1. Rigidity, gap theorems and maximum principles for Ricci solitons Manuel Fernández López Consellería de Educación e Ordenación Universitaria Xunta de Galicia Galicia SPAIN (joint work with Eduardo García Río) Ricci Solitons Days in Pisa 4-8th April 2011
2. Outline Rigidity of Ricci solitons Rigidity: compact case Rigidity: non-compact case Locally conformally ﬂat case Gap theorems Diameter bounds Gap theorems: compact case Gap theorems: non-compact case Maximum principles Introduction Omori-Yau maximum principle Applications Steady solitons Lower bound for the curvature of a steady soliton
3. 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 R is constant and sec(E, f ) ≥ 0 (sec(E, f ) ≤ 0) R is constant and 0 ≤ Ric ≤ λg (λg ≤ Ric ≤ 0) The curvature tensor is harmonic Ric ≥ 0 (Ric ≤ 0) and sec(E, f) = 0
4. 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 R is constant and sec(E, f ) ≥ 0 (sec(E, f ) ≤ 0) R is constant and 0 ≤ Ric ≤ λg (λg ≤ Ric ≤ 0) The curvature tensor is harmonic Ric ≥ 0 (Ric ≤ 0) and sec(E, f) = 0
5. Outline Rigidity of Ricci solitons Rigidity: compact case Rigidity: non-compact case Locally conformally ﬂat case Gap theorems Diameter bounds Gap theorems: compact case Gap theorems: non-compact case Maximum principles Introduction Omori-Yau maximum principle Applications Steady solitons Lower bound for the curvature of a steady soliton
6. Theorem (Eminenti, LaNave and Mantegazza, 2008)Let (M n , 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 (M n , 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
7. Theorem (Eminenti, LaNave and Mantegazza, 2008)Let (M n , 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 (M n , 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
8. Theorem (Eminenti, LaNave and Mantegazza, 2008)Let (M n , 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 (M n , 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
9. RThe Schouten tensor S = Rc − g is a Codazzi tensor 2(n − 1) X (R) Y (R)( X Rc)(Y , Z ) − ( Y Rc)(X , Z ) = g(Y , Z ) − g(X , Z ) 2(n − 1) 2(n − 1) 1 1Rm(X , Y , Z , f ) = Rc(X , f )g(Y , Z ) − Rc(Y , f )g(X , Z ) n−1 n−1 f is an eigenvector of Rc (div Rm)(X , Y , Z ) = Rm(X , Y , Z , f) 1 |div Rm|2 = | R|2 2(n − 1)
10. RThe Schouten tensor S = Rc − g is a Codazzi tensor 2(n − 1) X (R) Y (R)( X Rc)(Y , Z ) − ( Y Rc)(X , Z ) = g(Y , Z ) − g(X , Z ) 2(n − 1) 2(n − 1) 1 1Rm(X , Y , Z , f ) = Rc(X , f )g(Y , Z ) − Rc(Y , f )g(X , Z ) n−1 n−1 f is an eigenvector of Rc (div Rm)(X , Y , Z ) = Rm(X , Y , Z , f) 1 |div Rm|2 = | R|2 2(n − 1)
11. RThe Schouten tensor S = Rc − g is a Codazzi tensor 2(n − 1) X (R) Y (R)( X Rc)(Y , Z ) − ( Y Rc)(X , Z ) = g(Y , Z ) − g(X , Z ) 2(n − 1) 2(n − 1) 1 1Rm(X , Y , Z , f ) = Rc(X , f )g(Y , Z ) − Rc(Y , f )g(X , Z ) n−1 n−1 f is an eigenvector of Rc (div Rm)(X , Y , Z ) = Rm(X , Y , Z , f) 1 |div Rm|2 = | R|2 2(n − 1)
12. RThe Schouten tensor S = Rc − g is a Codazzi tensor 2(n − 1) X (R) Y (R)( X Rc)(Y , Z ) − ( Y Rc)(X , Z ) = g(Y , Z ) − g(X , Z ) 2(n − 1) 2(n − 1) 1 1Rm(X , Y , Z , f ) = Rc(X , f )g(Y , Z ) − Rc(Y , f )g(X , Z ) n−1 n−1 f is an eigenvector of Rc (div Rm)(X , Y , Z ) = Rm(X , Y , Z , f) 1 |div Rm|2 = | R|2 2(n − 1)
13. RThe Schouten tensor S = Rc − g is a Codazzi tensor 2(n − 1) X (R) Y (R)( X Rc)(Y , Z ) − ( Y Rc)(X , Z ) = g(Y , Z ) − g(X , Z ) 2(n − 1) 2(n − 1) 1 1Rm(X , Y , Z , f ) = Rc(X , f )g(Y , Z ) − Rc(Y , f )g(X , Z ) n−1 n−1 f is an eigenvector of Rc (div Rm)(X , Y , Z ) = Rm(X , Y , Z , f) 1 |div Rm|2 = | R|2 2(n − 1)
14. RThe Schouten tensor S = Rc − g is a Codazzi tensor 2(n − 1) X (R) Y (R)( X Rc)(Y , Z ) − ( Y Rc)(X , Z ) = g(Y , Z ) − g(X , Z ) 2(n − 1) 2(n − 1) 1 1Rm(X , Y , Z , f ) = Rc(X , f )g(Y , Z ) − Rc(Y , f )g(X , Z ) n−1 n−1 f is an eigenvector of Rc (div Rm)(X , Y , Z ) = Rm(X , Y , Z , f) 1 |div Rm|2 = | R|2 2(n − 1)
15. |div Rm|2 e−f = | Ric|2 e−f M M X. Cao, B. Wang and Z. Zhang; On Locally Conformally Flat Gradient Shrinking Ricci Solitons 1 1 | R|2 e−f ≥ | R|2 e−f 2(n − 1) M n MFor 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 EinsteinWhat about the noncompact case?
16. |div Rm|2 e−f = | Ric|2 e−f M M X. Cao, B. Wang and Z. Zhang; On Locally Conformally Flat Gradient Shrinking Ricci Solitons 1 1 | R|2 e−f ≥ | R|2 e−f 2(n − 1) M n MFor 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 EinsteinWhat about the noncompact case?
