on gradient ricci solitons
TRANSCRIPT
![Page 1: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/1.jpg)
Rigidity, gap theorems and maximumprinciples for Ricci solitons
Manuel Fernández López
Consellería de Educación e Ordenación UniversitariaXunta de Galicia
Galicia SPAIN
(joint work with Eduardo García Río)
Ricci Solitons Days in Pisa4-8th April 2011
![Page 2: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/2.jpg)
Outline
Rigidity of Ricci solitonsRigidity: compact caseRigidity: non-compact caseLocally conformally flat case
Gap theoremsDiameter boundsGap theorems: compact caseGap theorems: non-compact case
Maximum principlesIntroductionOmori-Yau maximum principleApplications
Steady solitonsLower bound for the curvature of a steady soliton
![Page 3: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/3.jpg)
Definition (Petersen and Wylie, 2007)A Ricci soliton is said to be rigid if it is of the form N ×Γ Rk ,where N is an Einstein manifold and Γ acts freely on N and byorthogonal transformations on Rk .
Theorem (Petersen and Wylie, 2007)The following conditions for a shrinking (expanding) gradientsoliton Ric + Hf = λg all imply that the metric is radially flat andhas constant scalar curvature
I R is constant and sec(E ,∇f ) ≥ 0 (sec(E ,∇f ) ≤ 0)
I R is constant and 0 ≤ Ric ≤ λg (λg ≤ Ric ≤ 0)
I The curvature tensor is harmonicI Ric ≥ 0 (Ric ≤ 0) and sec(E ,∇f ) = 0
![Page 4: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/4.jpg)
Definition (Petersen and Wylie, 2007)A Ricci soliton is said to be rigid if it is of the form N ×Γ Rk ,where N is an Einstein manifold and Γ acts freely on N and byorthogonal transformations on Rk .
Theorem (Petersen and Wylie, 2007)The following conditions for a shrinking (expanding) gradientsoliton Ric + Hf = λg all imply that the metric is radially flat andhas constant scalar curvature
I R is constant and sec(E ,∇f ) ≥ 0 (sec(E ,∇f ) ≤ 0)
I R is constant and 0 ≤ Ric ≤ λg (λg ≤ Ric ≤ 0)
I The curvature tensor is harmonicI Ric ≥ 0 (Ric ≤ 0) and sec(E ,∇f ) = 0
![Page 5: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/5.jpg)
Outline
Rigidity of Ricci solitonsRigidity: compact caseRigidity: non-compact caseLocally conformally flat case
Gap theoremsDiameter boundsGap theorems: compact caseGap theorems: non-compact case
Maximum principlesIntroductionOmori-Yau maximum principleApplications
Steady solitonsLower bound for the curvature of a steady soliton
![Page 6: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/6.jpg)
Theorem (Eminenti, LaNave and Mantegazza, 2008)Let (Mn,g) be an n-dimensional compact Ricci soliton. If (M,g)is locally conformally flat then it is Einstein (in fact, a spaceform).
Theorem (E. García Río and MFL, 2009)Let (Mn,g) be an n-dimensional compact Ricci soliton. Then(M,g) is rigid if an only if it has harmonic Weyl tensor.
A gradient Ricci soliton is a Riemannian manifold such that
Ric + Hf = λg
![Page 7: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/7.jpg)
Theorem (Eminenti, LaNave and Mantegazza, 2008)Let (Mn,g) be an n-dimensional compact Ricci soliton. If (M,g)is locally conformally flat then it is Einstein (in fact, a spaceform).
Theorem (E. García Río and MFL, 2009)Let (Mn,g) be an n-dimensional compact Ricci soliton. Then(M,g) is rigid if an only if it has harmonic Weyl tensor.
A gradient Ricci soliton is a Riemannian manifold such that
Ric + Hf = λg
![Page 8: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/8.jpg)
Theorem (Eminenti, LaNave and Mantegazza, 2008)Let (Mn,g) be an n-dimensional compact Ricci soliton. If (M,g)is locally conformally flat then it is Einstein (in fact, a spaceform).
Theorem (E. García Río and MFL, 2009)Let (Mn,g) be an n-dimensional compact Ricci soliton. Then(M,g) is rigid if an only if it has harmonic Weyl tensor.
A gradient Ricci soliton is a Riemannian manifold such that
Ric + Hf = λg
![Page 9: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/9.jpg)
The Schouten tensor S = Rc − R2(n − 1)
g is a Codazzi tensor
(∇X Rc)(Y ,Z )− (∇Y Rc)(X ,Z ) =X (R)
2(n − 1)g(Y ,Z )− Y (R)
2(n − 1)g(X ,Z )
Rm(X ,Y ,Z ,∇f ) =1
n − 1Rc(X ,∇f )g(Y ,Z )− 1
n − 1Rc(Y ,∇f )g(X ,Z )
∇f is an eigenvector of Rc
(div Rm)(X ,Y ,Z ) = Rm(X ,Y ,Z ,∇f )
|div Rm|2 =1
2(n − 1)|∇R|2
![Page 10: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/10.jpg)
The Schouten tensor S = Rc − R2(n − 1)
g is a Codazzi tensor
(∇X Rc)(Y ,Z )− (∇Y Rc)(X ,Z ) =X (R)
2(n − 1)g(Y ,Z )− Y (R)
2(n − 1)g(X ,Z )
Rm(X ,Y ,Z ,∇f ) =1
n − 1Rc(X ,∇f )g(Y ,Z )− 1
n − 1Rc(Y ,∇f )g(X ,Z )
∇f is an eigenvector of Rc
(div Rm)(X ,Y ,Z ) = Rm(X ,Y ,Z ,∇f )
|div Rm|2 =1
2(n − 1)|∇R|2
![Page 11: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/11.jpg)
The Schouten tensor S = Rc − R2(n − 1)
g is a Codazzi tensor
(∇X Rc)(Y ,Z )− (∇Y Rc)(X ,Z ) =X (R)
2(n − 1)g(Y ,Z )− Y (R)
2(n − 1)g(X ,Z )
Rm(X ,Y ,Z ,∇f ) =1
n − 1Rc(X ,∇f )g(Y ,Z )− 1
n − 1Rc(Y ,∇f )g(X ,Z )
∇f is an eigenvector of Rc
(div Rm)(X ,Y ,Z ) = Rm(X ,Y ,Z ,∇f )
|div Rm|2 =1
2(n − 1)|∇R|2
![Page 12: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/12.jpg)
The Schouten tensor S = Rc − R2(n − 1)
g is a Codazzi tensor
(∇X Rc)(Y ,Z )− (∇Y Rc)(X ,Z ) =X (R)
2(n − 1)g(Y ,Z )− Y (R)
2(n − 1)g(X ,Z )
Rm(X ,Y ,Z ,∇f ) =1
n − 1Rc(X ,∇f )g(Y ,Z )− 1
n − 1Rc(Y ,∇f )g(X ,Z )
∇f is an eigenvector of Rc
(div Rm)(X ,Y ,Z ) = Rm(X ,Y ,Z ,∇f )
|div Rm|2 =1
2(n − 1)|∇R|2
![Page 13: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/13.jpg)
The Schouten tensor S = Rc − R2(n − 1)
g is a Codazzi tensor
(∇X Rc)(Y ,Z )− (∇Y Rc)(X ,Z ) =X (R)
2(n − 1)g(Y ,Z )− Y (R)
2(n − 1)g(X ,Z )
Rm(X ,Y ,Z ,∇f ) =1
n − 1Rc(X ,∇f )g(Y ,Z )− 1
n − 1Rc(Y ,∇f )g(X ,Z )
∇f is an eigenvector of Rc
(div Rm)(X ,Y ,Z ) = Rm(X ,Y ,Z ,∇f )
|div Rm|2 =1
2(n − 1)|∇R|2
![Page 14: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/14.jpg)
The Schouten tensor S = Rc − R2(n − 1)
g is a Codazzi tensor
(∇X Rc)(Y ,Z )− (∇Y Rc)(X ,Z ) =X (R)
2(n − 1)g(Y ,Z )− Y (R)
2(n − 1)g(X ,Z )
Rm(X ,Y ,Z ,∇f ) =1
n − 1Rc(X ,∇f )g(Y ,Z )− 1
n − 1Rc(Y ,∇f )g(X ,Z )
∇f is an eigenvector of Rc
(div Rm)(X ,Y ,Z ) = Rm(X ,Y ,Z ,∇f )
|div Rm|2 =1
2(n − 1)|∇R|2
![Page 15: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/15.jpg)
∫M|div Rm|2e−f =
∫M|∇Ric|2e−f
X. Cao, B. Wang and Z. Zhang; On Locally ConformallyFlat Gradient Shrinking Ricci Solitons
12(n − 1)
∫M|∇R|2e−f ≥ 1
n
∫M|∇R|2e−f
For n ≤ 3 compact shrinking Ricci solitons are Einstein (n = 2Hamilton, n = 3 Ivey)
Since n ≥ 4 one has that R is constant
(M,g) is Einstein
What about the noncompact case?
![Page 16: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/16.jpg)
∫M|div Rm|2e−f =
∫M|∇Ric|2e−f
X. Cao, B. Wang and Z. Zhang; On Locally ConformallyFlat Gradient Shrinking Ricci Solitons
12(n − 1)
∫M|∇R|2e−f ≥ 1
n
∫M|∇R|2e−f
For n ≤ 3 compact shrinking Ricci solitons are Einstein (n = 2Hamilton, n = 3 Ivey)
Since n ≥ 4 one has that R is constant
(M,g) is Einstein
What about the noncompact case?
