Compact Gradient Shrinking Ricci Solitons with Positive Curvature Operator

0.1 MB·English

by  Xiaodong Cao

Compact Gradient Shrinking Ricci Solitons
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0 with Positive Curvature Operator
0
2

n
Xiaodong Cao
a
J
[email protected]
5
2
February 2, 2008

]
G
D
. Abstract
h
t
a In this paper, we first derive several identities on a compact shrinking Ricci
m
soliton. We then show that a compact gradient shrinking soliton must be
[
Einstein, if it admits a Riemannian metric with positive curvature operator

1 and satisfies an integral inequality. Furthermore, such a soliton must be of
v
constant curvature.
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9
5
1 1 Introduction and Main Theorems
0
6
0 Hamilton started the study of the Ricci flow in [2]. In [3], Hamilton has classified all
/
h compact manifolds with positive curvature operator in dimension four. Since then,
t
a the Ricci flow has become a powerful tool for the study of Riemannian manifolds,
m
especially for those manifolds with positive curvature. Perelman made significant
: progress in his recent work [5] and [6].
v
i Suppose we have a solution to the Ricci flow
X
r
a ∂
g = −2R (1.1)
ij ij
∂t
on a compact Riemannian manifold M with Riemannian metric g(t). Ricci soliton
emerges as the limit of the solutions of the Ricci flow. A solution to the Ricci flow is
called a Ricci soliton if it moves only by a one-parameter groupof diffeomorphism and
scaling. If the vector field which induce the diffeomorphism is in fact the gradient of
a function, we call it a gradient Ricci soliton. For a gradient shrinking Ricci soliton,
we have the equation
1
R +∇ ∇ f = g , (1.2)
ij i j ij
2τ
1
where τ = T − t. T is the time the soliton becomes a point, and f is called Ricci
potential function. In the special case when f is a constant, then we have an Einstein
manifold.
Besides the above equation, a gradient shrinking Ricci soliton must also satisfies
the following equations,
n
R+△f = (1.3)
2τ
and
f −c
2
R+|∇f| = , (1.4)
τ
where c is a constant in space. The last equation (1.4) determines the value of f.
The Ricci potential function f satisfies the following evolution equation,
∂
2
f = |∇f| . (1.5)
∂t
Inspired by his own work in [3] and [4], Hamilton made the following conjecture:
Conjecture 1. (Hamilton) A compact gradient shrinking Ricci soliton with positive
curvature operator must be Einstein.
On the other hand, it is a well-known theorem of Tachibana [8] that any compact
Einstein manifold with positive sectional curvature must be of constant curvature.
Hence Hamilton’s conjecture is a generalization of the Tachibana theorem, since Ein-
stein manifolds are special Ricci solitons with constant Ricci potential functions.
In this paper, we first derive a sequence of identities on gradient shrinking Ricci
solitons. Then we show that the above conjecture is in fact true provided that the
Ricci soliton satisfies an integral inequality.
One of our main theorems is the following:
Theorem 1. Let (M,g(t)) be a compact gradient shrinking Ricci soliton, then M
must be of constant curvature if its curvature operator is positive and satisfies the
following inequality,
1
|Rc|2|∇f|2e−f ≤ Ke−f + R R f f e−f , (1.6)
2 Z Z Z ijkl ik j l
where
K = (∇ ∇ R −∇ ∇ R )R . (1.7)
i j ik j i ik jk
In Section Two, we first derive some integral identities about Riemannian curva-
ture on gradient shrinking Ricci solitons. More precisely, we prove the following two
identities,
2
Theorem 2. On a compact gradient shrinking Ricci soliton, we have
1 1
Rm(Rc,Rc)e−f = |Rc|2e−f + |div Rm|2e−f (1.8)
Z 2τ Z 2 Z
and
1 1
Rm(Rc,Rc)e−f = |Rc|2e−f + |∇Rc|2e−f − |div Rm|2e−f , (1.9)
