood Uδ,C of
u∞ deﬁned by
Uδ,C := Uδ ∩ {u |sup |du| < C}.
Σ
Proof. Choose a sequence tn → ∞ such that un converges to a harmonic map u∞ in
W 1,2 ∩ C 0 as in Lemma 5.1. By Theorem 4.3 with v = u∞ we have
Bu (V, V ) ≥ µ V 2
2

∀V ∈ Γ(u∗ T N)

(5.3)

for u ∈ Uδ,C . After passing to a subsequence, we can assume that
|||u(tn ) − u∞ ||| <

δ1
4

∀n.

Note that Theorem 1.6 implies that u is smooth on Σ×(0, ∞) except for ﬁnitely many points
of Σ × (0, ∞) and hence we can assume u ∈ C ∞ (Σ × [t1 , ∞), N). For each n, let
Tn := sup{t | |||u(s) − u∞ ||| < δ1 ∀s ∈ [tn , t]}.
If Tn = ∞ for some n, we are done. Suppose Tn < ∞ for each n. We then have Tn > tn
and |||u(tn ) − u∞ ||| = δ1 for each n, since |||u(s) − u∞ ||| is a continuous function in s on
42

[t1 , ∞). Note that Bu(s) satisﬁes the condition (5.3) for all s ∈ [tn , Tn ). Then by the
Sobolev inequality and (3.20) we have

|||u(s) − u(tn )||| ≤ c34 u(s) − u(tn ) 2,2 ≤ c35 e−µtn

∀s ∈ [tn , Tn )

(5.4)

for each n.
1
Choose n large enough that the right side of (5.4) is less than 4 δ1 . Then

|||u(s) − u∞ ||| ≤ |||u(s) − u(tn )||| + |||u(tn ) − u∞ ||| <

δ
δ1 δ1
+
= 1
4
4
2

1
for all s ∈ [tn , Tn ). Hence, we have |||u(Tn ) − u∞ ||| ≤ 2 δ1 , a contradiction.
Theorem A. Let u : Σ2 × [0, ∞) → N be a smooth harmonic map heat ﬂow such that u(t)
converges to a harmonic map u∞ weakly in W 1,2 . Suppose that
(a) the Hessian of the energy E is positive deﬁnite at u∞ and
(b) supΣ |du(t)| < C for all large t > T .
Then u(t) converges to u∞ exponentially fast in W 2,2 and hence in C 1 .
Proof. By Proposition 5.2 and Theorem 4.3, one can ﬁnd a T > 0 so that the symmetric
bilinear tensor B is positive deﬁnite along u(t) for all t > T . Hence Theorem A follows from
Theorem 3.6.

43

Chapter 6
Examples of geodesic heat ﬂows
A classical result in Riemannian geometry asserts that every map u0 : S 1 → M into a closed
Riemannian manifold (M, h) is homotopic to a closed geodesic. Intuitively, this can be
proven by deforming u0 by following the ﬂow of the downward gradient vector ﬁeld of the
energy function
ˆ
1
|du|2 dθ
E(u) =
2 S1

(6.1)

on the free loop space of maps u : S 1 → M. It is standard to do this using the W 1,2 gradient
ﬂow on the space of maps, but alternatively one can use the heat ﬂow. Speciﬁcally, given an
W 1,2 map u0 : S 1 → M, we can consider the “geodesic heat ﬂow”
u = ∇T T
˙

u(θ, 0) = u0 (θ)

(6.2)

One expects that this heat ﬂow produces maps u(t) : S 1 → M that converge to a closed
geodesic at t → ∞. Standard results (cf. [8], [1]) show that the ﬂow exists for all time, that
u(t) is smooth for each t > 0 and that there is a sequence tn → ∞ so that u(tn ) converges
44

in W 1,2 to a closed geodesic. But, surprisingly, it is not known whether the ﬂow converges
(without extracting a sequence).
Actually, convergence is claimed in [8]. Unfortunately, the proof relies on the assertion
that the energy E(t) = E(u(t)) along the ﬂow is a convex function of t, and the proof of this
assertion in [8] has an error. In this chapter we give examples of geodesic heat ﬂows that
show that

• the energy function E(t) need not be convex, and
• the geodesic ﬂow may not converge.
Convergence results for the geodesic ﬂow will be given in [1].