![Page 17: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/17.jpg)
∫M|div Rm|2e−f =
∫M|∇Ric|2e−f
X. Cao, B. Wang and Z. Zhang; On Locally ConformallyFlat Gradient Shrinking Ricci Solitons
12(n − 1)
∫M|∇R|2e−f ≥ 1
n
∫M|∇R|2e−f
For n ≤ 3 compact shrinking Ricci solitons are Einstein (n = 2Hamilton, n = 3 Ivey)
Since n ≥ 4 one has that R is constant
(M,g) is Einstein
What about the noncompact case?
![Page 18: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/18.jpg)
∫M|div Rm|2e−f =
∫M|∇Ric|2e−f
X. Cao, B. Wang and Z. Zhang; On Locally ConformallyFlat Gradient Shrinking Ricci Solitons
12(n − 1)
∫M|∇R|2e−f ≥ 1
n
∫M|∇R|2e−f
For n ≤ 3 compact shrinking Ricci solitons are Einstein (n = 2Hamilton, n = 3 Ivey)
Since n ≥ 4 one has that R is constant
(M,g) is Einstein
What about the noncompact case?
![Page 19: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/19.jpg)
∫M|div Rm|2e−f =
∫M|∇Ric|2e−f
X. Cao, B. Wang and Z. Zhang; On Locally ConformallyFlat Gradient Shrinking Ricci Solitons
12(n − 1)
∫M|∇R|2e−f ≥ 1
n
∫M|∇R|2e−f
For n ≤ 3 compact shrinking Ricci solitons are Einstein (n = 2Hamilton, n = 3 Ivey)
Since n ≥ 4 one has that R is constant
(M,g) is Einstein
What about the noncompact case?
![Page 20: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/20.jpg)
∫M|div Rm|2e−f =
∫M|∇Ric|2e−f
X. Cao, B. Wang and Z. Zhang; On Locally ConformallyFlat Gradient Shrinking Ricci Solitons
12(n − 1)
∫M|∇R|2e−f ≥ 1
n
∫M|∇R|2e−f
For n ≤ 3 compact shrinking Ricci solitons are Einstein (n = 2Hamilton, n = 3 Ivey)
Since n ≥ 4 one has that R is constant
(M,g) is Einstein
What about the noncompact case?
![Page 21: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/21.jpg)
∫M|div Rm|2e−f =
∫M|∇Ric|2e−f
X. Cao, B. Wang and Z. Zhang; On Locally ConformallyFlat Gradient Shrinking Ricci Solitons
12(n − 1)
∫M|∇R|2e−f ≥ 1
n
∫M|∇R|2e−f
For n ≤ 3 compact shrinking Ricci solitons are Einstein (n = 2Hamilton, n = 3 Ivey)
Since n ≥ 4 one has that R is constant
(M,g) is Einstein
What about the noncompact case?
![Page 22: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/22.jpg)
Outline
Rigidity of Ricci solitonsRigidity: compact caseRigidity: non-compact caseLocally conformally flat case
Gap theoremsDiameter boundsGap theorems: compact caseGap theorems: non-compact case
Maximum principlesIntroductionOmori-Yau maximum principleApplications
Steady solitonsLower bound for the curvature of a steady soliton
![Page 23: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/23.jpg)
Theorem (E. García Río and MFL, 2009)Let (Mn,g) be a complete noncompact gradient shrinking Riccisoliton whose curvature tensor has at most exponential growthand having Ricci tensor bounded from below. Then (M,g) isrigid if an only if it has harmonic Weyl tensor.∫
M|div Rm|2e−f =
∫M|∇Ric|2e−f
R is constant and Rm(∇f ,X ,X ,∇f ) = 0
P. Petersen and W. Wilye; Rigidity of gradient Ricci solitons
Theorem (Munteanu and Sesum, 2009)Let (M,g) be a complete noncompact gradient shrinking Riccisoliton. Then (M,g) is rigid if an only if it has harmonic Weyltensor.
![Page 24: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/24.jpg)
Theorem (E. García Río and MFL, 2009)Let (Mn,g) be a complete noncompact gradient shrinking Riccisoliton whose curvature tensor has at most exponential growthand having Ricci tensor bounded from below. Then (M,g) isrigid if an only if it has harmonic Weyl tensor.∫
M|div Rm|2e−f =
∫M|∇Ric|2e−f
R is constant and Rm(∇f ,X ,X ,∇f ) = 0
P. Petersen and W. Wilye; Rigidity of gradient Ricci solitons
Theorem (Munteanu and Sesum, 2009)Let (M,g) be a complete noncompact gradient shrinking Riccisoliton. Then (M,g) is rigid if an only if it has harmonic Weyltensor.
![Page 25: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/25.jpg)
Theorem (E. García Río and MFL, 2009)Let (Mn,g) be a complete noncompact gradient shrinking Riccisoliton whose curvature tensor has at most exponential growthand having Ricci tensor bounded from below. Then (M,g) isrigid if an only if it has harmonic Weyl tensor.∫
M|div Rm|2e−f =
∫M|∇Ric|2e−f
R is constant and Rm(∇f ,X ,X ,∇f ) = 0
P. Petersen and W. Wilye; Rigidity of gradient Ricci solitons
Theorem (Munteanu and Sesum, 2009)Let (M,g) be a complete noncompact gradient shrinking Riccisoliton. Then (M,g) is rigid if an only if it has harmonic Weyltensor.
![Page 26: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/26.jpg)
Theorem (E. García Río and MFL, 2009)Let (Mn,g) be a complete noncompact gradient shrinking Riccisoliton whose curvature tensor has at most exponential growthand having Ricci tensor bounded from below. Then (M,g) isrigid if an only if it has harmonic Weyl tensor.∫
M|div Rm|2e−f =
∫M|∇Ric|2e−f
R is constant and Rm(∇f ,X ,X ,∇f ) = 0
P. Petersen and W. Wilye; Rigidity of gradient Ricci solitons
Theorem (Munteanu and Sesum, 2009)Let (M,g) be a complete noncompact gradient shrinking Riccisoliton. Then (M,g) is rigid if an only if it has harmonic Weyltensor.
![Page 27: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/27.jpg)
Theorem (E. García Río and MFL, 2009)Let (Mn,g) be a complete noncompact gradient shrinking Riccisoliton whose curvature tensor has at most exponential growthand having Ricci tensor bounded from below. Then (M,g) isrigid if an only if it has harmonic Weyl tensor.∫
M|div Rm|2e−f =
∫M|∇Ric|2e−f
R is constant and Rm(∇f ,X ,X ,∇f ) = 0
P. Petersen and W. Wilye; Rigidity of gradient Ricci solitons
Theorem (Munteanu and Sesum, 2009)Let (M,g) be a complete noncompact gradient shrinking Riccisoliton. Then (M,g) is rigid if an only if it has harmonic Weyltensor.
![Page 28: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/28.jpg)
Outline
Rigidity of Ricci solitonsRigidity: compact caseRigidity: non-compact caseLocally conformally flat case
Gap theoremsDiameter boundsGap theorems: compact caseGap theorems: non-compact case
Maximum principlesIntroductionOmori-Yau maximum principleApplications
Steady solitonsLower bound for the curvature of a steady soliton
![Page 29: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/29.jpg)
Lemma (E. García Río and MFL, 2010)Let (Mn,g) be a locally conformally flat gradient Ricci soliton.Then it is locally (where ∇f 6= 0) isometric to a warped product
(M,g) = ((a,b)× N,dt2 + ψ(t)2gN),
where (N,gN) is a space form.
W (V ,Ei ,Ei ,V ) = − Rc(V ,V )
(n − 1)(n − 2)− Rc(Ei ,Ei )
n − 2+
R(n − 1)(n − 2)
whereV =
1|∇f |∇f
![Page 30: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/30.jpg)
Lemma (E. García Río and MFL, 2010)Let (Mn,g) be a locally conformally flat gradient Ricci soliton.Then it is locally (where ∇f 6= 0) isometric to a warped product
(M,g) = ((a,b)× N,dt2 + ψ(t)2gN),
where (N,gN) is a space form.