17. |div Rm|2 e−f = | Ric|2 e−f M M X. Cao, B. Wang and Z. Zhang; On Locally Conformally Flat Gradient Shrinking Ricci Solitons 1 1 | R|2 e−f ≥ | R|2 e−f 2(n − 1) M n MFor 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 EinsteinWhat about the noncompact case?
18. |div Rm|2 e−f = | Ric|2 e−f M M X. Cao, B. Wang and Z. Zhang; On Locally Conformally Flat Gradient Shrinking Ricci Solitons 1 1 | R|2 e−f ≥ | R|2 e−f 2(n − 1) M n MFor 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 EinsteinWhat about the noncompact case?
19. |div Rm|2 e−f = | Ric|2 e−f M M X. Cao, B. Wang and Z. Zhang; On Locally Conformally Flat Gradient Shrinking Ricci Solitons 1 1 | R|2 e−f ≥ | R|2 e−f 2(n − 1) M n MFor 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 EinsteinWhat about the noncompact case?
20. |div Rm|2 e−f = | Ric|2 e−f M M X. Cao, B. Wang and Z. Zhang; On Locally Conformally Flat Gradient Shrinking Ricci Solitons 1 1 | R|2 e−f ≥ | R|2 e−f 2(n − 1) M n MFor 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 EinsteinWhat about the noncompact case?
21. |div Rm|2 e−f = | Ric|2 e−f M M X. Cao, B. Wang and Z. Zhang; On Locally Conformally Flat Gradient Shrinking Ricci Solitons 1 1 | R|2 e−f ≥ | R|2 e−f 2(n − 1) M n MFor 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 EinsteinWhat about the noncompact case?
22. Outline Rigidity of Ricci solitons Rigidity: compact case Rigidity: non-compact case Locally conformally ﬂat case Gap theorems Diameter bounds Gap theorems: compact case Gap theorems: non-compact case Maximum principles Introduction Omori-Yau maximum principle Applications Steady solitons Lower bound for the curvature of a steady soliton
23. Theorem (E. García Río and MFL, 2009)Let (M n , 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. |div Rm|2 e−f = | Ric|2 e−f M MR is constant and Rm( f , X , X , f) = 0 P. Petersen and W. Wilye; Rigidity of gradient Ricci solitonsTheorem (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.
24. Theorem (E. García Río and MFL, 2009)Let (M n , 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. |div Rm|2 e−f = | Ric|2 e−f M MR is constant and Rm( f , X , X , f) = 0 P. Petersen and W. Wilye; Rigidity of gradient Ricci solitonsTheorem (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.
25. Theorem (E. García Río and MFL, 2009)Let (M n , 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. |div Rm|2 e−f = | Ric|2 e−f M MR is constant and Rm( f , X , X , f) = 0 P. Petersen and W. Wilye; Rigidity of gradient Ricci solitonsTheorem (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.
26. Theorem (E. García Río and MFL, 2009)Let (M n , 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. |div Rm|2 e−f = | Ric|2 e−f M MR is constant and Rm( f , X , X , f) = 0 P. Petersen and W. Wilye; Rigidity of gradient Ricci solitonsTheorem (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.
27. Theorem (E. García Río and MFL, 2009)Let (M n , 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. |div Rm|2 e−f = | Ric|2 e−f M MR is constant and Rm( f , X , X , f) = 0 P. Petersen and W. Wilye; Rigidity of gradient Ricci solitonsTheorem (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.
28. Outline Rigidity of Ricci solitons Rigidity: compact case Rigidity: non-compact case Locally conformally ﬂat case Gap theorems Diameter bounds Gap theorems: compact case Gap theorems: non-compact case Maximum principles Introduction Omori-Yau maximum principle Applications Steady solitons Lower bound for the curvature of a steady soliton
29. Lemma (E. García Río and MFL, 2010)Let (M n , g) be a locally conformally ﬂat gradient Ricci soliton.Then it is locally (where f = 0) isometric to a warped product (M, g) = ((a, b) × N, dt 2 + ψ(t)2 gN ),where (N, gN ) is a space form. Rc(V , V ) Rc(Ei , Ei ) R W (V , Ei , Ei , V ) = − − + (n − 1)(n − 2) n−2 (n − 1)(n − 2)where 1 V = f | f|
30. Lemma (E. García Río and MFL, 2010)Let (M n , g) be a locally conformally ﬂat gradient Ricci soliton.Then it is locally (where f = 0) isometric to a warped product (M, g) = ((a, b) × N, dt 2 + ψ(t)2 gN ),where (N, gN ) is a space form. Rc(V , V ) Rc(Ei , Ei ) R W (V , Ei , Ei , V ) = − − + (n − 1)(n − 2) n−2 (n − 1)(n − 2)where 1 V = f | f|
31. 1 Rc(Ei , Ei ) = (R − Rc(V , V )) n−1 1 Hf (Ei , Ei ) = (∆f − Hf (V , V )) n−1N = 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 productN is a space form Brozos-Vázquez, García-Río and Vázquez-Lorenzo; Complete locally conformally ﬂat manifolds of negative curvature
32. 1 Rc(Ei , Ei ) = (R − Rc(V , V )) n−1 1 Hf (Ei , Ei ) = (∆f − Hf (V , V )) n−1N = 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 productN is a space form Brozos-Vázquez, García-Río and Vázquez-Lorenzo; Complete locally conformally ﬂat manifolds of negative curvature
33. 1 Rc(Ei , Ei ) = (R − Rc(V , V )) n−1 1 Hf (Ei , Ei ) = (∆f − Hf (V , V )) n−1N = 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 productN is a space form Brozos-Vázquez, García-Río and Vázquez-Lorenzo; Complete locally conformally ﬂat manifolds of negative curvature
