Eigenvalues and energy functionals with monotonicity formulae under Ricci flow

0.2 MB·English

by  Junfang Li

Mathematische Annalen manuscript No.
(will be inserted by the editor)
Eigenvalues and energy functionals with
monotonicity formulae under Ricci flow
7 Jun-Fang Li
0
0 Received:date / Revised version: date– (cid:13)c Springer-Verlag2008
2
n Abstract. Inthisnote,weconstructfamiliesoffunctionalsofthetypeofF-functional
a andW-functionalofPerelman.Weprovethatthesenewfunctionalsarenondecreasing
J under the Ricci flow. As applications, we give a proof of the theorem that compact

6 steady Ricci breathers must be Ricci-flat. Using these new functionals, we also give a
2 new proof of Perelman’s no non-trivial expanding breather theorem. Furthermore, we
prove that compact expanding Ricci breathers must be Einstein by a direct method.

] Inthisnote,wealso extendCao’s methodsofeigenvalues[1]andimprovetheirresults.
G
Mathematics Subject Classification (2000): 53C44; 35K55
D
.
h
t
a
m 1. Introduction
[
Let (M,g) be a closed Riemannian manifold. In [15], Perelman intro-

2 duced a functional
v
8 (g,f) = (R+ f 2)e−fdµ. (1)
4 F |∇ |
5 ZM
1 If (M,g) is a solution to the Ricci flow equation, Perelman proved that
0
the -functional is nondecreasing under the Ricci flow and the Ricci flow
7 F
0 can be viewed as the gradient flow of this functional. He proved that
/ under the following coupled system:
h
t
a ∂ g = 2R
m ∂t∂ifj = −∆fij R+ f 2, (2)
: (cid:26) ∂t − − |∇ |
v
the -functional is nondecreasing. More precisely,
i F
X
d
r = 2 R + f 2e−fdµ 0. (3)
ij i j
a dtF | ∇ ∇ | ≥
ZM
If one defines
λ(g) = inf (g,f),
F
Jun-Fang Li
Mathematical Sciences Research Institute, Berkeley, CA 94720, USA
email : [email protected]
2 Jun-Fang Li
where the infimum is taken over all the smooth f which satisfies
e−fdµ = 1,
ZM
then λ(g) is the lowest eigenvalue of the operator
4∆+R,
−
and the nondecreasing of the functional implies the nondecreasing of
F
λ(g). As an application, Perelman was able to show that there is no non-
trivial steady or expanding Ricci breathers on closed manifolds.
In [1], X.D. Cao considered the eigenvalues of the operator
R
∆+ ,
− 2
on manifolds with nonnegative curvature operator. They showed that the
eigenvalues of these manifolds arenondecreasingalong theRicci flow. Us-
ing the monotonicity of the eigenvalues, they proved that the only steady
Riccibreatherwithnonnegative curvature operatoristhetrivialonewhich
is Ricci-flat.
In this note, we study monotonicity formulae of various energy func-
tionals and the eigenvalues of the operator 4∆+kR. In section 2,1 we
study a monotonicity formula of eigenvalues−of ∆+ R, which improves
− 2
the result in [1] based on the same technique Cao has used. Thefollowing
is one of the main theorems in this paper :
Theorem 11 Let g(t), t [0,T), be a solution to the Ricci flow on a
∈
closed Riemannian manifold Mn. Assume that there is a C1-family of
smooth functions f(t)> 0, which satisfy
1
λ(t)f(t) = ∆ f(t)+ R f(t) (4)
g(t) g(t)
− 2
f2(t)dµ = 1 (5)
g(t)
ZM
where λ(t) is a function of t only. Then
2dλ(t) =4 R if jfdµ+2 Rc2f2dµ
dt ij∇ ∇ | |
Z Z
= R + ϕ2 e−ϕdµ+ Rc2 e−ϕdµ (6)
ij i j
| ∇ ∇ | | |
Z Z
0,
≥
1 Theresultsinsection 2arerelatively independentoftherestofthepaperandcan
be treated separately.
Monotonicity formulae underRicci flow 3
where ϕ satisfies e−ϕ = f2.