W (V ,Ei ,Ei ,V ) = − Rc(V ,V )
(n − 1)(n − 2)− Rc(Ei ,Ei )
n − 2+
R(n − 1)(n − 2)
whereV =
1|∇f |∇f
![Page 31: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/31.jpg)
Rc(Ei ,Ei) =1
n − 1(R − Rc(V ,V ))
Hf (Ei ,Ei) =1
n − 1(∆f − Hf (V ,V ))
N = f−1(c) is a totally umbilical submanifold of (M,g)
∇f is an eigenvector of Hf ↔ the integral curves of V aregeodesics
(M,g) is locally a warped product
N is a space form
Brozos-Vázquez, García-Río and Vázquez-Lorenzo;Complete locally conformally flat manifolds of negativecurvature
![Page 32: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/32.jpg)
Rc(Ei ,Ei) =1
n − 1(R − Rc(V ,V ))
Hf (Ei ,Ei) =1
n − 1(∆f − Hf (V ,V ))
N = f−1(c) is a totally umbilical submanifold of (M,g)
∇f is an eigenvector of Hf ↔ the integral curves of V aregeodesics
(M,g) is locally a warped product
N is a space form
Brozos-Vázquez, García-Río and Vázquez-Lorenzo;Complete locally conformally flat manifolds of negativecurvature
![Page 33: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/33.jpg)
Rc(Ei ,Ei) =1
n − 1(R − Rc(V ,V ))
Hf (Ei ,Ei) =1
n − 1(∆f − Hf (V ,V ))
N = f−1(c) is a totally umbilical submanifold of (M,g)
∇f is an eigenvector of Hf ↔ the integral curves of V aregeodesics
(M,g) is locally a warped product
N is a space form
Brozos-Vázquez, García-Río and Vázquez-Lorenzo;Complete locally conformally flat manifolds of negativecurvature
![Page 34: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/34.jpg)
Rc(Ei ,Ei) =1
n − 1(R − Rc(V ,V ))
Hf (Ei ,Ei) =1
n − 1(∆f − Hf (V ,V ))
N = f−1(c) is a totally umbilical submanifold of (M,g)
∇f is an eigenvector of Hf ↔ the integral curves of V aregeodesics
(M,g) is locally a warped product
N is a space form
Brozos-Vázquez, García-Río and Vázquez-Lorenzo;Complete locally conformally flat manifolds of negativecurvature
![Page 35: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/35.jpg)
Rc(Ei ,Ei) =1
n − 1(R − Rc(V ,V ))
Hf (Ei ,Ei) =1
n − 1(∆f − Hf (V ,V ))
N = f−1(c) is a totally umbilical submanifold of (M,g)
∇f is an eigenvector of Hf ↔ the integral curves of V aregeodesics
(M,g) is locally a warped product
N is a space form
Brozos-Vázquez, García-Río and Vázquez-Lorenzo;Complete locally conformally flat manifolds of negativecurvature
![Page 36: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/36.jpg)
Rc(Ei ,Ei) =1
n − 1(R − Rc(V ,V ))
Hf (Ei ,Ei) =1
n − 1(∆f − Hf (V ,V ))
N = f−1(c) is a totally umbilical submanifold of (M,g)
∇f is an eigenvector of Hf ↔ the integral curves of V aregeodesics
(M,g) is locally a warped product
N is a space form
Brozos-Vázquez, García-Río and Vázquez-Lorenzo;Complete locally conformally flat manifolds of negativecurvature
![Page 37: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/37.jpg)
Theorem (E. García Río and MFL, 2010)Let (Mn,g) be a simply connected complete locally conformallyflat gradient shrinking or steady Ricci soliton. Then it isrotationally symmetric.Any complete ancient solution to the Ricci flow has nonnegativecurvature operator (n = 3 Chen, n ≥ 4 Zhang)
RmN(X ,Y ,Y ,X ) = RmM(X ,Y ,Y ,X )+II(X ,X )II(Y ,Y )−II(X ,Y )2
N is a standard sphere
(Mn,g) is rotationally symmetric
B. Kotschwar; On rotationally invariant shrinking gradientRicci solitons
H.-D. Cao and Q. Chen; On Locally Conformally FlatGradient Steady Ricci Solitons
![Page 38: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/38.jpg)
Theorem (E. García Río and MFL, 2010)Let (Mn,g) be a simply connected complete locally conformallyflat gradient shrinking or steady Ricci soliton. Then it isrotationally symmetric.Any complete ancient solution to the Ricci flow has nonnegativecurvature operator (n = 3 Chen, n ≥ 4 Zhang)
RmN(X ,Y ,Y ,X ) = RmM(X ,Y ,Y ,X )+II(X ,X )II(Y ,Y )−II(X ,Y )2
N is a standard sphere
(Mn,g) is rotationally symmetric
B. Kotschwar; On rotationally invariant shrinking gradientRicci solitons
H.-D. Cao and Q. Chen; On Locally Conformally FlatGradient Steady Ricci Solitons
![Page 39: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/39.jpg)
Theorem (E. García Río and MFL, 2010)Let (Mn,g) be a simply connected complete locally conformallyflat gradient shrinking or steady Ricci soliton. Then it isrotationally symmetric.Any complete ancient solution to the Ricci flow has nonnegativecurvature operator (n = 3 Chen, n ≥ 4 Zhang)
RmN(X ,Y ,Y ,X ) = RmM(X ,Y ,Y ,X )+II(X ,X )II(Y ,Y )−II(X ,Y )2
N is a standard sphere
(Mn,g) is rotationally symmetric
B. Kotschwar; On rotationally invariant shrinking gradientRicci solitons
H.-D. Cao and Q. Chen; On Locally Conformally FlatGradient Steady Ricci Solitons
![Page 40: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/40.jpg)
Theorem (E. García Río and MFL, 2010)Let (Mn,g) be a simply connected complete locally conformallyflat gradient shrinking or steady Ricci soliton. Then it isrotationally symmetric.Any complete ancient solution to the Ricci flow has nonnegativecurvature operator (n = 3 Chen, n ≥ 4 Zhang)
RmN(X ,Y ,Y ,X ) = RmM(X ,Y ,Y ,X )+II(X ,X )II(Y ,Y )−II(X ,Y )2
N is a standard sphere
(Mn,g) is rotationally symmetric
B. Kotschwar; On rotationally invariant shrinking gradientRicci solitons
H.-D. Cao and Q. Chen; On Locally Conformally FlatGradient Steady Ricci Solitons
![Page 41: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/41.jpg)
Theorem (E. García Río and MFL, 2010)Let (Mn,g) be a simply connected complete locally conformallyflat gradient shrinking or steady Ricci soliton. Then it isrotationally symmetric.Any complete ancient solution to the Ricci flow has nonnegativecurvature operator (n = 3 Chen, n ≥ 4 Zhang)
RmN(X ,Y ,Y ,X ) = RmM(X ,Y ,Y ,X )+II(X ,X )II(Y ,Y )−II(X ,Y )2
N is a standard sphere
(Mn,g) is rotationally symmetric
B. Kotschwar; On rotationally invariant shrinking gradientRicci solitons
H.-D. Cao and Q. Chen; On Locally Conformally FlatGradient Steady Ricci Solitons
![Page 42: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/42.jpg)
Theorem (E. García Río and MFL, 2010)Let (Mn,g) be a simply connected complete locally conformallyflat gradient shrinking or steady Ricci soliton. Then it isrotationally symmetric.Any complete ancient solution to the Ricci flow has nonnegativecurvature operator (n = 3 Chen, n ≥ 4 Zhang)
RmN(X ,Y ,Y ,X ) = RmM(X ,Y ,Y ,X )+II(X ,X )II(Y ,Y )−II(X ,Y )2
N is a standard sphere
(Mn,g) is rotationally symmetric
B. Kotschwar; On rotationally invariant shrinking gradientRicci solitons
H.-D. Cao and Q. Chen; On Locally Conformally FlatGradient Steady Ricci Solitons
![Page 43: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/43.jpg)
Outline
Rigidity of Ricci solitonsRigidity: compact caseRigidity: non-compact caseLocally conformally flat case
Gap theoremsDiameter boundsGap theorems: compact caseGap theorems: non-compact case
Maximum principlesIntroductionOmori-Yau maximum principleApplications
Steady solitonsLower bound for the curvature of a steady soliton
![Page 44: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/44.jpg)
Theorem (E. García Río and MFL, 2008)Let (Mn,g) be a compact gradient Ricci soliton. Then
diam2(M,g) ≥ 2max
fmax − fmin
λ− c,fmax − fmin
C − λ,4
fmax − fmin
C − c
where c ≤ Ric ≤ C.
Theorem (E. García Río and MFL, 2008)Let (Mn,g) be a compact gradient Ricci soliton with Ric > 0.Then
diam2(M,g) ≥ max
Rmax − Rmin
λ(λ− c),Rmax − Rmin
λ(C − λ),4
Rmax − Rmin
λ(C − c)
where c ≤ Ric ≤ C.
![Page 45: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/45.jpg)
Theorem (E. García Río and MFL, 2008)Let (Mn,g) be a compact gradient Ricci soliton. Then
diam2(M,g) ≥ 2max
fmax − fmin
λ− c,fmax − fmin
C − λ,4
fmax − fmin
C − c
where c ≤ Ric ≤ C.
Theorem (E. García Río and MFL, 2008)Let (Mn,g) be a compact gradient Ricci soliton with Ric > 0.Then
diam2(M,g) ≥ max
Rmax − Rmin
λ(λ− c),Rmax − Rmin
λ(C − λ),4
Rmax − Rmin
λ(C − c)
where c ≤ Ric ≤ C.
![Page 46: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/46.jpg)
Outline
Rigidity of Ricci solitonsRigidity: compact caseRigidity: non-compact caseLocally conformally flat case
Gap theoremsDiameter boundsGap theorems: compact caseGap theorems: non-compact case
Maximum principlesIntroductionOmori-Yau maximum principleApplications
Steady solitonsLower bound for the curvature of a steady soliton
![Page 47: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/47.jpg)
Theorem (A. Futaki and Y. Sano, 2010)Let (M,g) be an n-dimensional compact shrinking Ricci soliton.Then
diam(M,g) ≥ 10π13√λ.