34. 1 Rc(Ei , Ei ) = (R − Rc(V , V )) n−1 1 Hf (Ei , Ei ) = (∆f − Hf (V , V )) n−1N = 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 productN is a space form Brozos-Vázquez, García-Río and Vázquez-Lorenzo; Complete locally conformally ﬂat manifolds of negative curvature
35. 1 Rc(Ei , Ei ) = (R − Rc(V , V )) n−1 1 Hf (Ei , Ei ) = (∆f − Hf (V , V )) n−1N = 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 productN is a space form Brozos-Vázquez, García-Río and Vázquez-Lorenzo; Complete locally conformally ﬂat manifolds of negative curvature
36. 1 Rc(Ei , Ei ) = (R − Rc(V , V )) n−1 1 Hf (Ei , Ei ) = (∆f − Hf (V , V )) n−1N = 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 productN is a space form Brozos-Vázquez, García-Río and Vázquez-Lorenzo; Complete locally conformally ﬂat manifolds of negative curvature
37. Theorem (E. García Río and MFL, 2010)Let (M n , 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 )2N is a standard sphere(M n , g) is rotationally symmetric B. Kotschwar; On rotationally invariant shrinking gradient Ricci solitons H.-D. Cao and Q. Chen; On Locally Conformally Flat Gradient Steady Ricci Solitons
38. Theorem (E. García Río and MFL, 2010)Let (M n , 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 )2N is a standard sphere(M n , g) is rotationally symmetric B. Kotschwar; On rotationally invariant shrinking gradient Ricci solitons H.-D. Cao and Q. Chen; On Locally Conformally Flat Gradient Steady Ricci Solitons
39. Theorem (E. García Río and MFL, 2010)Let (M n , 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 )2N is a standard sphere(M n , g) is rotationally symmetric B. Kotschwar; On rotationally invariant shrinking gradient Ricci solitons H.-D. Cao and Q. Chen; On Locally Conformally Flat Gradient Steady Ricci Solitons
40. Theorem (E. García Río and MFL, 2010)Let (M n , 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 )2N is a standard sphere(M n , g) is rotationally symmetric B. Kotschwar; On rotationally invariant shrinking gradient Ricci solitons H.-D. Cao and Q. Chen; On Locally Conformally Flat Gradient Steady Ricci Solitons
41. Theorem (E. García Río and MFL, 2010)Let (M n , 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 )2N is a standard sphere(M n , g) is rotationally symmetric B. Kotschwar; On rotationally invariant shrinking gradient Ricci solitons H.-D. Cao and Q. Chen; On Locally Conformally Flat Gradient Steady Ricci Solitons
42. Theorem (E. García Río and MFL, 2010)Let (M n , 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 )2N is a standard sphere(M n , g) is rotationally symmetric B. Kotschwar; On rotationally invariant shrinking gradient Ricci solitons H.-D. Cao and Q. Chen; On Locally Conformally Flat Gradient Steady Ricci Solitons
43. Outline Rigidity of Ricci solitons Rigidity: compact case Rigidity: non-compact case Locally conformally ﬂat case Gap theorems Diameter bounds Gap theorems: compact case Gap theorems: non-compact case Maximum principles Introduction Omori-Yau maximum principle Applications Steady solitons Lower bound for the curvature of a steady soliton
44. Theorem (E. García Río and MFL, 2008)Let (M n , g) be a compact gradient Ricci soliton. Then fmax − fmin fmax − fmin fmax − fmin diam2 (M, g) ≥ 2max , ,4 λ−c C−λ C−cwhere c ≤ Ric ≤ C.Theorem (E. García Río and MFL, 2008)Let (M n , g) be a compact gradient Ricci soliton with Ric > 0.Then Rmax − Rmin Rmax − Rmin Rmax − Rmindiam2 (M, g) ≥ max , ,4 λ(λ − c) λ(C − λ) λ(C − c)where c ≤ Ric ≤ C.
45. Theorem (E. García Río and MFL, 2008)Let (M n , g) be a compact gradient Ricci soliton. Then fmax − fmin fmax − fmin fmax − fmin diam2 (M, g) ≥ 2max , ,4 λ−c C−λ C−cwhere c ≤ Ric ≤ C.Theorem (E. García Río and MFL, 2008)Let (M n , g) be a compact gradient Ricci soliton with Ric > 0.Then Rmax − Rmin Rmax − Rmin Rmax − Rmindiam2 (M, g) ≥ max , ,4 λ(λ − c) λ(C − λ) λ(C − c)where c ≤ Ric ≤ C.
46. Outline Rigidity of Ricci solitons Rigidity: compact case Rigidity: non-compact case Locally conformally ﬂat case Gap theorems Diameter bounds Gap theorems: compact case Gap theorems: non-compact case Maximum principles Introduction Omori-Yau maximum principle Applications Steady solitons Lower bound for the curvature of a steady soliton
47. Theorem (A. Futaki and Y. Sano, 2010)Let (M, g) be an n-dimensional compact shrinking Ricci soliton.Then 10π diam(M, g) ≥ √ . 13 λTheorem (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: Rmax − Rmin(i) Ric ≥ 1 − λg, (n − 1)λπ 2 + Rmax − Rmin c(Rmax − Rmin )(ii) cg ≤ Ric ≤ λ + g, for some c > 0 (n − 1)λπ 2 4(Rmax − Rmin )(iii) cg ≤ Ric ≤ 1 + cg, for some c > 0. (n − 1)λπ 2
48. Theorem (A. Futaki and Y. Sano, 2010)Let (M, g) be an n-dimensional compact shrinking Ricci soliton.Then 10π diam(M, g) ≥ √ . 13 λTheorem (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: Rmax − Rmin(i) Ric ≥ 1 − λg, (n − 1)λπ 2 + Rmax − Rmin c(Rmax − Rmin )(ii) cg ≤ Ric ≤ λ + g, for some c > 0 (n − 1)λπ 2 4(Rmax − Rmin )(iii) cg ≤ Ric ≤ 1 + cg, for some c > 0. (n − 1)λπ 2
49. Assume (i) holds (n − 1)λ2 π 2 c= (n − 1)λπ 2 + Rmax − Rmin Rmax − Rmin (n − 1)π 2 diam2 (M, g) ≥ ≥ λ(λ − c) cMyers’ theorem: n−1 Ric ≥ cg > 0 =⇒ diam(M, g) ≤ π cBy Cheng M must be the standard sphereCONTRADICTION!