In section 3, we introduce a functional and derive its first variation
E
along the Ricci flow. In section 4, we define new functionals and we
k
F
prove that although the Ricci flow is not a gradient flow of in the
k
F
sense of Perelman’s, are still nondecreasing under the Ricci flow. The
k
F
monotonicity is strict unless we are on a Ricci-flat solution.
In section 5, by using the monotonicity of the eigenvalues, we rule out
compact steady Ricci breathers.
In section 6, we introduce a family of new functionals and .
e ek
W W
These functionals are not scale invariant in contrast to Perelman’s -
W
entropy. We obtain their first variations :
Theorem 12 Under the following coupled system
∂
g = 2R
ij ij
∂t −
 ∂ f = ∆f + f 2 R
 ∂dt − |∇ | −
 τ = 1,
dt


we have the followingmonotonicity,2
d 1
=2τ2 R + f + g 2e−fdµ
ek ij i j ij
dtW ∇ ∇ 2τ
+Z2M(k(cid:12)(cid:12) 1)τ2 Rij + 1 gi(cid:12)(cid:12)j 2e−fdµ,
− 2τ
ZM
(cid:12) (cid:12)
where k 1. The monotonicity is strict(cid:12) unless we a(cid:12)re on an expanding
≥
gradient Ricci soliton (k = 1) or an Einstein manifold (k > 1).
In section 7, we use the non-scale-invariant functional to prove
e
W
Perelman’s no non-trivial expanding breather theorem.
In section 8, we use functionals to rule out non-Einstein compact
ek
W
expanding Ricci breathers.
Remark 13 Thenonon-Einsteinsteady orexpanding Riccibreather the-
orem was first proved by T. Ivey in [11]. We give a different proof here
which is based on a global functional method and does not use maximal
principles.
Remark 14 The no non-trivial expanding Ricci breather theorem was
proved by Perelman in [15] by discussing the monotonicity of a scale in-
variant eigenvalue λ¯(g) = λ(g)V n2(g). We present a different proof in
section 7.
2 Seethedefinition of W in (34)
ek
4 Jun-Fang Li
Remark 15 For compact shrinking Ricci breathers, one can find refer-
ences in [15], [4], [5], [10], [11] and others.
Throughoutthispaper,weuseEinsteinconvention, i.e.repeatedindex
implies summation.
2. Eigenvalues of (−∆+ R) on manifolds
2
Let (M,g(t) be a compact Riemannian manifold, where g(t) is a
smooth solution to the Ricci flow equation on 0 t < T. In [1], Cao
≤
considers the problem from a viewpoint different from the entropy func-
tional method. They study the eigenvalues λ and eigenfunctions f of the
operator ∆+ R with the normalization f2dv = 1. First, they define
− 2
: Zm
R
λ(h,t) = ( ∆h+ h)hdµ, (7)
− 2
ZM
where h is a smooth function satisfying d( h2dµ) = 0 and h2dµ =
dt
ZM ZM
1.
They then obtain the monotonicity formula under the non-negative
curvature operator assumption as following:
Theorem 21 [1] On a compact Riemannian manifold with nonnegative
curvature operator, the eigenvalues of the operator ∆+ R are nonde-
− 2
creasing under the Ricci flow, i.e.
d
λ(f,t) = 2 R f f dµ+ Rc2f2dµ 0. (8)
dt |t=t0 ij i j | | ≥
Z Z
In this theorem, f denotes the covariant derivative of f with respect
i
to ∂ ,(also denotedas ∂ )andin(8), dλ(f,t)is evaluated attime t = t
∂xi i dt 0
and f is the correspondingeigenvalue at time t . As a direct consequence
0
of Theorem 21, they prove the following
Corollary 22 [1] There is no compact steady Ricci breather with non-
negative curvature operator, other than the one which is Ricci-flat.
Remark 23 Clearly, at time t, if f is the eigenfunction of the eigenvalue
λ(t), then λ(f,t) = λ(t).
In this section, based on (8), we will drop the curvature assumption
on the manifold and prove Theorem 11.