Theorem (E. García Río and MFL, 2010)Let (M,g) be an n-dimensional compact shrinking Ricci soliton.Then (M,g) is Einstein if and only if one of the followingconditions holds:
(i) Ric ≥(
1− Rmax − Rmin
(n − 1)λπ2 + Rmax − Rmin
)λg,
(ii) cg ≤ Ric ≤(λ+
c(Rmax − Rmin)
(n − 1)λπ2
)g, for some c > 0
(iii) cg ≤ Ric ≤(
1 +4(Rmax − Rmin)
(n − 1)λπ2
)cg, for some c > 0.
![Page 48: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/48.jpg)
Theorem (A. Futaki and Y. Sano, 2010)Let (M,g) be an n-dimensional compact shrinking Ricci soliton.Then
diam(M,g) ≥ 10π13√λ.
Theorem (E. García Río and MFL, 2010)Let (M,g) be an n-dimensional compact shrinking Ricci soliton.Then (M,g) is Einstein if and only if one of the followingconditions holds:
(i) Ric ≥(
1− Rmax − Rmin
(n − 1)λπ2 + Rmax − Rmin
)λg,
(ii) cg ≤ Ric ≤(λ+
c(Rmax − Rmin)
(n − 1)λπ2
)g, for some c > 0
(iii) cg ≤ Ric ≤(
1 +4(Rmax − Rmin)
(n − 1)λπ2
)cg, for some c > 0.
![Page 49: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/49.jpg)
Assume (i) holds
c =(n − 1)λ2π2
(n − 1)λπ2 + Rmax − Rmin
diam2(M,g) ≥ Rmax − Rmin
λ(λ− c)≥ (n − 1)π2
c
Myers’ theorem:
Ric ≥ cg > 0 =⇒ diam(M,g) ≤ π√
n − 1c
By Cheng M must be the standard sphere
CONTRADICTION!
![Page 50: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/50.jpg)
Assume (i) holds
c =(n − 1)λ2π2
(n − 1)λπ2 + Rmax − Rmin
diam2(M,g) ≥ Rmax − Rmin
λ(λ− c)≥ (n − 1)π2
c
Myers’ theorem:
Ric ≥ cg > 0 =⇒ diam(M,g) ≤ π√
n − 1c
By Cheng M must be the standard sphere
CONTRADICTION!
![Page 51: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/51.jpg)
Assume (i) holds
c =(n − 1)λ2π2
(n − 1)λπ2 + Rmax − Rmin
diam2(M,g) ≥ Rmax − Rmin
λ(λ− c)≥ (n − 1)π2
c
Myers’ theorem:
Ric ≥ cg > 0 =⇒ diam(M,g) ≤ π√
n − 1c
By Cheng M must be the standard sphere
CONTRADICTION!
![Page 52: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/52.jpg)
Assume (i) holds
c =(n − 1)λ2π2
(n − 1)λπ2 + Rmax − Rmin
diam2(M,g) ≥ Rmax − Rmin
λ(λ− c)≥ (n − 1)π2
c
Myers’ theorem:
Ric ≥ cg > 0 =⇒ diam(M,g) ≤ π√
n − 1c
By Cheng M must be the standard sphere
CONTRADICTION!
![Page 53: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/53.jpg)
Assume (i) holds
c =(n − 1)λ2π2
(n − 1)λπ2 + Rmax − Rmin
diam2(M,g) ≥ Rmax − Rmin
λ(λ− c)≥ (n − 1)π2
c
Myers’ theorem:
Ric ≥ cg > 0 =⇒ diam(M,g) ≤ π√
n − 1c
By Cheng M must be the standard sphere
CONTRADICTION!
![Page 54: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/54.jpg)
Assume (i) holds
c =(n − 1)λ2π2
(n − 1)λπ2 + Rmax − Rmin
diam2(M,g) ≥ Rmax − Rmin
λ(λ− c)≥ (n − 1)π2
c
Myers’ theorem:
Ric ≥ cg > 0 =⇒ diam(M,g) ≤ π√
n − 1c
By Cheng M must be the standard sphere
CONTRADICTION!
![Page 55: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/55.jpg)
Theorem (E. García Río and MFL, 2010)Let (M,g) be an n-dimensional compact shrinking Ricci soliton.Then (M,g) is Einstein if and only if
Rmax − nλ ≤(
1 +2n
)1
vol (M,g)
∫M|∇f |2.
Theorem (E. García Río and MFL, 2010)Let (M,g) be an n-dimensional compact shrinking Ricci soliton.Then (M,g) is Einstein if and only if
|Ric − λg| ≤ c ≤ −Λ +√
Λ2 + 8(n − 1)λΛ
4(n − 1),
where Λ = 1vol(M,g)
∫M |∇f |2 denotes the average of the L2-norm
of |∇f |.
![Page 56: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/56.jpg)
Theorem (E. García Río and MFL, 2010)Let (M,g) be an n-dimensional compact shrinking Ricci soliton.Then (M,g) is Einstein if and only if
Rmax − nλ ≤(
1 +2n
)1
vol (M,g)
∫M|∇f |2.
Theorem (E. García Río and MFL, 2010)Let (M,g) be an n-dimensional compact shrinking Ricci soliton.Then (M,g) is Einstein if and only if
|Ric − λg| ≤ c ≤ −Λ +√
Λ2 + 8(n − 1)λΛ
4(n − 1),
where Λ = 1vol(M,g)
∫M |∇f |2 denotes the average of the L2-norm
of |∇f |.
![Page 57: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/57.jpg)
(i)∫
M(∆f )2 =
∫M
((n + 2)λ− R) |∇f |2
(ii) |∇f |2 ≤ Rmax − R∫M
(∆f )2 = (n + 2)λ
∫M|∇f |2 −
∫M
R|∇f |2
≥ (n + 2)λ
∫M|∇f |2 − nλRmaxvol (M,g) +
∫M
R2
= (n + 2)λ
∫M|∇f |2 − nλRmaxvol (M,g)
+n2λ2vol (M,g) +∫
M(∆f )2
Rmax − nλ ≥ n + 2n
1vol (M,g)
∫M|∇f |2
2λf − R = |∇f |2 = Rmax − R ⇒ f is constant
![Page 58: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/58.jpg)
(i)∫
M(∆f )2 =
∫M
((n + 2)λ− R) |∇f |2
(ii) |∇f |2 ≤ Rmax − R∫M
(∆f )2 = (n + 2)λ
∫M|∇f |2 −
∫M
R|∇f |2
≥ (n + 2)λ
∫M|∇f |2 − nλRmaxvol (M,g) +
∫M
R2
= (n + 2)λ
∫M|∇f |2 − nλRmaxvol (M,g)
+n2λ2vol (M,g) +∫
M(∆f )2
Rmax − nλ ≥ n + 2n
1vol (M,g)
∫M|∇f |2
2λf − R = |∇f |2 = Rmax − R ⇒ f is constant
![Page 59: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/59.jpg)
(i)∫
M(∆f )2 =
∫M
((n + 2)λ− R) |∇f |2
(ii) |∇f |2 ≤ Rmax − R∫M
(∆f )2 = (n + 2)λ
∫M|∇f |2 −
∫M
R|∇f |2
≥ (n + 2)λ
∫M|∇f |2 − nλRmaxvol (M,g) +
∫M
R2
= (n + 2)λ
∫M|∇f |2 − nλRmaxvol (M,g)
+n2λ2vol (M,g) +∫
M(∆f )2
Rmax − nλ ≥ n + 2n
1vol (M,g)
∫M|∇f |2
2λf − R = |∇f |2 = Rmax − R ⇒ f is constant
![Page 60: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/60.jpg)
(i)∫
M(∆f )2 =
∫M
((n + 2)λ− R) |∇f |2
(ii) |∇f |2 ≤ Rmax − R∫M
(∆f )2 = (n + 2)λ
∫M|∇f |2 −
∫M
R|∇f |2
≥ (n + 2)λ
∫M|∇f |2 − nλRmaxvol (M,g) +
∫M
R2
= (n + 2)λ
∫M|∇f |2 − nλRmaxvol (M,g)
+n2λ2vol (M,g) +∫
M(∆f )2
Rmax − nλ ≥ n + 2n
1vol (M,g)
∫M|∇f |2
2λf − R = |∇f |2 = Rmax − R ⇒ f is constant
![Page 61: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/61.jpg)
Outline
Rigidity of Ricci solitonsRigidity: compact caseRigidity: non-compact caseLocally conformally flat case
Gap theoremsDiameter boundsGap theorems: compact caseGap theorems: non-compact case
Maximum principlesIntroductionOmori-Yau maximum principleApplications
Steady solitonsLower bound for the curvature of a steady soliton
![Page 62: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/62.jpg)
Theorem (E. García Río and MFL, 2010)Let (M,g) be a complete gradient shrinking Ricci soliton withbounded scalar curvature. Then (M,g) is compact Einstein if
Ric(∇f ,∇f ) ≥ ε
r(x)2 g(∇f ,∇f ),
for sufficiently large r(x), where ε > 0 and r(x) denotes thedistance from a fixed point.
Theorem (E. García Río and MFL, 2010)Let (M,g) be an n-dimensional complete gradient steady Riccisoliton. If
Ric(∇f ,∇f ) ≥ εg(∇f ,∇f ),
where ε is any positive constant, then (M,g) is Ricci flat.