50. Assume (i) holds (n − 1)λ2 π 2 c= (n − 1)λπ 2 + Rmax − Rmin Rmax − Rmin (n − 1)π 2 diam2 (M, g) ≥ ≥ λ(λ − c) cMyers’ theorem: n−1 Ric ≥ cg > 0 =⇒ diam(M, g) ≤ π cBy Cheng M must be the standard sphereCONTRADICTION!
51. Assume (i) holds (n − 1)λ2 π 2 c= (n − 1)λπ 2 + Rmax − Rmin Rmax − Rmin (n − 1)π 2 diam2 (M, g) ≥ ≥ λ(λ − c) cMyers’ theorem: n−1 Ric ≥ cg > 0 =⇒ diam(M, g) ≤ π cBy Cheng M must be the standard sphereCONTRADICTION!
52. Assume (i) holds (n − 1)λ2 π 2 c= (n − 1)λπ 2 + Rmax − Rmin Rmax − Rmin (n − 1)π 2 diam2 (M, g) ≥ ≥ λ(λ − c) cMyers’ theorem: n−1 Ric ≥ cg > 0 =⇒ diam(M, g) ≤ π cBy Cheng M must be the standard sphereCONTRADICTION!
53. Assume (i) holds (n − 1)λ2 π 2 c= (n − 1)λπ 2 + Rmax − Rmin Rmax − Rmin (n − 1)π 2 diam2 (M, g) ≥ ≥ λ(λ − c) cMyers’ theorem: n−1 Ric ≥ cg > 0 =⇒ diam(M, g) ≤ π cBy Cheng M must be the standard sphereCONTRADICTION!
54. Assume (i) holds (n − 1)λ2 π 2 c= (n − 1)λπ 2 + Rmax − Rmin Rmax − Rmin (n − 1)π 2 diam2 (M, g) ≥ ≥ λ(λ − c) cMyers’ theorem: n−1 Ric ≥ cg > 0 =⇒ diam(M, g) ≤ π cBy Cheng M must be the standard sphereCONTRADICTION!
55. 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 2 1 Rmax − nλ ≤ 1+ | f |2 . n vol (M, g) MTheorem (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 −Λ + Λ2 + 8(n − 1)λΛ |Ric − λg| ≤ c ≤ , 4(n − 1) 1where Λ = vol(M,g) M | f |2 denotes the average of the L2 -normof | f |.
56. 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 2 1 Rmax − nλ ≤ 1+ | f |2 . n vol (M, g) MTheorem (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 −Λ + Λ2 + 8(n − 1)λΛ |Ric − λg| ≤ c ≤ , 4(n − 1) 1where Λ = vol(M,g) M | f |2 denotes the average of the L2 -normof | f |.
57. (i) (∆f )2 = ((n + 2)λ − R) | f |2 M M (ii) | f |2 ≤ Rmax − R (∆f )2 = (n + 2)λ | f |2 − R| f |2M M M ≥ (n + 2)λ | f |2 − nλRmax vol (M, g) + R2 M M = (n + 2)λ | f |2 − nλRmax vol (M, g) M +n2 λ2 vol (M, g) + M (∆f ) 2 n+2 1 Rmax − nλ ≥ | f |2 n vol (M, g) M 2λf − R = | f |2 = Rmax − R ⇒ f is constant
58. (i) (∆f )2 = ((n + 2)λ − R) | f |2 M M (ii) | f |2 ≤ Rmax − R (∆f )2 = (n + 2)λ | f |2 − R| f |2M M M ≥ (n + 2)λ | f |2 − nλRmax vol (M, g) + R2 M M = (n + 2)λ | f |2 − nλRmax vol (M, g) M +n2 λ2 vol (M, g) + M (∆f ) 2 n+2 1 Rmax − nλ ≥ | f |2 n vol (M, g) M 2λf − R = | f |2 = Rmax − R ⇒ f is constant
59. (i) (∆f )2 = ((n + 2)λ − R) | f |2 M M (ii) | f |2 ≤ Rmax − R (∆f )2 = (n + 2)λ | f |2 − R| f |2M M M ≥ (n + 2)λ | f |2 − nλRmax vol (M, g) + R2 M M = (n + 2)λ | f |2 − nλRmax vol (M, g) M +n2 λ2 vol (M, g) + M (∆f ) 2 n+2 1 Rmax − nλ ≥ | f |2 n vol (M, g) M 2λf − R = | f |2 = Rmax − R ⇒ f is constant
60. (i) (∆f )2 = ((n + 2)λ − R) | f |2 M M (ii) | f |2 ≤ Rmax − R (∆f )2 = (n + 2)λ | f |2 − R| f |2M M M ≥ (n + 2)λ | f |2 − nλRmax vol (M, g) + R2 M M = (n + 2)λ | f |2 − nλRmax vol (M, g) M +n2 λ2 vol (M, g) + M (∆f ) 2 n+2 1 Rmax − nλ ≥ | f |2 n vol (M, g) M 2λf − R = | f |2 = Rmax − R ⇒ f is constant
61. Outline Rigidity of Ricci solitons Rigidity: compact case Rigidity: non-compact case Locally conformally ﬂat case Gap theorems Diameter bounds Gap theorems: compact case Gap theorems: non-compact case Maximum principles Introduction Omori-Yau maximum principle Applications Steady solitons Lower bound for the curvature of a steady soliton
62. 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 ) ≥ g( f , f ), r (x)2for sufﬁciently large r (x), where > 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 ) ≥ g( f , f ),where is any positive constant, then (M, g) is Ricci ﬂat.
63. 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 ) ≥ g( f , f ), r (x)2for sufﬁciently large r (x), where > 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 ) ≥ g( f , f ),where is any positive constant, then (M, g) is Ricci ﬂat.