Monotonicity formulae underRicci flow 5
Proof. (Theorem 11) Let ϕ be a function satisfying f2(x) = e−ϕ(x) and
plug it into (8), we have
2dλ(t) = 2dλ(f,t)= R iϕ jϕe−ϕdµ+2 Rc2e−ϕdµ. (9)
dt dt ij∇ ∇ | |
Z Z
Using the divergence theorem and rearrangements, we derive the first
term of the last identity
R iϕ jϕ e−ϕdµ = iR jϕ e−ϕdµ+ R i jϕ e−ϕdµ.
ij ij ij
∇ ∇ ∇ ∇ ∇ ∇
Z Z Z
(10)
By the contracted second Bianchi identity “ iR = 1 R” and inte-
∇ ij 2∇j
gration by parts, we have
(10) = 1 R jϕ e−ϕdµ+ R i jϕ e−ϕdµ
2 ∇j ∇ ij∇ ∇
Z Z (11)
= 1 R∆e−ϕdµ+ R i jϕ e−ϕdµ.
2 ij∇ ∇
Z Z
This also implies the following
1
R i jϕ e−ϕdµ = R iϕ jϕ e−ϕdµ R∆e−ϕdµ. (12)
ij ij
∇ ∇ ∇ ∇ − 2
Z Z Z
On the other hand, using integration by parts and symmetry of the hes-
sian of functions, we have
ϕ2 e−ϕdµ = ϕ i jϕ e−ϕdµ ϕ i jϕ e−ϕdµ
j i j i
|∇∇ | − ∇ ∇ ∇ ∇ − ∇ ∇ ∇ ∇
Z Z Z 1
= ϕ j iϕ e−ϕdµ+ ϕ2∆e−ϕdµ.
j i
− ∇ ∇ ∇ ∇ 2|∇ |
Z Z
(13)
By the commutator formulae for covariant derivatives which are known
as Ricci identities, see page 286 in [3], we have
iϕ= iϕ Rk kϕ = iϕ+R kϕ, (14)
i j j i iji j i kj
∇ ∇ ∇ ∇ ∇ ∇ − ∇ ∇ ∇ ∇ ∇
where Rl represents the Riemann curvature (3,1)-tensor and R de-
ijk kj
notes the Ricci curvature tensor. We use the following convention of Rie-
mann curvature tensor through out this paper:
R (X,Y)Z [ , ]Z Z,
m X Y [X,Y]
≡ ∇ ∇ −∇
where X,Y,Z are vector fields on the manifold.
{ }
6 Jun-Fang Li
Combing (13), (14), and the contracted second Bianchi identity, we
have
1
ϕ2 e−ϕdµ = ϕ( i iϕ R kϕ) e−ϕdµ+ ϕ2∆e−ϕdµ
j j jk
|∇∇ | − ∇ ∇ ∇ ∇ − ∇ 2|∇ |
Z = ϕ jZ∆ϕ e−ϕdµ+ R jϕ kϕ e−ϕdµ+ Z 1 ϕ2∆e−ϕdµ
j jk
− ∇ ∇ ∇ ∇ 2|∇ |
Z Z Z
= ∆e−ϕ∆ϕdµ+ ( R kϕ+R j kϕ) e−ϕdµ
j jk jk
− ∇ ∇ ∇ ∇
Z 1 Z
+ ϕ2∆e−ϕdµ
2|∇ |
Z 1
= ∆e−ϕ∆ϕdµ R kϕ e−ϕdµ R j kϕ e−ϕdµ
k jk
− − 2∇ ∇ − ∇ ∇
Z 1 Z Z
+ ϕ2∆e−ϕdµ
2|∇ |
Z 1 1
= ∆e−ϕ ∆ϕ+ R ϕ2 dµ R j kϕ e−ϕdµ.
jk
− 2 − 2|∇ | − ∇ ∇
Z Z
(cid:0) (cid:1) (15)
We notice that we are free to change the dummy index from i,j,k,l
{ }
to other index or exchange among them whenever necessary.