Z 2τ Z Z 2 Z
where
Rm(Rc,Rc) = R R R . (1.10)
ijkl ik jl
As a corollary of Theorem 2, we have
Corollary 1. On a compact gradient shrinking Ricci soliton, we have
|∇Rc|2e−f = |div Rm|2e−f . (1.11)
Z Z
Moreover, (1.8) and (1.9) can be written as follows,
1 1
Rm(Rc,Rc)e−f = |Rc|2e−f + |∇Rc|2e−f . (1.12)
Z 2τ Z 2 Z
In Section Three, we derive some identities about Ricci curvature, i.e., we show
the following theorem,
Theorem 3. On a compact gradient shrinking Ricci soliton, we have
1 1
|Rc|2△(e−f) = |∇Rc|2e−f + Ke−f + R R f f e−f . (1.13)
2 Z 2 Z Z Z kljp kj l p
In Section Four, we prove Theorem 1 under the hypothesis of positive curvature
operator and inequality (1.6).
Acknowledgement:We would like to thank Professor Gang Tian, for first bring-
ing this problem to our attention, and for many valuable suggestions during this
work. We would also like to thank Professor Richard Hamilton, for his patience and
guidance during this work.
We are indebt to Professor Bennett Chow, who shared his own notes in this
direction with us. We would like to thank him for his generous comments. We would
also like to thank Professor Tom Ilmanen and Professor Duong H. Phong for many
helpful discussion.
3
2 Identities of Riemannian Curvature
In this section, we prove Theorem 2. On a gradient shrinking Ricci soliton, we have
the following identities:
(div Rm) = R = ∇ R = ∇ R
jkl ijkl,i i ijkl i klij
=−∇ R −∇ R = ∇ R −∇ R
k ijli l ijik k jl l jk
=−∇ f +∇ f = ∇ ∇ f −∇ ∇ f
k jl l jk l k j k l j
=R f . (2.1)
lkjp p
Hence we have the following two identities,
∇ (R e−f) = 0 (2.2)
i ijkl
and
∇ (R e−f) = 0 . (2.3)
i ik
Using integration by parts, we derive that
|div R |2e−f
Z m
= R f (−R +R )e−f
Z lkjp p jk,l jl,k
= R f R e−f − R f R e−f
Z lkjp p jl,k Z lkjp p jk,l
=− R f R e−f + R f R e−f
Z lkjp pk jl Z lkjp pl jk
=− R R f e−f − R R f e−f
Z lkjp lj kp Z kljp kj lp
=−2 R R f e−f .
Z lkjp lj kp
Hence we have the following lemma:
Lemma 1. On a gradient shrinking Ricci soliton, we have
1
R R f e−f = − |div Rm|2e−f ≤ 0 . (2.4)
Z lkjp lj kp 2 Z
Now we can prove (1.8) in Theorem 2.
Proof. By the above lemma and the gradient shrinking Ricci soliton equation:
1
f = g −R ,
kp kp kp
2τ
4
we can derive
|div R |2e−f
Z m
1
=−2 R R ( g −R )e−f
Z lkjp lj 2τ kp kp
1
=− |Rc|2e−f +2 Rm(Rc,Rc)e−f ,
τ Z Z
so we have
1 1
Rm(Rc,Rc)e−f = |Rc|2e−f + |div Rm|2e−f .
Z 2τ Z 2 Z
Before we prove (1.9), we first prove the following two lemmas:
Lemma 2.
∇ ∇ R −∇ ∇ R = R R −R R (2.5)
i j ik j i ik jm mk ijmk im
Proof. Using the formula
∇ ∇ R −∇ ∇ R = −R R −R R ,
i j lk j i lk ijml mk ijmk lm
and let i = l in the above formula and take the sum.
Lemma 3. On a gradient shrinking Ricci soliton,
1
−2 ∇ R ∇ R e−f = |Rc|2e−f −2 Rm(Rc,Rc)e−f . (2.6)
Z k jl l jk τ Z Z
Proof.
−2 ∇ R ∇ R e−f
Z k jl l jk
=2 R (∇ ∇ R −∇ R f )e−f
Z jk i j ik j ik i
=2 R (∇ ∇ R )e−f −2 R ∇ R f e−f
Z jk i j ik Z jk j ik i
=2 R (∇ ∇ R +R R −R R )e−f +2 R R f e−f
Z jk j i ik mj mk ijmk im Z ik jk ij
=0+2 R R R e−f +2 R R f e−f −2 R R R e−f
Z jk mj mk Z ik jk ij Z ijmk im jk
=2 R R (f +R )e−f −2 R R R e−f
Z jk ki ij ij Z ijmk im jk
1
= |Rc|2e−f −2 Rm(Rc,Rc)e−f .
τ Z Z
This finishes the proof of the lemma.
5
We used the following lemma in the above,
Lemma 4. On a gradient shrinking Ricci soliton, we have
∇ ∇ R R e−f = 0 . (2.7)
Z j i ik jk
Proof.
∇ ∇ R R e−f = − ∇ R ∇ (R e−f) = 0 .
Z j i ik jk Z i ik j jk
Now we can prove (1.9) in Theorem 2.
Proof.
|div R |2e−f
Z m
= |∇ R −∇ R |2e−f
Z k jl l jk
=2 |∇Rc|2e−f −2 ∇ R ∇ R e−f
Z Z k jl l jk
1
=2 |∇Rc|2e−f + |Rc|2e−f −2 Rm(Rc,Rc)e−f ,
Z τ Z Z
so
1 1
Rm(Rc,Rc)e−f = |Rc|2e−f + |∇Rc|2e−f − |div R |2e−f .
Z 2τ Z Z 2 Z m
By (1.8) and (1.9) we have the corollary 1.
3 Identities of Ricci Curvature
