Gradient Ricci solitons - scgas/scgas-2012/Talks/Munteanu.pdf · Gradient Ricci solitons Ovidiu Munteanu Department of Mathematics, Columbia University, New York Conference in honor

Gradient Ricci solitons

Ovidiu Munteanu

Department of Mathematics,Columbia University, New York

Conference in honor of Peter Li’s research, U.C. Irvine,January 15, 2012

Gradient Ricci solitons

Ovidiu Munteanu

Department of Mathematics,Columbia University, New York

Conference in honor of Peter Li’s research, U.C. Irvine,January 15, 2012

Happy Birthday, Peter!

I wish to thank Peter for the memorable years I spent in Irvine. Iam very grateful for allowing me to be his student and forgenerously sharing with me so many great things over time.Indeed, I will always look up to him as to a person of great styleboth in mathematics and in life.

Before coming here I have learned that I have to prepare a joke forPeter. Since I worked so hard to find one, I decided to tell it.

Happy Birthday, Peter!

I wish to thank Peter for the memorable years I spent in Irvine. Iam very grateful for allowing me to be his student and forgenerously sharing with me so many great things over time.Indeed, I will always look up to him as to a person of great styleboth in mathematics and in life.

Before coming here I have learned that I have to prepare a joke forPeter. Since I worked so hard to find one, I decided to tell it.

True story

A shepherd and a math professor travel by train together. Theypass a herd of sheep and the professor starts counting how manysheep are in the herd. Before finishing, the shepherd tells thecorrect number. So the professor is really surpised.This continues once again and then once again, which makes theprofessor even more curious how the shepherd does it so fast. Sohe asked him and the shepherd repplied:It’s very easy, I first count the legs and then divide by four.

True story


True story


Overview of the talk

1 Introduction to Ricci solitons:

Definitions, setupImportance of gradient Ricci solitonsBasic examplesClassification results

2 The structure of Ricci solitons:

The volume growth rateCurvature propertiesThe topology of Ricci solitons

Definition of Ricci solitons

(M, g) denotes a complete (noncompact) Riemannian manifold.

Definition. M is called gradient Ricci soliton if

Ric + Hess (f ) = λg ,

where λ ∈ R.

Normalization: We can take λ ∈{−1

2 , 0,12

}by rescaling the

metric.

Definition: We call the soliton a

shrinker if λ = 12 i.e., Ric + Hess(f ) = 1

2g

steady if λ = 0 i.e., Ric + Hess(f ) = 0

expanding if λ = −12 i.e., Ric + Hess(f ) = −1

2g





where λ ∈ R.


2 , 0,12

}by rescaling the

metric.



2g



2g





where λ ∈ R.


2 , 0,12

}by rescaling the

metric.



2g



2g





where λ ∈ R.


2 , 0,12

}by rescaling the

metric.



2g



2g





where λ ∈ R.


2 , 0,12

}by rescaling the

metric.



2g



2g





where λ ∈ R.


2 , 0,12

}by rescaling the

metric.



2g



2g

Importance of Ricci solitons

Ricci solitons are important because

They are natural generalizations of Einstein manifolds: take fconstant. However, they may be quite different than Einsteinmanifolds. For example, there are many noncompactshrinkers, but all positive Einstein manifolds are compact.

Important in the theory of Ricci flowThe Ricci flow is defined by

∂g

∂t= −2Ric

g (0) = g0,

for g0 a fixed metric on M.





∂g

∂t= −2Ric

g (0) = g0,






∂g

∂t= −2Ric

g (0) = g0,


Ricci solitons and Ricci flow

Ricci solitons in the study of Ricci flow.

On one hand, they are self similar solutions of the Ricci flow.Example: For a gradient steady Ricci soliton (M, g) so thatRic + Hess (f ) = 0 define ϕ by

dϕ

dt= ∇f (ϕ (t))

and letg (t) := ϕ (t)∗ g .

Then g (t) is a solution of RF; also a steady soliton.The shrinker and expanding solitons change bydiffeomorphisms and rescaling.




dϕ

dt= ∇f (ϕ (t))






dϕ

dt= ∇f (ϕ (t))



Ricci solitons and Ricci flowHow Ricci solitons appear in the study of Ricci flow.