6.1

Geodesic heat ﬂows on surfaces of revolution

In this section, we derive the equation for the harmonic map heat ﬂow u(t) from S 1 into a
surface of revolution M ⊂ R3 , for u(t) being equivariant for the standard S 1 action that
ﬁxes the z-axis.
Consider a surface of revolution M in R3 obtained by rotating a curve r = f (z) ≥ 0
about the z-axis and imagine a circle S 1 sitting in M as in Figure 6.1 below. Introducing
the angle coordinate θ to S 1 and the cylindrical coordinates (r, θ, z) to R3 , one writes

u(θ, t) = ( f (z(t)) , θ , z(t) ) ∈ M ⊂ R3

(6.3)

Since the surface of revolution M is the graph of the map ϕ : [0, 2π) × R → M given by
45

z

(M, h)

(S 1 , g)
eiθ

p(r, θ, z)
u
y
x

Figure 6.1: u : (S 1 , g) → (M, h)

ϕ(θ, z) = (f (z), θ, z), we can use ϕ−1 as (global) coordinates for M as
ϕ−1 ◦ u(θ, t) = ( θ , z(t) ) := ( u1 (θ, t) , u2 (θ, t) ).

(6.4)

In coordinates θ for S 1 and (u1 , u2 ) for M, the metric on the domain is given by

g = dθ2 ,

(6.5)

and the metric on M is given by

h =
=

dr 2 + r 2 dθ2 + dz 2 |M
df 2 2
dz + f 2 dθ2 + dz 2
dz

= f 2 dθ2 + (f ′ )2 + 1 dz 2
46

(6.6)

df
for f ′ = . Then the geodesic heat ﬂow equation u = ∆u is given in coordinates by
˙
dz

u = g θθ
˙

2
i=1




∂ 2 ui
∂θ2

2
+
j,k=1


j ∂uk
∂u
 ∂ .
Γi (u)
jk
∂θ ∂θ
∂ui

(6.7)

Using the deﬁnition of u1 and u2 , the right side of (6.7) reduces to

∂
2 ∂
u = Γ1
˙
11 ∂u1 + Γ11 ∂u2
∂
1
∂
= 0
− h22 h11,2
2
∂u1
∂u2
f f′
∂
= −
′ )2 + 1 ∂u2
(f

(6.8)

where we used the formula for the Christoﬀel symbols for M.
On the other hand, we have u = z ∂ 2 . Hence the harmonic map heat ﬂow equation is
˙
˙
∂u
a nonlinear ﬁrst order ODE: for z = z(t),

z = −
˙

6.2

f (z) f ′ (z)
.
f ′ (z) 2 + 1

(6.9)

The energy function E(t) needn’t be convex.

We will ﬁnd the solution of a S 1 equivariant harmonic map heat ﬂow u(t) from S 1 into S 2
whose energy function E(u(t)) is not convex. For future use, we present two ways of ﬁnding
the solution via cylindrical and spherical coordinates.
(Method 1) In cylindrical coordinates (r, θ, z), S 2 as a surface of revolution about the z-axis
is obtained by rotating the graph r =

1 − z 2 . Applying (6.9) with f (z) =
47

1 − z2 ,

the S 1 equivariant harmonic map heat ﬂow equation is

z = z(1 − z 2 ).
˙

(6.10)

Separating variables and taking integrals, we have

dz
=
z(1 − z 2 )

ˆ

ˆ

dt = t + C

(6.11)

for some constant C. Using the partial fraction expansion, the left integral of (6.11) is

ˆ

dz
= ln
z(1 − z 2 )

|z|

1 − z2

.

Noting z = cos α in spherical coordinates (θ, α) on S 2 , we have

et+C =

± cos α
= cot α.
1 − cos2 α

Thus we have, for a constant C,

α = arccot et+C ,
z = cos arccot et+C .

(6.12)

(Method 2) Let (M, h) be the unit sphere S 2 in R3 . In spherical coordinates (ρ, θ, α) in R3 ,
48

the harmonic map heat ﬂow equation for maps u : S 1 × [0, ∞) → S 2 is
u = τ (u)
˙
2

= g θθ

i=1




∂ 2 ui
∂θ2

2
+
j,k=1


j ∂uk
∂u
 ∂
Γi (u)
jk
∂θ ∂θ
∂ui

(6.13)

where u(θ, t) = ( θ , α(t) ) := ( u1 (θ, t) , u2 (θ, t) ) in spherical coordinates (θ, α) of S 2
(see Figure 6.2).
Since ρ ≡ 1 on S 2 , the metric h on S 2 is given by
h =

dρ2 + ρ2 sin2 α dθ2 + ρ2 dα2 |M

= sin2 α dθ2 + dα2 .