Combing (12) and (15), we have the following
2R i jϕe−ϕdµ + ϕ2e−ϕdµ
ij
∇ ∇ |∇∇ |
Z Z 1 1
= R j kϕ e−ϕdµ ∆e−ϕ ∆ϕ+ R ϕ2 dµ
jk
∇ ∇ − 2 − 2|∇ |
Z Z 1
= R iϕ jϕ e−ϕdµ ∆e−ϕ(cid:0) ∆ϕ+R ϕ2 (cid:1)dµ
ij
∇ ∇ − − 2|∇ |
Z Z
(cid:0) (16)(cid:1)
Recall (4), then a simple calculation yields
1
2λ(t) = ∆ ϕ+R ϕ2 (17)
g(t) g(t) − 2|∇ |g(t)
Plugging (17) into (16), by divergence theorem on closed manifolds,
we have
2R i jϕe−ϕdµ + ϕ2e−ϕdµ
ij
∇ ∇ |∇∇ |
Z Z
= R iϕ jϕ e−ϕdµ 2λ(t)∆e−ϕdµ (18)
ij
∇ ∇ −
Z Z
= R iϕ jϕ e−ϕdµ.
ij
∇ ∇
Z
Monotonicity formulae underRicci flow 7
In the end, we plug (18) into (9) and have the following
2dλ(t)= R iϕ jϕe−ϕdµ+ Rc2e−ϕdµ+ Rc2e−ϕdµ
dt ij∇ ∇ | | | |
Z Z Z
= R + i jϕ2 e−ϕdµ+ Rc2 e−ϕdµ (19)
ij
| ∇ ∇ | | |
Z Z
0
≥
Remark 24 One can find applications of Theorem 11 in [1] ( e.g. The-
orem 3). When the lowest eigenvalues are concerned, the eigenfunctions
are always smooth and positive, see in [4]. For the monotonicity of eigen-
values of ordinary Laplace operator under Ricci flow, one can see a result
in [13].
Remark 25 For evolution of Yamabe constant under Ricci flow, see a
recent preprint [2]. We thank Xiao-Dong Cao for pointing out the refer-
ence to us.
3. Construct entropy functional from first variation formula
In this section, we will construct a functional and show that it is
E
monotone along the Ricci flow.
Under the following coupled system:
∂ g = 2(R + f)
∂t ij − ij ∇i∇j (20)
∂ f = ∆f R,
(cid:26) ∂t − −
one has the following first variation formula, (see [15])
d
= 2 R + f 2e−fdµ 0, (21)
ij i j
dtF | ∇ ∇ | ≥
ZM
where is definedas in (1). Hence,themodifiedRicci flowcan beviewed
F
as an L2 gradient Ricci flow of Perelman’s -functional.
F
One natural question to ask is : can we find a functional such that
E
the‘honest’RicciflowistheL2 gradientRicciflowofitinacertainsense?
Namely, one expects that under the following coupled system :
∂ g = 2R
∂t ij − ij (22)
∂ f = ∆f R+ f 2,
(cid:26) ∂t − − |∇ |
there is a functional such that the following is true,
E
d
= 2 Rc2e−fdµ 0. (23)
dtE | | ≥
ZM
8 Jun-Fang Li
It turns out that this can be done by integrating the above first variation
formula over a time interval [0,t]. We carry out the computations as the
following
t d t
(t) (0) = ds = 2 Rc2e−fdµds
E −E dsE | |
Z0 t Z0 ZM
= 2 gijgklR R e−f det g ds dxn (24)
ik jl
= ZM Zt0∂gjlR e−f det pg ds dxn
jl
∂t
ZM Z0
p
The last step follows from a lemma in [3], see page 67, Lemma 3.1.
Integration by parts yields
s t ∂R
(t) (0) = gjlR e−f det g gjl jle−f det g ds
jl
E −E − ∂t
ZM (cid:20)t ∂e−pf√det(cid:12)(cid:12)g0 Z0 p
gjlR (cid:12) ds dxn
− jl ∂t (cid:12)
Z0 s t (cid:21)∂R
= Re−fdµ gjl jle−f det g ds
− ∂t
ZM t (cid:12)(cid:12)∂0e−fZ√Md(cid:20)etZg0 p
+ gjlR (cid:12) ds dxn
jl(cid:12) ∂t
Z0 (cid:21)
(25)
Using Lemma 3.5 in [3] and the fact that metric tensor g is a covariant
constant, by letting h = 2Rc, we have the following 3
−
∂R
gjl jl = gjlgpq( R + R R R )
∂t − ∇q∇j lp ∇q∇l jp−∇q∇p jl−∇j∇l pq (26)
= ∆R.