On the other hand, they are possible singularity models for theRicci flow.Example: The Ricci flow typically exists on [0,T ), forT <∞. The Riemann curvature tensor becomes unboundedas t → T . We call the singularity of type I if

maxx∈M|Rm| (x , t) ≤ C

T − t.

It is known that if one rescales the Ricci flow conveniently i.e.,

gi (t) := λig

(T +

t

λi

), t ∈ [−λiT , 0),

it will converge to a nontrivial gradient shrinking Ricci solitonas λi →∞.




T − t.


gi (t) := λig

(T +

t

λi

), t ∈ [−λiT , 0),





T − t.


gi (t) := λig

(T +

t

λi

), t ∈ [−λiT , 0),


Examples of gradient Ricci solitons

1 Some examples of shrinkers

The Gaussian shrinker(Rn, dx2

)with potential f = 1

4 |x |2.

Round cylinders Rk × Sn−k .Kahler examples on line bundles over CPn−1, obtained byCalabi’s anszatz (Koiso, H.D.Cao, Feldman-Ilmanen-Knopf,A.Dancer-M.Wang, etc).

2 Some examples of steady solitons

Cigar: (Σ, g) =(R2, dx2+dy2

1+x2+y2

)and f := − ln

(1 + x2 + y2

).

Higher dim analogue found by Bryant.Examples on Cn or line bundles (H.D.Cao, F-I-K).

3 Some examples of gradient expanding solitons

The Gaussian expanding(Rn, dx2

)with f := − 1

4 |x |2.

Products of it with negative Einstein manifolds.Kahler examples.





4 |x |2.




1+x2+y2

)and f := − ln

(1 + x2 + y2

).




)with f := − 1

4 |x |2.






4 |x |2.




1+x2+y2

)and f := − ln

(1 + x2 + y2

).




)with f := − 1

4 |x |2.






4 |x |2.




1+x2+y2

)and f := − ln

(1 + x2 + y2

).




)with f := − 1

4 |x |2.






4 |x |2.




1+x2+y2

)and f := − ln

(1 + x2 + y2

).




)with f := − 1

4 |x |2.






4 |x |2.




1+x2+y2

)and f := − ln

(1 + x2 + y2

).




)with f := − 1

4 |x |2.






4 |x |2.




1+x2+y2

)and f := − ln

(1 + x2 + y2

).




)with f := − 1

4 |x |2.






4 |x |2.




1+x2+y2

)and f := − ln

(1 + x2 + y2

).




)with f := − 1

4 |x |2.


Questions for Ricci solitons

Classification of solitons

Most results known in dimension n = 2 or n = 3.

Example: 3− dim shrinkers are either R3, or R× S2, or S3:Ivey, Perelman, L.Ni-N.Wallach, H.D.Cao-B.L.Chen-X.Zhu.

Partial results in dimension n ≥ 4 : classified locallyconformally flat ones (or with harmonic Weyl tensor):Ni-Wallach, Petersen-Wylie, H.D.Cao-Q.Chen,X.Cao-B.Wang-Z.Zhang, Z.H.Zhang, Catino-Mantegazza,M.-Sesum.The shrinkers are Rn or R× Sn−1 or Sn.Other results in dimension 4 by Ni-Wallach, Naber.
















The structure of Ricci solitons

Theorem

For any gradient shrinking Ricci soliton (M, g) we have

1 Sharp volume uper bound (H.D. Cao and D. Zhou, 2009)

Vol (Bp (r)) ≤ c1rn, for all r > 0

2 Sharp volume lower bound (Munteanu and J. Wang, 2011)

Vol (Bp (r)) ≥ c0r , for all r > 0.

Remarks:

These estimates correspond to well known estimates of Bishop(upper bound) and Yau (lower bound) for Riemannianmanifolds with Ric ≥ 0.

There are known examples of shrinking solitons with Riccicurvature of alternating sign.


Theorem






Remarks:




Theorem






Remarks:



Comments about the volume estimate

The two volume estimates for solitons use differenttechniques, but none of them uses comparison geometrydirectly (i.e., Laplace comparison theorem).

Cao and Zhou used an integration argument on the level setsof the potential f .