![Page 63: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/63.jpg)
Theorem (E. García Río and MFL, 2010)Let (M,g) be a complete gradient shrinking Ricci soliton withbounded scalar curvature. Then (M,g) is compact Einstein if
Ric(∇f ,∇f ) ≥ ε
r(x)2 g(∇f ,∇f ),
for sufficiently large r(x), where ε > 0 and r(x) denotes thedistance from a fixed point.
Theorem (E. García Río and MFL, 2010)Let (M,g) be an n-dimensional complete gradient steady Riccisoliton. If
Ric(∇f ,∇f ) ≥ εg(∇f ,∇f ),
where ε is any positive constant, then (M,g) is Ricci flat.
![Page 64: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/64.jpg)
Theorem (P. Li)If a complete manifold has Ricci curvature bounded from belowby εr(x)−2, for some constant ε > 1/4 and all r(x) > 1, then Mmust be compact.
2λf = R + |∇f |2
There exists c such that f (x) ≥ 14(r(x)− c)2
H.-D. Cao and D. Zhou; On complete gradient shrinkingsolitons
γ : [0,+∞)→ M an integral curve of ∇f (note that ∇f is acomplete vector field)
Z.-H. Zhang; On the completeness of gradient Ricci solitons
![Page 65: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/65.jpg)
Theorem (P. Li)If a complete manifold has Ricci curvature bounded from belowby εr(x)−2, for some constant ε > 1/4 and all r(x) > 1, then Mmust be compact.
2λf = R + |∇f |2
There exists c such that f (x) ≥ 14(r(x)− c)2
H.-D. Cao and D. Zhou; On complete gradient shrinkingsolitons
γ : [0,+∞)→ M an integral curve of ∇f (note that ∇f is acomplete vector field)
Z.-H. Zhang; On the completeness of gradient Ricci solitons
![Page 66: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/66.jpg)
Theorem (P. Li)If a complete manifold has Ricci curvature bounded from belowby εr(x)−2, for some constant ε > 1/4 and all r(x) > 1, then Mmust be compact.
2λf = R + |∇f |2
There exists c such that f (x) ≥ 14(r(x)− c)2
H.-D. Cao and D. Zhou; On complete gradient shrinkingsolitons
γ : [0,+∞)→ M an integral curve of ∇f (note that ∇f is acomplete vector field)
Z.-H. Zhang; On the completeness of gradient Ricci solitons
![Page 67: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/67.jpg)
Theorem (P. Li)If a complete manifold has Ricci curvature bounded from belowby εr(x)−2, for some constant ε > 1/4 and all r(x) > 1, then Mmust be compact.
2λf = R + |∇f |2
There exists c such that f (x) ≥ 14(r(x)− c)2
H.-D. Cao and D. Zhou; On complete gradient shrinkingsolitons
γ : [0,+∞)→ M an integral curve of ∇f (note that ∇f is acomplete vector field)
Z.-H. Zhang; On the completeness of gradient Ricci solitons
![Page 68: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/68.jpg)
For r ≥ r1
(R γ)′(t) = 2Ric(∇f (γ(t)),∇f (γ(t)) ≥ 2εr(x)2 |∇f |2
Since R is bounded, for some k1 > 0 and k2 > 0
|∇f |2 = 2λf − R ≥ λ2(r(x)− c)2 − R ≥ k1r(x)2 ≥ k2R2r(x)2
for r(x) ≥ r2 ≥ r1
p ∈ Ω = x ∈ M /2λf (x) ≥ k3 such that B(x0, r2) ⊂ Ω
Since f is increasing along the integral curves of ∇f , if wesuppose that γ(0) = p, then γ(t) ∈ M \ Ω for all t ≥ 0
![Page 69: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/69.jpg)
For r ≥ r1
(R γ)′(t) = 2Ric(∇f (γ(t)),∇f (γ(t)) ≥ 2εr(x)2 |∇f |2
Since R is bounded, for some k1 > 0 and k2 > 0
|∇f |2 = 2λf − R ≥ λ2(r(x)− c)2 − R ≥ k1r(x)2 ≥ k2R2r(x)2
for r(x) ≥ r2 ≥ r1
p ∈ Ω = x ∈ M /2λf (x) ≥ k3 such that B(x0, r2) ⊂ Ω
Since f is increasing along the integral curves of ∇f , if wesuppose that γ(0) = p, then γ(t) ∈ M \ Ω for all t ≥ 0
![Page 70: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/70.jpg)
For r ≥ r1
(R γ)′(t) = 2Ric(∇f (γ(t)),∇f (γ(t)) ≥ 2εr(x)2 |∇f |2
Since R is bounded, for some k1 > 0 and k2 > 0
|∇f |2 = 2λf − R ≥ λ2(r(x)− c)2 − R ≥ k1r(x)2 ≥ k2R2r(x)2
for r(x) ≥ r2 ≥ r1
p ∈ Ω = x ∈ M /2λf (x) ≥ k3 such that B(x0, r2) ⊂ Ω
Since f is increasing along the integral curves of ∇f , if wesuppose that γ(0) = p, then γ(t) ∈ M \ Ω for all t ≥ 0
![Page 71: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/71.jpg)
For r ≥ r1
(R γ)′(t) = 2Ric(∇f (γ(t)),∇f (γ(t)) ≥ 2εr(x)2 |∇f |2
Since R is bounded, for some k1 > 0 and k2 > 0
|∇f |2 = 2λf − R ≥ λ2(r(x)− c)2 − R ≥ k1r(x)2 ≥ k2R2r(x)2
for r(x) ≥ r2 ≥ r1
p ∈ Ω = x ∈ M /2λf (x) ≥ k3 such that B(x0, r2) ⊂ Ω
Since f is increasing along the integral curves of ∇f , if wesuppose that γ(0) = p, then γ(t) ∈ M \ Ω for all t ≥ 0
![Page 72: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/72.jpg)
We have that
(R γ)′(t) = g(∇R(γ(t)), γ′(t)) ≥ 2εk2R2(γ(t)),
along γ ∫ t
0
(R γ)′(t)(R γ)2(t)
ds ≥∫ t
02εk2dt
1R(γ(0))
− 1R(γ(t))
≥ 2εk2t
Contradiction for t going to infinite.
![Page 73: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/73.jpg)
We have that
(R γ)′(t) = g(∇R(γ(t)), γ′(t)) ≥ 2εk2R2(γ(t)),
along γ ∫ t
0
(R γ)′(t)(R γ)2(t)
ds ≥∫ t
02εk2dt
1R(γ(0))
− 1R(γ(t))
≥ 2εk2t
Contradiction for t going to infinite.
![Page 74: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/74.jpg)
We have that
(R γ)′(t) = g(∇R(γ(t)), γ′(t)) ≥ 2εk2R2(γ(t)),
along γ ∫ t
0
(R γ)′(t)(R γ)2(t)
ds ≥∫ t
02εk2dt
1R(γ(0))
− 1R(γ(t))
≥ 2εk2t
Contradiction for t going to infinite.
![Page 75: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/75.jpg)
We have that
(R γ)′(t) = g(∇R(γ(t)), γ′(t)) ≥ 2εk2R2(γ(t)),
along γ ∫ t
0
(R γ)′(t)(R γ)2(t)
ds ≥∫ t
02εk2dt
1R(γ(0))
− 1R(γ(t))
≥ 2εk2t
Contradiction for t going to infinite.
![Page 76: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/76.jpg)
Outline
Rigidity of Ricci solitonsRigidity: compact caseRigidity: non-compact caseLocally conformally flat case
Gap theoremsDiameter boundsGap theorems: compact caseGap theorems: non-compact case
Maximum principlesIntroductionOmori-Yau maximum principleApplications
Steady solitonsLower bound for the curvature of a steady soliton
![Page 77: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/77.jpg)
A Riemannian manifold (M,g) is said to satisfy the Omori-Yaumaximum principle if given any function u ∈ C2(M) withu∗ = supM u < +∞, there exists a sequence (xk ) of points onM satisfying
i) u(xk ) > u∗ − 1k, ii) |(∇u)(xk )| < 1
k, iii) (∆u)(xk ) <
1k,
for each k ∈ N. If, instead of iii) we assume that
Hu(xk ) <1k
g,
in the sense of quadratic forms, then it is said that theRiemannian manifold satisfies the Omori-Yau maximumprinciple for the Hessian.The f -Laplacian is
∆f = ef div(e−f∇) = ∆− g(∇f , ·)
![Page 78: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/78.jpg)
A Riemannian manifold (M,g) is said to satisfy the Omori-Yaumaximum principle if given any function u ∈ C2(M) withu∗ = supM u < +∞, there exists a sequence (xk ) of points onM satisfying
i) u(xk ) > u∗ − 1k, ii) |(∇u)(xk )| < 1
k, iii) (∆u)(xk ) <
1k,
for each k ∈ N. If, instead of iii) we assume that
Hu(xk ) <1k
g,
in the sense of quadratic forms, then it is said that theRiemannian manifold satisfies the Omori-Yau maximumprinciple for the Hessian.The f -Laplacian is
∆f = ef div(e−f∇) = ∆− g(∇f , ·)
![Page 79: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/79.jpg)
In 1967 Omori showed that the Omori-Yau maximum principlefor the Hessian is satisfied by Riemannian manifolds withcurvature bounded from below.