64. Theorem (P. Li)If a complete manifold has Ricci curvature bounded from belowby r (x)−2 , for some constant > 1/4 and all r (x) > 1, then Mmust be compact. 2λf = R + | f |2There exists c such that f (x) ≥ 1 (r (x) − c)2 4 H.-D. Cao and D. Zhou; On complete gradient shrinking solitonsγ : [0, +∞) → M an integral curve of f (note that f is acomplete vector ﬁeld) Z.-H. Zhang; On the completeness of gradient Ricci solitons
65. Theorem (P. Li)If a complete manifold has Ricci curvature bounded from belowby r (x)−2 , for some constant > 1/4 and all r (x) > 1, then Mmust be compact. 2λf = R + | f |2There exists c such that f (x) ≥ 1 (r (x) − c)2 4 H.-D. Cao and D. Zhou; On complete gradient shrinking solitonsγ : [0, +∞) → M an integral curve of f (note that f is acomplete vector ﬁeld) Z.-H. Zhang; On the completeness of gradient Ricci solitons
66. Theorem (P. Li)If a complete manifold has Ricci curvature bounded from belowby r (x)−2 , for some constant > 1/4 and all r (x) > 1, then Mmust be compact. 2λf = R + | f |2There exists c such that f (x) ≥ 1 (r (x) − c)2 4 H.-D. Cao and D. Zhou; On complete gradient shrinking solitonsγ : [0, +∞) → M an integral curve of f (note that f is acomplete vector ﬁeld) Z.-H. Zhang; On the completeness of gradient Ricci solitons
67. Theorem (P. Li)If a complete manifold has Ricci curvature bounded from belowby r (x)−2 , for some constant > 1/4 and all r (x) > 1, then Mmust be compact. 2λf = R + | f |2There exists c such that f (x) ≥ 1 (r (x) − c)2 4 H.-D. Cao and D. Zhou; On complete gradient shrinking solitonsγ : [0, +∞) → M an integral curve of f (note that f is acomplete vector ﬁeld) Z.-H. Zhang; On the completeness of gradient Ricci solitons
68. For r ≥ r1 2 (R ◦ γ) (t) = 2Ric( f (γ(t)), f (γ(t)) ≥ | f |2 r (x)2Since R is bounded, for some k1 > 0 and k2 > 0 | f |2 = 2λf − R ≥ λ2 (r (x) − c)2 − R ≥ k1 r (x)2 ≥ k2 R 2 r (x)2for r (x) ≥ r2 ≥ r1p ∈ Ω = {x ∈ M / 2λf (x) ≥ k3 } such that B(x0 , r2 ) ⊂ ΩSince f is increasing along the integral curves of f , if wesuppose that γ(0) = p, then γ(t) ∈ M Ω for all t ≥ 0
69. For r ≥ r1 2 (R ◦ γ) (t) = 2Ric( f (γ(t)), f (γ(t)) ≥ | f |2 r (x)2Since R is bounded, for some k1 > 0 and k2 > 0 | f |2 = 2λf − R ≥ λ2 (r (x) − c)2 − R ≥ k1 r (x)2 ≥ k2 R 2 r (x)2for r (x) ≥ r2 ≥ r1p ∈ Ω = {x ∈ M / 2λf (x) ≥ k3 } such that B(x0 , r2 ) ⊂ ΩSince f is increasing along the integral curves of f , if wesuppose that γ(0) = p, then γ(t) ∈ M Ω for all t ≥ 0
70. For r ≥ r1 2 (R ◦ γ) (t) = 2Ric( f (γ(t)), f (γ(t)) ≥ | f |2 r (x)2Since R is bounded, for some k1 > 0 and k2 > 0 | f |2 = 2λf − R ≥ λ2 (r (x) − c)2 − R ≥ k1 r (x)2 ≥ k2 R 2 r (x)2for r (x) ≥ r2 ≥ r1p ∈ Ω = {x ∈ M / 2λf (x) ≥ k3 } such that B(x0 , r2 ) ⊂ ΩSince f is increasing along the integral curves of f , if wesuppose that γ(0) = p, then γ(t) ∈ M Ω for all t ≥ 0
71. For r ≥ r1 2 (R ◦ γ) (t) = 2Ric( f (γ(t)), f (γ(t)) ≥ | f |2 r (x)2Since R is bounded, for some k1 > 0 and k2 > 0 | f |2 = 2λf − R ≥ λ2 (r (x) − c)2 − R ≥ k1 r (x)2 ≥ k2 R 2 r (x)2for r (x) ≥ r2 ≥ r1p ∈ Ω = {x ∈ M / 2λf (x) ≥ k3 } such that B(x0 , r2 ) ⊂ ΩSince f is increasing along the integral curves of f , if wesuppose that γ(0) = p, then γ(t) ∈ M Ω for all t ≥ 0
72. We have that (R ◦ γ) (t) = g( R(γ(t)), γ (t)) ≥ 2 k2 R 2 (γ(t)),along γ t t (R ◦ γ) (t) ds ≥ 2 k2 dt 0 (R ◦ γ)2 (t) 0 1 1 − ≥ 2 k2 t R(γ(0)) R(γ(t))Contradiction for t going to inﬁnite.
73. We have that (R ◦ γ) (t) = g( R(γ(t)), γ (t)) ≥ 2 k2 R 2 (γ(t)),along γ t t (R ◦ γ) (t) ds ≥ 2 k2 dt 0 (R ◦ γ)2 (t) 0 1 1 − ≥ 2 k2 t R(γ(0)) R(γ(t))Contradiction for t going to inﬁnite.
74. We have that (R ◦ γ) (t) = g( R(γ(t)), γ (t)) ≥ 2 k2 R 2 (γ(t)),along γ t t (R ◦ γ) (t) ds ≥ 2 k2 dt 0 (R ◦ γ)2 (t) 0 1 1 − ≥ 2 k2 t R(γ(0)) R(γ(t))Contradiction for t going to inﬁnite.
75. We have that (R ◦ γ) (t) = g( R(γ(t)), γ (t)) ≥ 2 k2 R 2 (γ(t)),along γ t t (R ◦ γ) (t) ds ≥ 2 k2 dt 0 (R ◦ γ)2 (t) 0 1 1 − ≥ 2 k2 t R(γ(0)) R(γ(t))Contradiction for t going to inﬁnite.