We proved the volume lower bound using a log-Sobolevinequality of J.Carillo and L.Ni∫M

u2 ln u2−(∫

Mu2

)ln

(∫M

u2

)≤ 4

∫M|∇u|2+

∫M

Ru2+C

∫M

u2.





u2 ln u2−(∫

Mu2

)ln

(∫M

u2

)≤ 4

∫M|∇u|2+

∫M

Ru2+C

∫M

u2.





u2 ln u2−(∫

Mu2

)ln

(∫M

u2

)≤ 4

∫M|∇u|2+

∫M

Ru2+C

∫M

u2.

About the volume lower bound

Perelman has used the relation between log Sobolev andvolume growth in his famous non-collapsing results for theRicci flow on compact manifolds. His ideas applied here implythat if Ric is bounded on M, then the volume grows at leastlinearly (Carillo-Ni).

Ricci bounded is used in two places in the argument of Carilloand Ni (following Perelman):

1 On one hand, in the log-Sobolev take u with support in Bx(2)such that u = 1 on Bx(1). Need volume comparison of theform

Vol(Bx(2)) ≤ CVol(Bx(1)).

2 Another major difficulty is to control the scalar curvature term∫M

Ru2 in the log-Sobolev. If Ricci is bounded, this is clear.






















Our proofOur proof uses some specific properties of solitons, thus avoidingany additional assumptions.

Scalar curvature bound (H.D. Cao and D. Zhou)

1

V (r)

∫B(r)

R ≤ c,

where B (r) is a ball of radius r and V (r) its volume.

New volume estimates (Munteanu-J.Wang)

V (r + 1) ≤ 2V (r)

V (r + 1)− V (r) ≤ cV (r)

r.

In the log-Sobolev we took a cut-off function u with supportin B(r + 3)\B(r), so it is defined on an annulus. This relatesthe question to our volume growth estimates.



1

V (r)

∫B(r)

R ≤ c,



V (r + 1) ≤ 2V (r)

V (r + 1)− V (r) ≤ cV (r)

r.




1

V (r)

∫B(r)

R ≤ c,



V (r + 1) ≤ 2V (r)

V (r + 1)− V (r) ≤ cV (r)

r.




1

V (r)

∫B(r)

R ≤ c,



V (r + 1) ≤ 2V (r)

V (r + 1)− V (r) ≤ cV (r)

r.


The curvature of gradient shrinkers

The curvature

Controlling the curvature is a central theme in the theory of Ricciflow (for example, in convergence theorems).

For Ricci solitons in particular, curvature estimates have alsoappeared in classification problems: in Petersen-Wylie’sclassification of locally conformally flat shrinkers they needed∫

M|Ric |2 e−f <∞.

In general, when the manifold is not locally conformally flat, wewant to bound the full curvature tensor.


The curvature



M|Ric |2 e−f <∞.



The curvature



M|Ric |2 e−f <∞.



The curvature



M|Ric |2 e−f <∞.


The curvature of shrinkers

Theorem (Munteanu and M.-T. Wang, 2010)

Let (M, g) be a complete gradient shrinking Ricci soliton withbounded Ricci curvature. Then there exist a constant a > 0,depending only on dimension n and supM |Ric |, and a constantC > 0, such that

|Rm| (x) ≤ C (r (x) + 1)a on M.

Remarks

Somewhat related estimates have been known for self shrinkersof the mean curvature flow by the work of Colding andMinicozzi, following Schoen-Simon-Yau for minimal surfaces.

What is difficult here: both the soliton equation and thehypothesis are conditions on the Ricci curvature. Theconclusion is a bound on the Riemann curvature tensor.




|Rm| (x) ≤ C (r (x) + 1)a on M.

Remarks






|Rm| (x) ≤ C (r (x) + 1)a on M.

Remarks



A main curvature estimate

Important achievement in the proof: We proved a weightedintegral bound of the type∫

M|Rm|α f −a <∞.

This is done by integration by parts and uses a useful identity forgradient shrinking Ricci solitons, which relates the Ricci curvatureand the Riemann curvature tensor:

∇lRijkl = Rijkl fl = ∇jRik −∇iRjk .

A main curvature estimate

Important achievement in the proof: We proved a weightedintegral bound of the type∫

M|Rm|α f −a <∞.

This is done by integration by parts and uses a useful identity forgradient shrinking Ricci solitons, which relates the Ricci curvatureand the Riemann curvature tensor:

∇lRijkl = Rijkl fl = ∇jRik −∇iRjk .

Gap Theorem

As a consequence we obtained a gap theorem for the Gaussian.