Intheabovecomputations,wehaveusedthecontractedsecondBianchi
identity. With the help of Lemma 3.9 in [3], We obtain
∂e−f√det g
= ( ∂ f R)e−f√det g (27)
∂t −∂t −
3 As suggested by thereferee, a simpler way to obtain equation (26) is :
∂∂Rt = ∆R+2|Rc|2 =gij∂R∂tij +Rij∂∂gtij
= gij∂Rij +2|Rc|2.
∂t
Monotonicity formulae underRicci flow 9
Plug (26) and (27) into (25), we have
s t ∂
(t) (0) = Re−fdµ ∆R e−f +R( f R)e−f dµ ds
E −E − −∂t −
= ZM Re−fdµ(cid:12)(cid:12)(cid:12)s0 Z0tZM R(cid:20) ∆f + f 2 ∂ f R (cid:21)e−f dµ ds
(cid:12) − − |∇ | − ∂t −
ZM (cid:12)0 Z0 ZM (cid:20) (cid:21)
= Re−fdµ(cid:12)(cid:12) Re−fdµ .
(cid:12) −
ZM (cid:12)s = t ZM (cid:12)s = 0
(cid:12) (cid:12) (28)
(cid:12) (cid:12)
The last step follows from(cid:12)the third equation i(cid:12)n the coupled system (22).
The computation above suggests that we could define -functional as
E
= Re−fdµ. (29)
E
ZM
Next, we prove the monotonicity property for -functional directly.
E
Proposition 31 Assume g(t) satisfies the Ricci flow equation over the
time interval [0,T), and also function f satisfies the evolution equation
(2) (also (22)), then
d
E = 2 Rc2e−fdµ.
dt | |
ZM
Proof. (First proof of Proposition 31) Using ∂R = ∆R+2Rc2, we
∂t | |
derive the following
dR d
d Re−fdµ = e−fdµ+R (e−fdµ)
dt dt dt
ZM ZM
= (∆R+2Rc2)e−fdµ+ R( f R)e−fdµ
t
| | − −
ZM ZM
= 2 Rc2e−fdµ+ R[ ∆f + f 2 f R]e−fdµ
t
| | − |∇ | − −
ZM ZM
= 2 Rc2e−fdµ.
| |
ZM
The last equality comes from the second equation of the coupled sys-
tem (2).
(Second proof) Under the following coupled system with modified
Ricci flow (see also similar system in (35)),
∂
g = 2 R + f
ij ij i j
∂t − ∇ ∇ (30)
 (cid:18) (cid:19)
∂ f = ∆f R,
 ∂t − −

10 Jun-Fang Li
we have the following
d Re−fdµ = (∆R R f +2Rc2)e−f dµ
dt −∇ ∇ | |
ZM ZM
=2 Rc2e−f dµ.
| |
ZM
By using the diffeomorphism invariance, see e.g. Proposition 1.2 in [15],
we prove that under the original coupled system we still have
d Re−fdµ = 2 Rc2e−f dµ.
dt | |
ZM ZM
Remark 32 We will apply the diffeomorphism invariance principle gen-
erally in this paper. See details in the proof of Corollary 66 and Corollary
67.
4. Entropy functionals Fk and their monotonicity
Definition 41 We define the following variations of -functional,
F
(g,f) = (kR+ f 2)e−fdµ, (31)
k
F |∇ |
ZM
where k 1. When k = 1, this is the -functional.
≥ F
Nextwederivethemonotonicityformulaforthesefunctionals (g,f).
k
F
Theorem 42 Suppose the Ricci flow of g(t) exists for [0,T), then all the
functionals (g,f) will be monotone under the coupled system (22), i.e.
k
F
d (g ,f)= 2(k 1) Rc2e−fdµ+2 R + f 2e−fdµ 0
dtFk ij − | | | ij ∇i∇j | ≥
ZM ZM
(32)
Furthermore, the monotonicity is strict unless the Ricci flow is a trivial
Ricci soliton and f is a constant function. Namely, the metric is Ricci-
flat.
Proof. (Theorem 42) Under the coupled system (2)
∂ g = 2R
∂t ij − ij
( ∂∂tf = −∆f −R+|∇f|2,
we have shown in Proposition 31 that,
d
Re−fdµ = 2 Rc2e−fdµ.
dt | |
ZM ZM