76. Outline Rigidity of Ricci solitons Rigidity: compact case Rigidity: non-compact case Locally conformally ﬂat case Gap theorems Diameter bounds Gap theorems: compact case Gap theorems: non-compact case Maximum principles Introduction Omori-Yau maximum principle Applications Steady solitons Lower bound for the curvature of a steady soliton
77. A Riemannian manifold (M, g) is said to satisfy the Omori-Yaumaximum principle if given any function u ∈ C 2 (M) withu ∗ = supM u < +∞, there exists a sequence (xk ) of points onM satisfying 1 1 1 i) u(xk ) > u ∗ − , ii) |( u)(xk )| < , iii) (∆u)(xk ) < , k k kfor each k ∈ N. If, instead of iii) we assume that 1 Hu (xk ) < g, kin 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 , ·)
78. A Riemannian manifold (M, g) is said to satisfy the Omori-Yaumaximum principle if given any function u ∈ C 2 (M) withu ∗ = supM u < +∞, there exists a sequence (xk ) of points onM satisfying 1 1 1 i) u(xk ) > u ∗ − , ii) |( u)(xk )| < , iii) (∆u)(xk ) < , k k kfor each k ∈ N. If, instead of iii) we assume that 1 Hu (xk ) < g, kin 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 , ·)
79. 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 manifoldsIn 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 Riemannian manifoldsFrom now on we will work with Ricci solitons normalized in thesense 1 Rc + Hf = ± g 2
80. 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 manifoldsIn 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 Riemannian manifoldsFrom now on we will work with Ricci solitons normalized in thesense 1 Rc + Hf = ± g 2
81. 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 manifoldsIn 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 Riemannian manifoldsFrom now on we will work with Ricci solitons normalized in thesense 1 Rc + Hf = ± g 2
82. Outline Rigidity of Ricci solitons Rigidity: compact case Rigidity: non-compact case Locally conformally ﬂat case Gap theorems Diameter bounds Gap theorems: compact case Gap theorems: non-compact case Maximum principles Introduction Omori-Yau maximum principle Applications Steady solitons Lower bound for the curvature of a steady soliton
83. Theorem (E. García Río and MFL, 2010)Let (M n , 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 (M n , g) be an n-dimensional complete noncompact gradientshrinking Ricci soliton. Then (M, g) satisﬁes the Omori-Yaumaximum principle for the f -Laplacian.
84. Theorem (E. García Río and MFL, 2010)Let (M n , 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 (M n , g) be an n-dimensional complete noncompact gradientshrinking Ricci soliton. Then (M, g) satisﬁes the Omori-Yaumaximum principle for the f -Laplacian.
85. S. Pigola, M. Rigoli and A. Setti; Maximum principles on Riemannian manifolds and applications(M, g) satisﬁes the Omori-Yau maximum principle if ∃0 ≤ ϕ ∈ C 2 , 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, +∞), √ ∞ dt tG( t) (4) iii) = ∞, iv ) lim sup < ∞. 0 G(t) t→∞ G(t) √ √ ∃B > 0 s. t. Hϕ ≤ B ϕ G( ϕ), off a compact set (5)(M, g) satisﬁes the Omori-Yau maximum principle for Hessian.
86. S. Pigola, M. Rigoli and A. Setti; Maximum principles on Riemannian manifolds and applications(M, g) satisﬁes the Omori-Yau maximum principle if ∃0 ≤ ϕ ∈ C 2 , 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, +∞), √ ∞ dt tG( t) (4) iii) = ∞, iv ) lim sup < ∞. 0 G(t) t→∞ G(t) √ √ ∃B > 0 s. t. Hϕ ≤ B ϕ G( ϕ), off a compact set (5)(M, g) satisﬁes the Omori-Yau maximum principle for Hessian.
87. 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 ∈ C 2 (M) with u ∗ = supM u < +∞, thereexists a sequence (xk ) of points on M satisfying 1 1 i) u(xk ) > u ∗ − , ii) (∆f u)(xk ) < , k kfor 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.
88. 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 ∈ C 2 (M) with u ∗ = supM u < +∞, thereexists a sequence (xk ) of points on M satisfying 1 1 i) u(xk ) > u ∗ − , ii) (∆f u)(xk ) < , k kfor 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.
89. 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 ∈ C 2 (M) with u ∗ = supM u < +∞, thereexists a sequence (xk ) of points on M satisfying 1 1 i) u(xk ) > u ∗ − , ii) (∆f u)(xk ) < , k kfor 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.
90. Outline Rigidity of Ricci solitons Rigidity: compact case Rigidity: non-compact case Locally conformally ﬂat case Gap theorems Diameter bounds Gap theorems: compact case Gap theorems: non-compact case Maximum principles Introduction Omori-Yau maximum principle Applications Steady solitons Lower bound for the curvature of a steady soliton
91. Theorem (E. García Río and MFL, 2010)Let (M n , g) be an n-dimensional complete gradient shrinkingRicci soliton. Then: (i) (M, g) has constant scalar curvature if and only if | R|2 2|Ric|2 ≤ R + c , for some c ≥ 0. R+1(ii) (M, g) is isometric to (Rn , geuc ) if and only if | R|2 2|Ric|2 ≤ (1 − )R + c , for some c ≥ 0 and > 0. R+1
92. 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|22(R 2 −|Ric|2 ) .We consider an orthonormal frame {E1 , . . . , En } formed byeigenvectors of the Ricci operator. ∆Rii = g( Rii , f ) + Rii − 2Rijji R jj ,where Rii = Ric(Ei , Ei ), Rijji = R(Ei , Ej , Ej , Ei )∆f |Ric|2 = 2|Ric|2 −4R ii Rijji R jj +2 Rii R ii ≥ 2|Ric|2 −4R ii Rijji R jj .
93. 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|22(R 2 −|Ric|2 ) .We consider an orthonormal frame {E1 , . . . , En } formed byeigenvectors of the Ricci operator. ∆Rii = g( Rii , f ) + Rii − 2Rijji R jj ,where Rii = Ric(Ei , Ei ), Rijji = R(Ei , Ej , Ej , Ei )∆f |Ric|2 = 2|Ric|2 −4R ii Rijji R jj +2 Rii R ii ≥ 2|Ric|2 −4R ii Rijji R jj .
94. 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|22(R 2 −|Ric|2 ) .We consider an orthonormal frame {E1 , . . . , En } formed byeigenvectors of the Ricci operator. ∆Rii = g( Rii , f ) + Rii − 2Rijji R jj ,where Rii = Ric(Ei , Ei ), Rijji = R(Ei , Ej , Ej , Ei )∆f |Ric|2 = 2|Ric|2 −4R ii Rijji R jj +2 Rii R ii ≥ 2|Ric|2 −4R ii Rijji R jj .