Corollary (Munteanu and M.-T. Wang)

Let (M, g , f ) be a complete noncompact gradient shrinking Riccisoliton. Assume that |Ric | ≤ c (n) := 1

100n ; then M is isometric tothe Gaussian soliton.

=⇒ It is not possible to produce complete shrinkers which are veryclose to the Gaussian. Yokota proved a gap result along the samelines, if the weighted volume is close to the corresponding value forthe Gaussian, then the shrinker is flat.

Since the type I limits are not flat, our result tells someinformation about the Ricci curvature of the evolving manifolds:the maximum of their Ricci curvature blows up like C

T−t .

Gap Theorem







T−t .

Gap Theorem







T−t .

Gap Theorem







T−t .

Ends of gradient steady solitons

Topology at infinity

Goal: Study the asymptotic structure of Ricci solitons. Do theyalways have exactly one end?

This information is also useful in constructing (or ruling out) newexamples e.g., by connected sum.

ApproachWe believe that studying geometric analysis of manifolds withdensities is very useful in understanding solitons. In some sense,solitons are canonical smooth metric measure spaces.












Definition.A smooth metric measure space is a complete manifold with aconformal change in volume

(M, g , e−f dv

).

The natural operator is ∆f := ∆− 〈∇f ,∇〉 , which is self adjoint:∫M

(∆f u) ve−f = −∫M〈∇u,∇v〉 e−f .

It has non-negative spectrum and its infimum of spectrum is

λ1 (∆f ) := infϕ∈C∞

0 (M)

∫M |∇ϕ|

2 e−f∫M ϕ2e−f

.



(M, g , e−f dv

).


(∆f u) ve−f = −∫M〈∇u,∇v〉 e−f .


λ1 (∆f ) := infϕ∈C∞

0 (M)

∫M |∇ϕ|


.



(M, g , e−f dv

).


(∆f u) ve−f = −∫M〈∇u,∇v〉 e−f .


λ1 (∆f ) := infϕ∈C∞

0 (M)

∫M |∇ϕ|


.


Our favorite curvature quantity is the Bakry-Emery Ricci curvature:

Ricf := Ric + Hess (f ) .

It can be motivated by the Bochner formula

1

2∆f |∇u|2 = |Hess (u)|2 + 〈∇∆f u,∇u〉+ Ricf (∇u,∇u) .

The following theorem shows that gradient steady Ricci solitonshave maximal bottom of spectrum among all smooth metricmeasure spaces with nonnegative Bakry-Emery curvature andlinear potential.





1







1



Connectedness of gradient steady solitons

Theorem (Munteanu and J. Wang, 2011)

Let(M, g , e−f dv

)be a smooth metric measure space so that

Ricf ≥ 0 and |f | (x) ≤ ar (x) + b. Then

λ1 (∆f ) ≤ 1

4a2.

Moreover, equality is achieved by any gradient steady Ricci soliton.

Remarks:

When f is constant, λ1 = 0. One motivation for our resultwas S.Y.Cheng’s upper bound for Riemannian manifolds withRic ≥ − (n − 1): there is a sharp estimate λ1 ≤ 1

4 (n − 1)2 .

The result I stated above is interesting because the equality isachieved by any gradient steady Ricci soliton. Related toCheng’s estimate, it is not true that any Einstein manifold hasmaximal λ1.



Let(M, g , e−f dv



λ1 (∆f ) ≤ 1

4a2.


Remarks:


4 (n − 1)2 .




Let(M, g , e−f dv



λ1 (∆f ) ≤ 1

4a2.


Remarks:


4 (n − 1)2 .




Let(M, g , e−f dv



λ1 (∆f ) ≤ 1

4a2.


Remarks:


4 (n − 1)2 .


Related comments

Remark There is a sharp Kahler version of Cheng’s estimate(Munteanu, 2008):

λ1 (M) ≤ m2 if Ric ≥ −2 (m + 1) .

Equality achieved for CHm. (P. Li and J. Wang previouslyproved it with BK ≥ −1).

Important achievement in my proof: an integral gradientestimate for harmonic functions, which is sharp for Kahlermanifolds. This extends Yau’s famous gradient estimate.

Related comments


λ1 (M) ≤ m2 if Ric ≥ −2 (m + 1) .



Related comments


λ1 (M) ≤ m2 if Ric ≥ −2 (m + 1) .