H. Omori; Isometric immersions of Riemannian manifolds
In 1975 Yau proved that the Omori-Yau maximum principle issatisfied by Riemannian manifolds with Ricci curvaturebounded from below.
S. T. Yau; Harmonic functions on complete Riemannianmanifolds
From now on we will work with Ricci solitons normalized in thesense
Rc + Hf = ±12
g
![Page 80: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/80.jpg)
In 1967 Omori showed that the Omori-Yau maximum principlefor the Hessian is satisfied by Riemannian manifolds withcurvature bounded from below.
H. Omori; Isometric immersions of Riemannian manifolds
In 1975 Yau proved that the Omori-Yau maximum principle issatisfied by Riemannian manifolds with Ricci curvaturebounded from below.
S. T. Yau; Harmonic functions on complete Riemannianmanifolds
From now on we will work with Ricci solitons normalized in thesense
Rc + Hf = ±12
g
![Page 81: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/81.jpg)
In 1967 Omori showed that the Omori-Yau maximum principlefor the Hessian is satisfied by Riemannian manifolds withcurvature bounded from below.
H. Omori; Isometric immersions of Riemannian manifolds
In 1975 Yau proved that the Omori-Yau maximum principle issatisfied by Riemannian manifolds with Ricci curvaturebounded from below.
S. T. Yau; Harmonic functions on complete Riemannianmanifolds
From now on we will work with Ricci solitons normalized in thesense
Rc + Hf = ±12
g
![Page 82: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/82.jpg)
Outline
Rigidity of Ricci solitonsRigidity: compact caseRigidity: non-compact caseLocally conformally flat case
Gap theoremsDiameter boundsGap theorems: compact caseGap theorems: non-compact case
Maximum principlesIntroductionOmori-Yau maximum principleApplications
Steady solitonsLower bound for the curvature of a steady soliton
![Page 83: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/83.jpg)
Theorem (E. García Río and MFL, 2010)Let (Mn,g) be an n-dimensional complete noncompact gradientshrinking Ricci soliton. Then (M,g) satisfies the Omori-Yaumaximum principle.Moreover, if there exists C > 0 such that Ric ≥ −Cr(x)2, wherer(x) denotes the distance to a fixed point, then the Omori-Yaumaximum principle for the Hessian holds on (M,g).
Theorem (E. García Río and MFL, 2010)Let (Mn,g) be an n-dimensional complete noncompact gradientshrinking Ricci soliton. Then (M,g) satisfies the Omori-Yaumaximum principle for the f -Laplacian.
![Page 84: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/84.jpg)
Theorem (E. García Río and MFL, 2010)Let (Mn,g) be an n-dimensional complete noncompact gradientshrinking Ricci soliton. Then (M,g) satisfies the Omori-Yaumaximum principle.Moreover, if there exists C > 0 such that Ric ≥ −Cr(x)2, wherer(x) denotes the distance to a fixed point, then the Omori-Yaumaximum principle for the Hessian holds on (M,g).
Theorem (E. García Río and MFL, 2010)Let (Mn,g) be an n-dimensional complete noncompact gradientshrinking Ricci soliton. Then (M,g) satisfies the Omori-Yaumaximum principle for the f -Laplacian.
![Page 85: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/85.jpg)
S. Pigola, M. Rigoli and A. Setti; Maximum principles onRiemannian manifolds and applications
(M,g) satisfies the Omori-Yau maximum principle if ∃0 ≤ ϕ ∈ C2, s. t.
ϕ(x) −→ +∞ as x −→∞, (1)
∃A < 0 such that |∇ϕ| ≤ A√ϕ off a compact set, and (2)
∃B > 0 s. t. ∆ϕ ≤ B√ϕ√
G(√ϕ), off a compact set, (3)
where G is a smooth function on [0,+∞) satisfying
i) G(0) > 0, ii) G′(t) ≥ 0, on [0,+∞),
iii)∫ ∞
0
dt√G(t)
=∞, iv) lim supt→∞
tG(√
t)G(t)
<∞.(4)
∃B > 0 s. t. Hϕ ≤ B√ϕ√
G(√ϕ), off a compact set (5)
(M,g) satisfies the Omori-Yau maximum principle for Hessian.
![Page 86: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/86.jpg)
S. Pigola, M. Rigoli and A. Setti; Maximum principles onRiemannian manifolds and applications
(M,g) satisfies the Omori-Yau maximum principle if ∃0 ≤ ϕ ∈ C2, s. t.
ϕ(x) −→ +∞ as x −→∞, (1)
∃A < 0 such that |∇ϕ| ≤ A√ϕ off a compact set, and (2)
∃B > 0 s. t. ∆ϕ ≤ B√ϕ√
G(√ϕ), off a compact set, (3)
where G is a smooth function on [0,+∞) satisfying
i) G(0) > 0, ii) G′(t) ≥ 0, on [0,+∞),
iii)∫ ∞
0
dt√G(t)
=∞, iv) lim supt→∞
tG(√
t)G(t)
<∞.(4)
∃B > 0 s. t. Hϕ ≤ B√ϕ√
G(√ϕ), off a compact set (5)
(M,g) satisfies the Omori-Yau maximum principle for Hessian.
![Page 87: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/87.jpg)
Theorem (S. Pigola, M. Rimoldi and A. Setti, 2009)Let (M,g) be a complete gradient (shrinking, steady orexpanding) Ricci solitons. Then, the weak maximum principleat infinity for the f -Laplacian holds.Given any function u ∈ C2(M) with u∗ = supM u < +∞, thereexists a sequence (xk ) of points on M satisfying
i) u(xk ) > u∗ − 1k, ii) (∆f u)(xk ) <
1k,
for each k ∈ N.
Theorem (S. Pigola, M. Rimoldi and A. Setti, 2009)Let (M,g) be a complete gradient (shrinking, steady orexpanding) Ricci solitons. Then it is stochastically complete.
![Page 88: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/88.jpg)
Theorem (S. Pigola, M. Rimoldi and A. Setti, 2009)Let (M,g) be a complete gradient (shrinking, steady orexpanding) Ricci solitons. Then, the weak maximum principleat infinity for the f -Laplacian holds.Given any function u ∈ C2(M) with u∗ = supM u < +∞, thereexists a sequence (xk ) of points on M satisfying
i) u(xk ) > u∗ − 1k, ii) (∆f u)(xk ) <
1k,
for each k ∈ N.
Theorem (S. Pigola, M. Rimoldi and A. Setti, 2009)Let (M,g) be a complete gradient (shrinking, steady orexpanding) Ricci solitons. Then it is stochastically complete.
![Page 89: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/89.jpg)
Theorem (S. Pigola, M. Rimoldi and A. Setti, 2009)Let (M,g) be a complete gradient (shrinking, steady orexpanding) Ricci solitons. Then, the weak maximum principleat infinity for the f -Laplacian holds.Given any function u ∈ C2(M) with u∗ = supM u < +∞, thereexists a sequence (xk ) of points on M satisfying
i) u(xk ) > u∗ − 1k, ii) (∆f u)(xk ) <
1k,
for each k ∈ N.
Theorem (S. Pigola, M. Rimoldi and A. Setti, 2009)Let (M,g) be a complete gradient (shrinking, steady orexpanding) Ricci solitons. Then it is stochastically complete.
![Page 90: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/90.jpg)
Outline
Rigidity of Ricci solitonsRigidity: compact caseRigidity: non-compact caseLocally conformally flat case
Gap theoremsDiameter boundsGap theorems: compact caseGap theorems: non-compact case
Maximum principlesIntroductionOmori-Yau maximum principleApplications
Steady solitonsLower bound for the curvature of a steady soliton
![Page 91: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/91.jpg)
Theorem (E. García Río and MFL, 2010)Let (Mn,g) be an n-dimensional complete gradient shrinkingRicci soliton. Then:
(i) (M,g) has constant scalar curvature if and only if
2|Ric|2 ≤ R + c|∇R|2
R + 1, for some c ≥ 0.
(ii) (M,g) is isometric to (Rn,geuc) if and only if
2|Ric|2 ≤ (1− ε)R + c|∇R|2
R + 1, for some c ≥ 0 and ε > 0.
![Page 92: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/92.jpg)
Theorem (E. García Río and MFL, 2010)Let (M,g) be a complete gradient shrinking Ricci soliton withbounded nonnegative Ricci tensor. Then (M,g) is rigid if andonly if the sectional curvature is bounded from above by|Ric|2
2(R2−|Ric|2).
We consider an orthonormal frame E1, . . . ,En formed byeigenvectors of the Ricci operator.
∆Rii = g(∇Rii ,∇f ) + Rii − 2RijjiR jj ,
where Rii = Ric(Ei ,Ei), Rijji = R(Ei ,Ej ,Ej ,Ei)
∆f |Ric|2 = 2|Ric|2−4R iiRijjiR jj+2∇Rii∇R ii ≥ 2|Ric|2−4R iiRijjiR jj .
![Page 93: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/93.jpg)
Theorem (E. García Río and MFL, 2010)Let (M,g) be a complete gradient shrinking Ricci soliton withbounded nonnegative Ricci tensor. Then (M,g) is rigid if andonly if the sectional curvature is bounded from above by|Ric|2
2(R2−|Ric|2).
We consider an orthonormal frame E1, . . . ,En formed byeigenvectors of the Ricci operator.