95. 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|22(R 2 −|Ric|2 ) .We consider an orthonormal frame {E1 , . . . , En } formed byeigenvectors of the Ricci operator. ∆Rii = g( Rii , f ) + Rii − 2Rijji R jj ,where Rii = Ric(Ei , Ei ), Rijji = R(Ei , Ej , Ej , Ei )∆f |Ric|2 = 2|Ric|2 −4R ii Rijji R jj +2 Rii R ii ≥ 2|Ric|2 −4R ii Rijji R jj .
96. Under our assumption one has 4|Ric|2 4 R ii Rijji R jj ≤ (R 2 − |Ric|2 ) = 2|Ric|2 . 2(R 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 ii Rijji R jj + 2 Rii R ii ≥ 0 n ⇒ | Rii |2 = 0 i=1The Ricci soliton is rigid.
97. Under our assumption one has 4|Ric|2 4 R ii Rijji R jj ≤ (R 2 − |Ric|2 ) = 2|Ric|2 . 2(R 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 ii Rijji R jj + 2 Rii R ii ≥ 0 n ⇒ | Rii |2 = 0 i=1The Ricci soliton is rigid.
98. Under our assumption one has 4|Ric|2 4 R ii Rijji R jj ≤ (R 2 − |Ric|2 ) = 2|Ric|2 . 2(R 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 ii Rijji R jj + 2 Rii R ii ≥ 0 n ⇒ | Rii |2 = 0 i=1The Ricci soliton is rigid.
99. Under our assumption one has 4|Ric|2 4 R ii Rijji R jj ≤ (R 2 − |Ric|2 ) = 2|Ric|2 . 2(R 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 ii Rijji R jj + 2 Rii R ii ≥ 0 n ⇒ | Rii |2 = 0 i=1The Ricci soliton is rigid.
100. 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 that 1 1R(xk ) < R∗ + k and (∆f R)(xk ) > − k . 1 2R(xk )2 ≥ 2|Ric(xk )|2 ≥ . k nTaking the limit when k goes to inﬁnity we get that R∗ = 0.
101. 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 that 1 1R(xk ) < R∗ + k and (∆f R)(xk ) > − k . 1 2R(xk )2 ≥ 2|Ric(xk )|2 ≥ . k nTaking the limit when k goes to inﬁnity we get that R∗ = 0.
102. 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 that 1 1R(xk ) < R∗ + k and (∆f R)(xk ) > − k . 1 2R(xk )2 ≥ 2|Ric(xk )|2 ≥ . k nTaking the limit when k goes to inﬁnity we get that R∗ = 0.
103. 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 that 1 1R(xk ) < R∗ + k and (∆f R)(xk ) > − k . 1 2R(xk )2 ≥ 2|Ric(xk )|2 ≥ . k nTaking the limit when k goes to inﬁnity we get that R∗ = 0.
104. 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 that 1 1R(xk ) < R∗ + k and (∆f R)(xk ) > − k . 1 2R(xk )2 ≥ 2|Ric(xk )|2 ≥ . k nTaking the limit when k goes to inﬁnity we get that R∗ = 0.
105. 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 ≤ R ≤ − 1 . 2 2 ∆f R = −R − 2|Ric|2 ≥ −R − 2R 2 = −R(1 + 2R) 1 R ∗ (1 + 2R ∗ ) ≥ 0 ⇒ R ∗ ≤ − 2
106. 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 ≤ R ≤ − 1 . 2 2 ∆f R = −R − 2|Ric|2 ≥ −R − 2R 2 = −R(1 + 2R) 1 R ∗ (1 + 2R ∗ ) ≥ 0 ⇒ R ∗ ≤ − 2
107. 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 ≤ R ≤ − 1 . 2 2 ∆f R = −R − 2|Ric|2 ≥ −R − 2R 2 = −R(1 + 2R) 1 R ∗ (1 + 2R ∗ ) ≥ 0 ⇒ R ∗ ≤ − 2
108. Outline Rigidity of Ricci solitons Rigidity: compact case Rigidity: non-compact case Locally conformally ﬂat case Gap theorems Diameter bounds Gap theorems: compact case Gap theorems: non-compact case Maximum principles Introduction Omori-Yau maximum principle Applications Steady solitons Lower bound for the curvature of a steady soliton
109. TheoremLet (M n , g, f ) be a complete noncompact nonﬂat shrinkinggradient Ricci soliton. Then for any given point O ∈ M there −1exists a constant CO > 0 such that R(x)d(x, O)2 ≥ COwherever d(x, O) ≥ CO . B. Chow, P. Lu and B. Yang; A lower bound for the scalar curvature of noncompact nonﬂat Ricci shrinkersTheoremLet (M n , g, f ) be a complete steady gradient Ricci solitons withRc = −Hf and R + | f |2 = 1. If limx→∞ f (x) = −∞ and f ≤ 0, 1then R ≥ √ n ef . 2 +2 B. Chow, P. Lu and B. Yang; A lower bound for the scalar curvature of certain steady gradient Ricci solitons
110. TheoremLet (M n , g, f ) be a complete noncompact nonﬂat shrinkinggradient Ricci soliton. Then for any given point O ∈ M there −1exists a constant CO > 0 such that R(x)d(x, O)2 ≥ COwherever d(x, O) ≥ CO . B. Chow, P. Lu and B. Yang; A lower bound for the scalar curvature of noncompact nonﬂat Ricci shrinkersTheoremLet (M n , g, f ) be a complete steady gradient Ricci solitons withRc = −Hf and R + | f |2 = 1. If limx→∞ f (x) = −∞ and f ≤ 0, 1then R ≥ √ n ef . 2 +2 B. Chow, P. Lu and B. Yang; A lower bound for the scalar curvature of certain steady gradient Ricci solitons