Motivation for our next result

The main motivation for our study was a theory developed byP.Li and J.Wang for Riemannian manifolds with Riccicurvature bounded below. They assumed equality in Cheng’sestimate:

RicM ≥ − (n − 1) and λ1 (M) =(n − 1)2

4

and that M has at least two ends. Then M = R×h N, for Ncompact and h (t) = et for n ≥ 3 and also h (t) = cosh t forn = 3.

Li-Wang’s work was developed in connection to the results byE.Witten-S.T.Yau, X. Wang, M.Cai-G.Galloway forconformally compact Riemannian manifolds.

We have adapted Li-Wang’s theory to the setting of smoothmetric measure spaces. A consequence of our results:



RicM ≥ − (n − 1) and λ1 (M) =(n − 1)2

4






RicM ≥ − (n − 1) and λ1 (M) =(n − 1)2

4






Let (M, g , f ) be any steady Ricci soliton. Then M is connected atinfinity i.e. has one end; or splits as a direct product M = R× N,for a compact Ricci flat manifold N.

Remarks:

The Theorem can be compared to Cheeger-Gromoll’s splittingtheorem:A Riemannian manifold M with Ric ≥ 0 and that contains aline splits as M = R× N. In particular, if it has two or moreends, it contains a line.

The idea of both proofs is to understand the properties of theBusemann function, which is a distance function to infinity. Inour proof, we combine this with the result that λ1 (∆f ) ismaximal.




Remarks:






Remarks:






Remarks:



A little about the proof

First, we separate the ends of M into f−nonparabolic andf−parabolic, according to whether they admit a positivesymmetric Green’s function for ∆f or not. Since ends have acompact boundary, the Green’s function is assumed withNeumann boundary condition there.

Since λ1 (∆f ) > 0, it is true that there exists at least onef−nonparabolic end.

P. Li and L.F. Tam have developed a theory that usesharmonic functions to count the number of ends. Forexample, two f−nonoparabolic ends will generate a boundedf−harmonic function with finite energy. This can be ruled outby a Bochner type argument, or by a Liouville theorem.









Finish the proof

The last case is the one that produces the splitting: M has atleast two ends, only one is f−nonparabolic, all others aref−parabolic. Here we use the Busemann function β:

β (x) = limt→∞

(t − d (x , γ (t))) ,

for γ a ray pointing to the infinity of an f−parabolic end.

Comparison geometry =⇒ ∆f β ≥ −a

Then β can be used to construct a test function for λ1 (∆f ) :

∆f e−12aβ ≥ −1

4a2e−

12aβ.

Important detail: the construction of β and some volume

estimates imply that e−12af is ”almost” in L2

(e−f dv

). In the

last step it can be justified that all these inequalities willbecome equalities.

Finish the proof


β (x) = limt→∞

(t − d (x , γ (t))) ,




∆f e−12aβ ≥ −1

4a2e−

12aβ.



(e−f dv

). In the


Finish the proof


β (x) = limt→∞

(t − d (x , γ (t))) ,




∆f e−12aβ ≥ −1

4a2e−

12aβ.



(e−f dv

). In the


Finish the proof


β (x) = limt→∞

(t − d (x , γ (t))) ,




∆f e−12aβ ≥ −1

4a2e−

12aβ.



(e−f dv

). In the


Final comments

What about the other solitons?


If (M, g) is an expanding Ricci soliton with R ≥ −n2 + 1

2 theneither has one end or it is isometric to R× N, where the R factoris the Gaussian expander and N is a compact Einstein manifold.

note that R×N has scalar curvature R = −n2 + 1

2 . In general,any expanding soliton has R ≥ −n

2 .

why is the expanding soliton case different than steadysoliton: the potential does not grow linearly, so it is not clearhow to establish spectral estimates anymore. This time, ourproof is more specialized to expanding solitons.

Finally, in the shrinking case, there are partial results by N.Sesum and myself, for Kahler shrinkers.

Final comments







2 .



Final comments







2 .



Final comments







2 .



Final comments







2 .



THANK YOU FOR YOUR TIME!!

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Gradient Ricci solitons - scgas/scgas-2012/Talks/Munteanu.pdf · Gradient Ricci solitons Ovidiu Munteanu Department of Mathematics, Columbia University, New York Conference in honor