Because of the soliton equation, there will be several identities for Ricci curvature on
the gradient shrinking Ricci solitons. We first prove Theorem 3. By using (2.1), we
derive that
∆R = ∇ (∇ R ) = ∇ (∇ R −R f ) = ∇ ∇ R −(∇ R )f −R f , (3.1)
jk i i jk i j ik ijkl l i j ik i ijkl l ijkl li
so
< ∆Rc,Rc >= ∇ ∇ R R = ∇ ∇ R R −(∇ R )f R −R f R , (3.2)
i i jk jk i j ik jk i ijkl l jk ijkl li jk
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and
1 1
2 2
∆|Rc| = ∆(R R ) = ∇ (∇ R R ) = (∆R R )+|∇Rc| . (3.3)
jk jk i i jk jk jk jk
2 2
Furthermore, we have
1 1
∆|Rc|2e−f = |Rc|2∆e−f
2 Z 2 Z
so
1
|Rc|2∆e−f
2 Z
= < ∆Rc,Rc > e−f + |∇Rc|2e−f
Z Z
= |∇Rc|2e−f + (∇ ∇ R R −∇ ∇ R R )e−f
Z Z i j ik jk j i ik jk
+ ∇ ∇ R R e−f − ∇ R f R e−f − R f R e−f
Z j i ik jk Z i ijkl l jk Z ijkl li jk
= |∇Rc|2e−f + Ke−f + ∇ ∇ R R e−f − ∇ R f R e−f − R f R e−f
Z Z Z j i ik jk Z i ijkl l jk Z ijkl li jk
= |∇Rc|2e−f + Ke−f − ∇ R f R e−f − R f R e−f . (3.4)
Z Z Z i ijkl l jk Z ijkl li jk
We used Lemma 4 in the last equation.
Plug (2.1) and (2.4) into (3.4), apply Corollary 1, we obtain
1 1
|Rc|2∆e−f = |∇Rc|2e−f + Ke−f − ∇ R f R e−f − |∇Rc|2e−f
2 Z Z Z Z i ijkl l jk 2 Z
1
= |∇Rc|2e−f + Ke−f + R R f f e−f . (3.5)
2 Z Z Z kljp jk l p
If we assume that the metric on the gradient shrinking Ricci soliton has positive
curvature, then
R R f f e−f
Z kljp jk p l
is a positive term. In fact, this is true for any metric with positive curvature operator.
We have
Lemma 5. Let (M,g) be a Riemannian manifold with positive curvature operator,
then
R R f f ≥ 0
ijkl ik j l
point-wise.
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Proof.
R = λ ωαωα ,
ikjl α ik jl
X
α
where
ωα = ωαdxi ∧dxk
ik
are 2-forms (in fact, they are the eigenfunctions of the curvature operator). And
λ ≥ 0 ,
α
so
R R f f = λ [ωαωαR f f ] ,
ikjl ij k l α ik jl ij k l
X
α
with
ωαωαR f f = R (ωαf )(ωαf ) = R γαγα ≥ 0
ik jl ij k l ij ik k jl l ij i j
and
γα = ωαf .
i ik k
Remark. It’s an easy calculation to see that
1 1
|Rc|2∆e−f = Rc(∇f,∇f)e−f ≥ 0 .
2 Z τ Z
4 Proof of Theorem 1
For a compact Riemannian manifold with positive curvature operator, we first need
the following lemma of Berger:
Lemma 6. (Berger) Assume T is a symmetric two tensor on a Riemannian mani-
fold (M,g) with non-negative sectional curvature, then
K = (∇ ∇ T −∇ ∇ T )T ≥ 0 .
i j ik j i ik jk
In fact,
2
K = R (λ −λ ) ,
ijij i j
X
i<j
where λ ’s are the eigenvalues of T.
i
8
We apply this lemma in the special case of
T = Rc .
Then we know that our K which is defined in (1.7) is non-negative.
By combining Theorem 3 and inequality (1.6), we can prove Theorem 1.
Proof. By
1
|Rc|2△(e−f)
2 Z
= ∇ R R f e−f
Z i jk jk i
1 1
≤ |∇Rc|2e−f + |Rc|2|∇f|2e−f
2 Z 2 Z
1
≤ |∇Rc|2e−f + Ke−f + R R f f e−f ,
2 Z Z Z ijkl ik j l
we show that for all i, j and k we have
∇ R = R f ,
i jk jk i
and
1
R R f f e−f = R ∇ R f e−f = − R R f e−f = |∇ Rc|2e−f .
Z kljp kj l p Z kljp l kj p Z kljp kj lp 2 Z
So
Ke−f = 0 ,
Z
hence
K ≡ 0
and f is a constant. Therefore, the soliton must be of constant curvature.
References
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[3] Richard S. Hamilton. Four-manifolds with positive curvature operator. J. Differ-
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[4] Richard S. Hamilton. The formation of singularities in the Ricci flow. In Surveys
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[6] Grisha Perelman. Ricci flow with surgery on three-manifolds. Preprint, 2003.
[7] Peter Petersen. Riemannian geometry, volume 171 of Graduate Texts in Mathe-
matics. Springer-Verlag, New York, 1998.
[8] Shun-ichi Tachibana. A theorem of Riemannian manifolds of positive curvature
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