∆Rii = g(∇Rii ,∇f ) + Rii − 2RijjiR jj ,
where Rii = Ric(Ei ,Ei), Rijji = R(Ei ,Ej ,Ej ,Ei)
∆f |Ric|2 = 2|Ric|2−4R iiRijjiR jj+2∇Rii∇R ii ≥ 2|Ric|2−4R iiRijjiR jj .
![Page 94: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/94.jpg)
Theorem (E. García Río and MFL, 2010)Let (M,g) be a complete gradient shrinking Ricci soliton withbounded nonnegative Ricci tensor. Then (M,g) is rigid if andonly if the sectional curvature is bounded from above by|Ric|2
2(R2−|Ric|2).
We consider an orthonormal frame E1, . . . ,En formed byeigenvectors of the Ricci operator.
∆Rii = g(∇Rii ,∇f ) + Rii − 2RijjiR jj ,
where Rii = Ric(Ei ,Ei), Rijji = R(Ei ,Ej ,Ej ,Ei)
∆f |Ric|2 = 2|Ric|2−4R iiRijjiR jj+2∇Rii∇R ii ≥ 2|Ric|2−4R iiRijjiR jj .
![Page 95: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/95.jpg)
Theorem (E. García Río and MFL, 2010)Let (M,g) be a complete gradient shrinking Ricci soliton withbounded nonnegative Ricci tensor. Then (M,g) is rigid if andonly if the sectional curvature is bounded from above by|Ric|2
2(R2−|Ric|2).
We consider an orthonormal frame E1, . . . ,En formed byeigenvectors of the Ricci operator.
∆Rii = g(∇Rii ,∇f ) + Rii − 2RijjiR jj ,
where Rii = Ric(Ei ,Ei), Rijji = R(Ei ,Ej ,Ej ,Ei)
∆f |Ric|2 = 2|Ric|2−4R iiRijjiR jj+2∇Rii∇R ii ≥ 2|Ric|2−4R iiRijjiR jj .
![Page 96: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/96.jpg)
Under our assumption one has
4(
R iiRijjiR jj)≤ 4|Ric|2
2(R2 − |Ric|2)(R2 − |Ric|2) = 2|Ric|2.
Then ∆f |Ric|2 ≥ 0, and by f -parabolicity it follows that |Ric|2 isconstant.
0 = ∆f |Ric|2 = 2|Ric|2 − 4R iiRijjiR jj + 2∇Rii∇R ii ≥ 0
⇒n∑
i=1
|∇Rii |2 = 0
The Ricci soliton is rigid.
![Page 97: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/97.jpg)
Under our assumption one has
4(
R iiRijjiR jj)≤ 4|Ric|2
2(R2 − |Ric|2)(R2 − |Ric|2) = 2|Ric|2.
Then ∆f |Ric|2 ≥ 0, and by f -parabolicity it follows that |Ric|2 isconstant.
0 = ∆f |Ric|2 = 2|Ric|2 − 4R iiRijjiR jj + 2∇Rii∇R ii ≥ 0
⇒n∑
i=1
|∇Rii |2 = 0
The Ricci soliton is rigid.
![Page 98: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/98.jpg)
Under our assumption one has
4(
R iiRijjiR jj)≤ 4|Ric|2
2(R2 − |Ric|2)(R2 − |Ric|2) = 2|Ric|2.
Then ∆f |Ric|2 ≥ 0, and by f -parabolicity it follows that |Ric|2 isconstant.
0 = ∆f |Ric|2 = 2|Ric|2 − 4R iiRijjiR jj + 2∇Rii∇R ii ≥ 0
⇒n∑
i=1
|∇Rii |2 = 0
The Ricci soliton is rigid.
![Page 99: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/99.jpg)
Under our assumption one has
4(
R iiRijjiR jj)≤ 4|Ric|2
2(R2 − |Ric|2)(R2 − |Ric|2) = 2|Ric|2.
Then ∆f |Ric|2 ≥ 0, and by f -parabolicity it follows that |Ric|2 isconstant.
0 = ∆f |Ric|2 = 2|Ric|2 − 4R iiRijjiR jj + 2∇Rii∇R ii ≥ 0
⇒n∑
i=1
|∇Rii |2 = 0
The Ricci soliton is rigid.
![Page 100: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/100.jpg)
Theorem (E. García Río and MFL, 2010)Let (M,g) be a complete gradient steady Ricci soliton. ThenR∗ = infM R = 0.
∆f R = −2|Ric|2.
There exists a sequence (xk ) of points of M such thatR(xk ) < R∗ + 1
k and (∆f R)(xk ) > − 1k .
1k≥ 2|Ric(xk )|2 ≥ 2R(xk )2
n.
Taking the limit when k goes to infinity we get that R∗ = 0.
![Page 101: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/101.jpg)
Theorem (E. García Río and MFL, 2010)Let (M,g) be a complete gradient steady Ricci soliton. ThenR∗ = infM R = 0.
∆f R = −2|Ric|2.
There exists a sequence (xk ) of points of M such thatR(xk ) < R∗ + 1
k and (∆f R)(xk ) > − 1k .
1k≥ 2|Ric(xk )|2 ≥ 2R(xk )2
n.
Taking the limit when k goes to infinity we get that R∗ = 0.
![Page 102: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/102.jpg)
Theorem (E. García Río and MFL, 2010)Let (M,g) be a complete gradient steady Ricci soliton. ThenR∗ = infM R = 0.
∆f R = −2|Ric|2.
There exists a sequence (xk ) of points of M such thatR(xk ) < R∗ + 1
k and (∆f R)(xk ) > − 1k .
1k≥ 2|Ric(xk )|2 ≥ 2R(xk )2
n.
Taking the limit when k goes to infinity we get that R∗ = 0.
![Page 103: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/103.jpg)
Theorem (E. García Río and MFL, 2010)Let (M,g) be a complete gradient steady Ricci soliton. ThenR∗ = infM R = 0.
∆f R = −2|Ric|2.
There exists a sequence (xk ) of points of M such thatR(xk ) < R∗ + 1
k and (∆f R)(xk ) > − 1k .
1k≥ 2|Ric(xk )|2 ≥ 2R(xk )2
n.
Taking the limit when k goes to infinity we get that R∗ = 0.
![Page 104: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/104.jpg)
Theorem (E. García Río and MFL, 2010)Let (M,g) be a complete gradient steady Ricci soliton. ThenR∗ = infM R = 0.
∆f R = −2|Ric|2.
There exists a sequence (xk ) of points of M such thatR(xk ) < R∗ + 1
k and (∆f R)(xk ) > − 1k .
1k≥ 2|Ric(xk )|2 ≥ 2R(xk )2
n.
Taking the limit when k goes to infinity we get that R∗ = 0.
![Page 105: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/105.jpg)
Theorem (E. García Río and MFL, 2010)Let (M,g) be a complete gradient expanding Ricci soliton withRic ≤ 0. If R∗ = supM R < 0 then −n
2 ≤ R ≤ −12 .
∆f R = −R − 2|Ric|2 ≥ −R − 2R2 = −R(1 + 2R)
R∗(1 + 2R∗) ≥ 0⇒ R∗ ≤ −12
![Page 106: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/106.jpg)
Theorem (E. García Río and MFL, 2010)Let (M,g) be a complete gradient expanding Ricci soliton withRic ≤ 0. If R∗ = supM R < 0 then −n
2 ≤ R ≤ −12 .
∆f R = −R − 2|Ric|2 ≥ −R − 2R2 = −R(1 + 2R)
R∗(1 + 2R∗) ≥ 0⇒ R∗ ≤ −12
![Page 107: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/107.jpg)
Theorem (E. García Río and MFL, 2010)Let (M,g) be a complete gradient expanding Ricci soliton withRic ≤ 0. If R∗ = supM R < 0 then −n
2 ≤ R ≤ −12 .
∆f R = −R − 2|Ric|2 ≥ −R − 2R2 = −R(1 + 2R)
R∗(1 + 2R∗) ≥ 0⇒ R∗ ≤ −12
![Page 108: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/108.jpg)
Outline
Rigidity of Ricci solitonsRigidity: compact caseRigidity: non-compact caseLocally conformally flat case
Gap theoremsDiameter boundsGap theorems: compact caseGap theorems: non-compact case
Maximum principlesIntroductionOmori-Yau maximum principleApplications
Steady solitonsLower bound for the curvature of a steady soliton
![Page 109: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/109.jpg)
TheoremLet (Mn,g, f ) be a complete noncompact nonflat shrinkinggradient Ricci soliton. Then for any given point O ∈ M thereexists a constant CO > 0 such that R(x)d(x ,O)2 ≥ C−1
Owherever d(x ,O) ≥ CO.
B. Chow, P. Lu and B. Yang; A lower bound for the scalarcurvature of noncompact nonflat Ricci shrinkers
TheoremLet (Mn,g, f ) be a complete steady gradient Ricci solitons withRc = −Hf and R + |∇f |2 = 1. If limx→∞ f (x) = −∞ and f ≤ 0,then R ≥ 1√
n2 +2
ef .
B. Chow, P. Lu and B. Yang; A lower bound for the scalarcurvature of certain steady gradient Ricci solitons
![Page 110: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/110.jpg)
TheoremLet (Mn,g, f ) be a complete noncompact nonflat shrinkinggradient Ricci soliton. Then for any given point O ∈ M thereexists a constant CO > 0 such that R(x)d(x ,O)2 ≥ C−1
Owherever d(x ,O) ≥ CO.
