On gradient Ricci solitons

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 by orthogonal transformations on Rk . Theorem (Petersen and Wylie, 2007) The following conditions for a shrinking (expanding) gradient soliton Ric + Hf = λg all imply that the metric is radially ﬂat and has 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 by orthogonal transformations on Rk . Theorem (Petersen and Wylie, 2007) The following conditions for a shrinking (expanding) gradient soliton Ric + Hf = λg all imply that the metric is radially ﬂat and has 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

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 space form). 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. R The 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 1 Rm(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 M For n ≤ 3 compact shrinking Ricci solitons are Einstein (n = 2 Hamilton, n = 3 Ivey) Since n ≥ 4 one has that R is constant (M, g) is Einstein What about the noncompact case?

23. Theorem (E. García Río and MFL, 2009) Let (M n , g) be a complete noncompact gradient shrinking Ricci soliton whose curvature tensor has at most exponential growth and having Ricci tensor bounded from below. Then (M, g) is rigid if an only if it has harmonic Weyl tensor. |div Rm|2 e−f = | Ric|2 e−f M M 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 Ricci soliton. Then (M, g) is rigid if an only if it has harmonic Weyl tensor.

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−1 N = f −1 (c) is a totally umbilical submanifold of (M, g) f is an eigenvector of Hf ↔ the integral curves of V are geodesics (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 negative curvature

37. Theorem (E. García Río and MFL, 2010) Let (M n , g) be a simply connected complete locally conformally ﬂat gradient shrinking or steady Ricci soliton. Then it is rotationally symmetric. Any complete ancient solution to the Ricci ﬂow has nonnegative curvature 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 (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

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−c where 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 − Rmin diam2 (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−c where 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 − Rmin diam2 (M, g) ≥ max , ,4 λ(λ − c) λ(C − λ) λ(C − c) where c ≤ Ric ≤ C.

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 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 following conditions 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 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 following conditions 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) c Myers’ theorem: n−1 Ric ≥ cg > 0 =⇒ diam(M, g) ≤ π c By Cheng M must be the standard sphere CONTRADICTION!

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) M 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 −Λ + Λ2 + 8(n − 1)λΛ |Ric − λg| ≤ c ≤ , 4(n − 1) 1 where Λ = vol(M,g) M | f |2 denotes the average of the L2 -norm of | 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) M 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 −Λ + Λ2 + 8(n − 1)λΛ |Ric − λg| ≤ c ≤ , 4(n − 1) 1 where Λ = vol(M,g) M | f |2 denotes the average of the L2 -norm of | 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 |2 M 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

62. Theorem (E. García Río and MFL, 2010) Let (M, g) be a complete gradient shrinking Ricci soliton with bounded scalar curvature. Then (M, g) is compact Einstein if Ric( f , f ) ≥ g( f , f ), r (x)2 for sufficiently large r (x), where > 0 and r (x) denotes the distance from a fixed point. Theorem (E. García Río and MFL, 2010) Let (M, g) be an n-dimensional complete gradient steady Ricci soliton. 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 with bounded scalar curvature. Then (M, g) is compact Einstein if Ric( f , f ) ≥ g( f , f ), r (x)2 for sufficiently large r (x), where > 0 and r (x) denotes the distance from a fixed point. Theorem (E. García Río and MFL, 2010) Let (M, g) be an n-dimensional complete gradient steady Ricci soliton. 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 below by r (x)−2 , for some constant > 1/4 and all r (x) > 1, then M must be compact. 2λf = R + | f |2 There exists c such that f (x) ≥ 1 (r (x) − c)2 4 H.-D. Cao and D. Zhou; On complete gradient shrinking solitons γ : [0, +∞) → M an integral curve of f (note that f is a complete 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)2 Since 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)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 we suppose 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.

77. A Riemannian manifold (M, g) is said to satisfy the Omori-Yau maximum principle if given any function u ∈ C 2 (M) with u ∗ = supM u < +∞, there exists a sequence (xk ) of points on M satisfying 1 1 1 i) u(xk ) > u ∗ − , ii) |( u)(xk )| < , iii) (∆u)(xk ) < , k k k for each k ∈ N. If, instead of iii) we assume that 1 Hu (xk ) < g, k in the sense of quadratic forms, then it is said that the Riemannian manifold satisﬁes the Omori-Yau maximum principle 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-Yau maximum principle if given any function u ∈ C 2 (M) with u ∗ = supM u < +∞, there exists a sequence (xk ) of points on M satisfying 1 1 1 i) u(xk ) > u ∗ − , ii) |( u)(xk )| < , iii) (∆u)(xk ) < , k k k for each k ∈ N. If, instead of iii) we assume that 1 Hu (xk ) < g, k in the sense of quadratic forms, then it is said that the Riemannian manifold satisﬁes the Omori-Yau maximum principle for the Hessian. The f -Laplacian is ∆f = ef div (e−f ) = ∆ − g( f , ·)