(6.14)

Then the same computation as in (6.8) shows

u = α
˙
˙

∂
∂u2

= −

∂
∂
1 22
1
h h11,2
= − sin 2α
.
2
2
2
∂u
∂u2

(6.15)

Thus the harmonic map heat ﬂow equation is

1
α = − sin 2α.
˙
2

(6.16)

Using the relation z = cos α between two coordinates on S 2 , it is straightforward that
(6.16) is equivalent to (6.10).
We compute the energy function E(t) with the solution α(t) = arccot et+C where the
constant C will be determined later. From (6.5) and (6.14), the energy density e(u(t)) of
49

(S 1 , g)

(S 2 , h)

z

eiθ

p(1, θ, α)

α
u

θ

y

x

Figure 6.2: u : (S 1 , g) → (S 2 , h)

u(θ, t) = ( u1 (θ, t) , u2 (θ, t) ) = (θ, α(t)) is

e(u(t)) =

1
1
1
|du(t)|2 = h11 =
sin2 α,
2
2
2

2
1
since ∂u = 1 and ∂u = 0. Consequently, the energy E(t) is
∂θ
∂θ
ˆ
π e−2(t+C)
1
.
|du(t)|2 dθ = π sin2 α(t) =
E(t) =
2 S1
1 + e−2(t+C)

This formula implies that if the initial map u(0) is a circle close to the north pole, say
α(0) = .01, then u(t) approaches the point map to the north pole exponential fast, and the
energy function is convex. However, if the initial map is a circle close to the equator in the
northern hemisphere, say α(0) = 0.49π = arccot eC , then z(t) increases slowly at ﬁrst, then
rapidly, and then approaches z = 1 exponentially as in Figure 6.3(a). Correspondingly, the
energy E(t) decreases slowly at ﬁrst, then rapidly, and then approaches 0 exponentially as
in Figure 6.3(b). In particular, the energy function E(t) is not convex.

50

z

E

1.0

3

0.8
2

0.6
0.4

1

0.2
2 3 4 5 6 t
(b) E(t) for the same ﬂow.

1 2 3 4 5 6 t
(a) z(t) for α(0) = 0.49π

1

Figure 6.3: E(t) is not convex.

6.3

A non-converging geodesic heat ﬂow

In this ﬁnal section, we construct an explicit geodesic heat ﬂow u(t) : S 1 → (M, h) that does
not converge as t → ∞. In this example, (M, h) is the 3-dimensional torus T 3 with a warped
product metric — essentially the metric used for the harmonic map heat ﬂow described in
Chapter 2. Again, this is based on the example of Topping in [20].
Let (w, z, θ) be the standard coordinates for the ﬂat 3-torus T 3 and ﬁx a smooth cutoﬀ
function η with support on [−2, 2], with 0 ≤ η ≤ 1 and with η ≡ 1 on [−1, 1]. Deﬁne a
metric h on T 3 as

h = dw 2 + 2η(w)w 2dw dz + 1 + η(w)w 4 dz 2 + ψ(w, z)dθ2

where ψ(w, z) is the smooth function given by

ψ(w, z) =




1


1


 1 + η(w)e− |w| 1 + sin 1 − z

w
51

if w = 0, ±π
otherwise.

1
If x = w − z and y = z with 0 < w < 1, the metric is written as
h = (x + y)−4 dx2 + dy 2 + ψ(x, y)dθ2

where ψ(x, y) = 1 + e−x−y (1 + sin x).
We consider the geodesic heat ﬂow for maps u : S 1 → T 3 of the form
u(t)(θ) = ( 0, y(t), θ)

(6.17)

in (x, y, θ) coordinates. Then the image of u(t) lies in the product of the curve {(0, y)} ⊂ T 2
∂y
= 0 and ∂θ = 1, we have
and S 1 . Noting
∂θ
∂θ
τ x (u(t)) = g θθ Γx
θθ

1
∂θ 2
= − hxx hθθ,x = 0,
∂θ
2

(6.18)

∂ψ
(0, y) = 0. Similarly,
since hθθ,x =
∂x
τ y (u(t)) = g θθ Γ

y
θθ

τ θ (u(t)) = g θθ Γθ
θθ

1
1 ∂ψ
∂θ 2
= − hxx hθθ,y = −
(0, y) = e−y
∂θ
2
2 ∂y
∂θ 2
= hθθ,θ = 0.
∂θ

Since u = (0, y, 0), the geodesic heat ﬂow equation is
˙
˙

y = e−y .
˙

(6.19)

1
We take y(0) = π. Then the geodesic heat ﬂow u(t) starts from (w, z) = ( π , π) and lies in
the set where η ≡ 1. Solving (6.19) for y, we have y(t) = ln(t + eπ ) and so the geodesic heat
52

ﬂow is
u(t)(θ) = (w, z, θ) =

1
, ln(t + eπ ), θ .
ln(t + eπ )

This ﬂow does not converge anywhere as t → ∞ although there exist sequences {tn } such
that un → u∞ in C ∞ as n → ∞.

53

BIBLIOGRAPHY

54

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[3] J. Eells and L. Lemaire, Selected topics in harmonic maps, C.B.M.S. Regional Conf.
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Vol. 20, (1988) 385–524.
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