B. Chow, P. Lu and B. Yang; A lower bound for the scalarcurvature of noncompact nonflat Ricci shrinkers
TheoremLet (Mn,g, f ) be a complete steady gradient Ricci solitons withRc = −Hf and R + |∇f |2 = 1. If limx→∞ f (x) = −∞ and f ≤ 0,then R ≥ 1√
n2 +2
ef .
B. Chow, P. Lu and B. Yang; A lower bound for the scalarcurvature of certain steady gradient Ricci solitons
![Page 111: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/111.jpg)
Theorem (E. García Río and MFL, 2011)Let (M,g) be a complete gradient steady Ricci soliton satisfying|Ric|2 ≤ R2
2 . Then
R(x) ≥ ksech2 r(x)
2,
where r(x) is the distance from a fixed point O ∈ M and k ≤ 1is a constant that only depends on O and R(O).
|Hf |2 = |∇∇f |2 ≥ |∇|∇f ||2 = |∇√
1− R|2 =|∇R|2
4|∇f |2
|∇R|2 ≤ 4|Hf |2|∇f |2
|Hf |2 = |Rc|2 ≤ R2
2⇒ |∇R|
R√
1− R≤ 1
![Page 112: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/112.jpg)
Theorem (E. García Río and MFL, 2011)Let (M,g) be a complete gradient steady Ricci soliton satisfying|Ric|2 ≤ R2
2 . Then
R(x) ≥ ksech2 r(x)
2,
where r(x) is the distance from a fixed point O ∈ M and k ≤ 1is a constant that only depends on O and R(O).
|Hf |2 = |∇∇f |2 ≥ |∇|∇f ||2 = |∇√
1− R|2 =|∇R|2
4|∇f |2
|∇R|2 ≤ 4|Hf |2|∇f |2
|Hf |2 = |Rc|2 ≤ R2
2⇒ |∇R|
R√
1− R≤ 1
![Page 113: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/113.jpg)
Theorem (E. García Río and MFL, 2011)Let (M,g) be a complete gradient steady Ricci soliton satisfying|Ric|2 ≤ R2
2 . Then
R(x) ≥ ksech2 r(x)
2,
where r(x) is the distance from a fixed point O ∈ M and k ≤ 1is a constant that only depends on O and R(O).
|Hf |2 = |∇∇f |2 ≥ |∇|∇f ||2 = |∇√
1− R|2 =|∇R|2
4|∇f |2
|∇R|2 ≤ 4|Hf |2|∇f |2
|Hf |2 = |Rc|2 ≤ R2
2⇒ |∇R|
R√
1− R≤ 1
![Page 114: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/114.jpg)
Theorem (E. García Río and MFL, 2011)Let (M,g) be a complete gradient steady Ricci soliton satisfying|Ric|2 ≤ R2
2 . Then
R(x) ≥ ksech2 r(x)
2,
where r(x) is the distance from a fixed point O ∈ M and k ≤ 1is a constant that only depends on O and R(O).
|Hf |2 = |∇∇f |2 ≥ |∇|∇f ||2 = |∇√
1− R|2 =|∇R|2
4|∇f |2
|∇R|2 ≤ 4|Hf |2|∇f |2
|Hf |2 = |Rc|2 ≤ R2
2⇒ |∇R|
R√
1− R≤ 1
![Page 115: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/115.jpg)
Integrating −(Rγ)′
R√
1−Ralong a minimizing geodesic γ(s)
[ln
1 +√
1− R1−√
1− R
]t
0= −
∫ l
0
(R γ)′
R√
1− Rds ≤
∫ t
0
|∇R|R√
1− Rds ≤ t
Writing c =1+√
1−R(O)
1−√
1−R(O)we get that
1 +√
1− R(γ(t)) ≤ cet (1−√
1− R(γ(t)))
R(γ(t)) ≥ 4cc2et + 2c + e−t
Since c ≥ 1 we have that
R(γ(t)) ≥ 4cc2et + 2c + e−t ≥
4cc2et + 2c2 + c2e−t =
1c
sech2 t2
![Page 116: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/116.jpg)
Integrating −(Rγ)′
R√
1−Ralong a minimizing geodesic γ(s)
[ln
1 +√
1− R1−√
1− R
]t
0= −
∫ l
0
(R γ)′
R√
1− Rds ≤
∫ t
0
|∇R|R√
1− Rds ≤ t
Writing c =1+√
1−R(O)
1−√
1−R(O)we get that
1 +√
1− R(γ(t)) ≤ cet (1−√
1− R(γ(t)))
R(γ(t)) ≥ 4cc2et + 2c + e−t
Since c ≥ 1 we have that
R(γ(t)) ≥ 4cc2et + 2c + e−t ≥
4cc2et + 2c2 + c2e−t =
1c
sech2 t2
![Page 117: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/117.jpg)
Integrating −(Rγ)′
R√
1−Ralong a minimizing geodesic γ(s)
[ln
1 +√
1− R1−√
1− R
]t
0= −
∫ l
0
(R γ)′
R√
1− Rds ≤
∫ t
0
|∇R|R√
1− Rds ≤ t
Writing c =1+√
1−R(O)
1−√
1−R(O)we get that
1 +√
1− R(γ(t)) ≤ cet (1−√
1− R(γ(t)))
R(γ(t)) ≥ 4cc2et + 2c + e−t
Since c ≥ 1 we have that
R(γ(t)) ≥ 4cc2et + 2c + e−t ≥
4cc2et + 2c2 + c2e−t =
1c
sech2 t2
![Page 118: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/118.jpg)
Integrating −(Rγ)′
R√
1−Ralong a minimizing geodesic γ(s)
[ln
1 +√
1− R1−√
1− R
]t
0= −
∫ l
0
(R γ)′
R√
1− Rds ≤
∫ t
0
|∇R|R√
1− Rds ≤ t
Writing c =1+√
1−R(O)
1−√
1−R(O)we get that
1 +√
1− R(γ(t)) ≤ cet (1−√
1− R(γ(t)))
R(γ(t)) ≥ 4cc2et + 2c + e−t
Since c ≥ 1 we have that
R(γ(t)) ≥ 4cc2et + 2c + e−t ≥
4cc2et + 2c2 + c2e−t =
1c
sech2 t2
![Page 119: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/119.jpg)
The scalar curvature of Hamilton’s cigar soliton(R2,
dx2 + dy2
1 + x2 + y2
)satisfies
R(x) = 4sech2r(x)
The scalar curvature of normalized Hamilton’s cigar soliton(R2,
4(dx2 + dy2)
1 + x2 + y2
)satisfies
R(x) = sech2 r(x)
2Our inequality is SHARP
![Page 120: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/120.jpg)
The scalar curvature of Hamilton’s cigar soliton(R2,
dx2 + dy2
1 + x2 + y2
)satisfies
R(x) = 4sech2r(x)
The scalar curvature of normalized Hamilton’s cigar soliton(R2,
4(dx2 + dy2)
1 + x2 + y2
)satisfies
R(x) = sech2 r(x)
2Our inequality is SHARP
![Page 121: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/121.jpg)
The scalar curvature of Hamilton’s cigar soliton(R2,
dx2 + dy2
1 + x2 + y2
)satisfies
R(x) = 4sech2r(x)
The scalar curvature of normalized Hamilton’s cigar soliton(R2,
4(dx2 + dy2)
1 + x2 + y2
)satisfies
R(x) = sech2 r(x)
2Our inequality is SHARP
![Page 122: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/122.jpg)
Theorem (E. García Río and MFL, 2011)Let (M,g) be a complete gradient steady Ricci soliton withnonnegative Ricci curvature normalized as before. Then
R(x) ≥ ksech2r(x),
where r(x) is the distance from a fixed point O ∈ M and k ≤ 1is a constant that only depends on O and R(O).
|∇R|2 ≤ 4|Hf |2|∇f |2
Since |Hf |2 = |Rc|2 ≤ R2 one has
|∇R|R√
1− R≤ 2
![Page 123: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/123.jpg)
Theorem (E. García Río and MFL, 2011)Let (M,g) be a complete gradient steady Ricci soliton withnonnegative Ricci curvature normalized as before. Then
R(x) ≥ ksech2r(x),
where r(x) is the distance from a fixed point O ∈ M and k ≤ 1is a constant that only depends on O and R(O).
|∇R|2 ≤ 4|Hf |2|∇f |2
Since |Hf |2 = |Rc|2 ≤ R2 one has
|∇R|R√
1− R≤ 2
![Page 124: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/124.jpg)
Theorem (E. García Río and MFL, 2011)Let (M,g) be a complete gradient steady Ricci soliton withnonnegative Ricci curvature normalized as before. Then
R(x) ≥ ksech2r(x),
where r(x) is the distance from a fixed point O ∈ M and k ≤ 1is a constant that only depends on O and R(O).
|∇R|2 ≤ 4|Hf |2|∇f |2
Since |Hf |2 = |Rc|2 ≤ R2 one has
|∇R|R√
1− R≤ 2
![Page 125: On gradient Ricci solitons](https://reader033.vdocuments.co/reader033/viewer/2022052901/5566c6ecd8b42aac288b5223/html5/thumbnails/125.jpg)
THANK YOU VERY MUCHFOR YOUR ATTENTION