79. In 1967 Omori showed that the Omori-Yau maximum principle for the Hessian is satisﬁed by Riemannian manifolds with curvature bounded from below. H. Omori; Isometric immersions of Riemannian manifolds In 1975 Yau proved that the Omori-Yau maximum principle is satisﬁed by Riemannian manifolds with Ricci curvature bounded from below. S. T. Yau; Harmonic functions on complete Riemannian manifolds From now on we will work with Ricci solitons normalized in the sense 1 Rc + Hf = ± g 2

83. Theorem (E. García Río and MFL, 2010) Let (M n , g) be an n-dimensional complete noncompact gradient shrinking Ricci soliton. Then (M, g) satisfies the Omori-Yau maximum principle. Moreover, if there exists C > 0 such that Ric ≥ −Cr (x)2 , where r (x) denotes the distance to a fixed point, then the Omori-Yau maximum 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 gradient shrinking Ricci soliton. Then (M, g) satisfies the Omori-Yau maximum principle for the f -Laplacian.

84. Theorem (E. García Río and MFL, 2010) Let (M n , g) be an n-dimensional complete noncompact gradient shrinking Ricci soliton. Then (M, g) satisfies the Omori-Yau maximum principle. Moreover, if there exists C > 0 such that Ric ≥ −Cr (x)2 , where r (x) denotes the distance to a fixed point, then the Omori-Yau maximum 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 gradient shrinking Ricci soliton. Then (M, g) satisfies the Omori-Yau maximum 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 or expanding) Ricci solitons. Then, the weak maximum principle at inﬁnity for the f -Laplacian holds. Given any function u ∈ C 2 (M) with u ∗ = supM u < +∞, there exists a sequence (xk ) of points on M satisfying 1 1 i) u(xk ) > u ∗ − , ii) (∆f u)(xk ) < , k k for each k ∈ N. Theorem (S. Pigola, M. Rimoldi and A. Setti, 2009) Let (M, g) be a complete gradient (shrinking, steady or expanding) Ricci solitons. Then it is stochastically complete.

91. Theorem (E. García Río and MFL, 2010) Let (M n , g) be an n-dimensional complete gradient shrinking Ricci 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 with bounded nonnegative Ricci tensor. Then (M, g) is rigid if and only if the sectional curvature is bounded from above by |Ric|2 2(R 2 −|Ric|2 ) . We consider an orthonormal frame {E1 , . . . , En } formed by eigenvectors 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 is constant. 0 = ∆f |Ric|2 = 2|Ric|2 − 4R ii Rijji R jj + 2 Rii R ii ≥ 0 n ⇒ | Rii |2 = 0 i=1 The Ricci soliton is rigid.

100. Theorem (E. García Río and MFL, 2010) Let (M, g) be a complete gradient steady Ricci soliton. Then R∗ = infM R = 0. ∆f R = −2|Ric|2 . There exists a sequence (xk ) of points of M such that 1 1 R(xk ) < R∗ + k and (∆f R)(xk ) > − k . 1 2R(xk )2 ≥ 2|Ric(xk )|2 ≥ . k n Taking 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 with Ric ≤ 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

109. Theorem Let (M n , g, f ) be a complete noncompact nonﬂat shrinking gradient Ricci soliton. Then for any given point O ∈ M there −1 exists a constant CO > 0 such that R(x)d(x, O)2 ≥ CO wherever d(x, O) ≥ CO . B. Chow, P. Lu and B. Yang; A lower bound for the scalar curvature of noncompact nonﬂat Ricci shrinkers Theorem Let (M n , g, f ) be a complete steady gradient Ricci solitons with Rc = −Hf and R + | f |2 = 1. If limx→∞ f (x) = −∞ and f ≤ 0, 1 then 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. Theorem Let (M n , g, f ) be a complete noncompact nonﬂat shrinking gradient Ricci soliton. Then for any given point O ∈ M there −1 exists a constant CO > 0 such that R(x)d(x, O)2 ≥ CO wherever d(x, O) ≥ CO . B. Chow, P. Lu and B. Yang; A lower bound for the scalar curvature of noncompact nonﬂat Ricci shrinkers Theorem Let (M n , g, f ) be a complete steady gradient Ricci solitons with Rc = −Hf and R + | f |2 = 1. If limx→∞ f (x) = −∞ and f ≤ 0, 1 then 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 , 2 where r (x) is the distance from a ﬁxed point O ∈ M and k ≤ 1 is 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−t Since 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 + y2 satisﬁes R(x) = 4sech2 r (x) The scalar curvature of normalized Hamilton’s cigar soliton 4(dx 2 + dy 2 ) R2 , 1 + x2 + y2 satisﬁes r (x) R(x) = sech2 2 Our inequality is SHARP

122. Theorem (E. García Río and MFL, 2011) Let (M, g) be a complete gradient steady Ricci soliton with nonnegative 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 ≤ 1 is a constant that only depends on O and R(O). | R|2 ≤ 4|Hf |2 | f |2 Since |Hf |2 = |Rc|2 ≤ R 2 one has | R| √ ≤2 R 1−R

125. THANK YOU VERY MUCH FOR YOUR ATTENTION

On gradient Ricci solitons

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