111. Theorem (E. García Río and MFL, 2011)Let (M, g) be a complete gradient steady Ricci soliton satisfying 2|Ric|2 ≤ R . Then 2 r (x) R(x) ≥ k sech2 , 2where 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 |Hf |2 = | f |2 ≥ | | f ||2 = | 1 − R|2 = 4| f |2 | R|2 ≤ 4|Hf |2 | f |2 R2 | R| |Hf |2 = |Rc|2 ≤ ⇒ √ ≤1 2 R 1−R
112. Theorem (E. García Río and MFL, 2011)Let (M, g) be a complete gradient steady Ricci soliton satisfying 2|Ric|2 ≤ R . Then 2 r (x) R(x) ≥ k sech2 , 2where 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 |Hf |2 = | f |2 ≥ | | f ||2 = | 1 − R|2 = 4| f |2 | R|2 ≤ 4|Hf |2 | f |2 R2 | R| |Hf |2 = |Rc|2 ≤ ⇒ √ ≤1 2 R 1−R
113. Theorem (E. García Río and MFL, 2011)Let (M, g) be a complete gradient steady Ricci soliton satisfying 2|Ric|2 ≤ R . Then 2 r (x) R(x) ≥ k sech2 , 2where 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 |Hf |2 = | f |2 ≥ | | f ||2 = | 1 − R|2 = 4| f |2 | R|2 ≤ 4|Hf |2 | f |2 R2 | R| |Hf |2 = |Rc|2 ≤ ⇒ √ ≤1 2 R 1−R
114. Theorem (E. García Río and MFL, 2011)Let (M, g) be a complete gradient steady Ricci soliton satisfying 2|Ric|2 ≤ R . Then 2 r (x) R(x) ≥ k sech2 , 2where 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 |Hf |2 = | f |2 ≥ | | f ||2 = | 1 − R|2 = 4| f |2 | R|2 ≤ 4|Hf |2 | f |2 R2 | R| |Hf |2 = |Rc|2 ≤ ⇒ √ ≤1 2 R 1−R
115. −(R◦γ)Integrating √ R 1−R along a minimizing geodesic γ(s) √ t l t 1+ 1−R (R ◦ γ) | R| ln √ =− √ ds ≤ √ ds ≤ t 1− 1−R 0 0 R 1−R 0 R 1−R √ 1+ 1−R(O)Writing c = √ we get that 1− 1−R(O) 1+ 1 − R(γ(t)) ≤ cet (1 − 1 − R(γ(t))) 4c R(γ(t)) ≥ c 2 et + 2c + e−tSince c ≥ 1 we have that 4c 4c 1 t R(γ(t)) ≥ −t ≥ 2 t 2 + c 2 e−t = sech2 c 2 et + 2c + e c e + 2c c 2
116. −(R◦γ)Integrating √ R 1−R along a minimizing geodesic γ(s) √ t l t 1+ 1−R (R ◦ γ) | R| ln √ =− √ ds ≤ √ ds ≤ t 1− 1−R 0 0 R 1−R 0 R 1−R √ 1+ 1−R(O)Writing c = √ we get that 1− 1−R(O) 1+ 1 − R(γ(t)) ≤ cet (1 − 1 − R(γ(t))) 4c R(γ(t)) ≥ c 2 et + 2c + e−tSince c ≥ 1 we have that 4c 4c 1 t R(γ(t)) ≥ −t ≥ 2 t 2 + c 2 e−t = sech2 c 2 et + 2c + e c e + 2c c 2
117. −(R◦γ)Integrating √ R 1−R along a minimizing geodesic γ(s) √ t l t 1+ 1−R (R ◦ γ) | R| ln √ =− √ ds ≤ √ ds ≤ t 1− 1−R 0 0 R 1−R 0 R 1−R √ 1+ 1−R(O)Writing c = √ we get that 1− 1−R(O) 1+ 1 − R(γ(t)) ≤ cet (1 − 1 − R(γ(t))) 4c R(γ(t)) ≥ c 2 et + 2c + e−tSince c ≥ 1 we have that 4c 4c 1 t R(γ(t)) ≥ −t ≥ 2 t 2 + c 2 e−t = sech2 c 2 et + 2c + e c e + 2c c 2
118. −(R◦γ)Integrating √ R 1−R along a minimizing geodesic γ(s) √ t l t 1+ 1−R (R ◦ γ) | R| ln √ =− √ ds ≤ √ ds ≤ t 1− 1−R 0 0 R 1−R 0 R 1−R √ 1+ 1−R(O)Writing c = √ we get that 1− 1−R(O) 1+ 1 − R(γ(t)) ≤ cet (1 − 1 − R(γ(t))) 4c R(γ(t)) ≥ c 2 et + 2c + e−tSince c ≥ 1 we have that 4c 4c 1 t R(γ(t)) ≥ −t ≥ 2 t 2 + c 2 e−t = sech2 c 2 et + 2c + e c e + 2c c 2
119. The scalar curvature of Hamilton’s cigar soliton dx 2 + dy 2 R2 , 1 + x2 + y2satisﬁes R(x) = 4sech2 r (x)The scalar curvature of normalized Hamilton’s cigar soliton 4(dx 2 + dy 2 ) R2 , 1 + x2 + y2satisﬁes r (x) R(x) = sech2 2Our inequality is SHARP
120. The scalar curvature of Hamilton’s cigar soliton dx 2 + dy 2 R2 , 1 + x2 + y2satisﬁes R(x) = 4sech2 r (x)The scalar curvature of normalized Hamilton’s cigar soliton 4(dx 2 + dy 2 ) R2 , 1 + x2 + y2satisﬁes r (x) R(x) = sech2 2Our inequality is SHARP
121. The scalar curvature of Hamilton’s cigar soliton dx 2 + dy 2 R2 , 1 + x2 + y2satisﬁes R(x) = 4sech2 r (x)The scalar curvature of normalized Hamilton’s cigar soliton 4(dx 2 + dy 2 ) R2 , 1 + x2 + y2satisﬁes r (x) R(x) = sech2 2Our inequality is SHARP
122. 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) ≥ k sech2 r (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 |2Since |Hf |2 = |Rc|2 ≤ R 2 one has | R| √ ≤2 R 1−R
123. 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) ≥ k sech2 r (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 |2Since |Hf |2 = |Rc|2 ≤ R 2 one has | R| √ ≤2 R 1−R
124. 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) ≥ k sech2 r (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 |2Since |Hf |2 = |Rc|2 ≤ R 2 one has | R| √ ≤2 R 1−R
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