On gradient Ricci solitons

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

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

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

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

Outline

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

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

(∇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 )

(div Rm)(X ,Y ,Z ) = Rm(X ,Y ,Z ,∇f )

|div Rm|2 =1

2(n − 1)|∇R|2

(∇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 )

(div Rm)(X ,Y ,Z ) = Rm(X ,Y ,Z ,∇f )

|div Rm|2 =1

2(n − 1)|∇R|2

(∇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 )

(div Rm)(X ,Y ,Z ) = Rm(X ,Y ,Z ,∇f )

|div Rm|2 =1

2(n − 1)|∇R|2

(∇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 )

(div Rm)(X ,Y ,Z ) = Rm(X ,Y ,Z ,∇f )

|div Rm|2 =1

2(n − 1)|∇R|2

(∇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 )

(div Rm)(X ,Y ,Z ) = Rm(X ,Y ,Z ,∇f )

|div Rm|2 =1

2(n − 1)|∇R|2

∫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

∫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?

∫M|∇Ric|2e−f

12(n − 1)

∫M|∇R|2e−f ≥ 1

∫M|∇R|2e−f

(M,g) is Einstein

∫M|∇Ric|2e−f

12(n − 1)

∫M|∇R|2e−f ≥ 1

∫M|∇R|2e−f

(M,g) is Einstein

∫M|∇Ric|2e−f

12(n − 1)

∫M|∇R|2e−f ≥ 1

∫M|∇R|2e−f

(M,g) is Einstein

∫M|∇Ric|2e−f

12(n − 1)

∫M|∇R|2e−f ≥ 1

∫M|∇R|2e−f

(M,g) is Einstein

∫M|∇Ric|2e−f

12(n − 1)

∫M|∇R|2e−f ≥ 1

∫M|∇R|2e−f

(M,g) is Einstein

∫M|∇Ric|2e−f

12(n − 1)

∫M|∇R|2e−f ≥ 1

∫M|∇R|2e−f

(M,g) is Einstein

Outline

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.

M|div Rm|2e−f =

∫M|∇Ric|2e−f

M|div Rm|2e−f =

∫M|∇Ric|2e−f

M|div Rm|2e−f =

∫M|∇Ric|2e−f

M|div Rm|2e−f =

∫M|∇Ric|2e−f

Outline

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

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

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

Rc(Ei ,Ei) =1

n − 1(R − Rc(V ,V ))

Hf (Ei ,Ei) =1

n − 1(∆f − Hf (V ,V ))

N is a space form

Rc(Ei ,Ei) =1

n − 1(R − Rc(V ,V ))

Hf (Ei ,Ei) =1

n − 1(∆f − Hf (V ,V ))

N is a space form

Rc(Ei ,Ei) =1

n − 1(R − Rc(V ,V ))

Hf (Ei ,Ei) =1

n − 1(∆f − Hf (V ,V ))

N is a space form

Rc(Ei ,Ei) =1

n − 1(R − Rc(V ,V ))

Hf (Ei ,Ei) =1

n − 1(∆f − Hf (V ,V ))

N is a space form

Rc(Ei ,Ei) =1

n − 1(R − Rc(V ,V ))

Hf (Ei ,Ei) =1

n − 1(∆f − Hf (V ,V ))

N is a space form

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

Outline

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)

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

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)

Outline

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

(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.

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

(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.

Assume (i) holds

c =(n − 1)λ2π2

diam2(M,g) ≥ Rmax − Rmin

λ(λ− c)≥ (n − 1)π2

Myers’ theorem:

Ric ≥ cg > 0 =⇒ diam(M,g) ≤ π√

n − 1c

By Cheng M must be the standard sphere

CONTRADICTION!

Assume (i) holds

c =(n − 1)λ2π2

λ(λ− c)≥ (n − 1)π2

Myers’ theorem:

n − 1c

CONTRADICTION!

Assume (i) holds

c =(n − 1)λ2π2

λ(λ− c)≥ (n − 1)π2

Myers’ theorem:

n − 1c

CONTRADICTION!

Assume (i) holds

c =(n − 1)λ2π2

λ(λ− c)≥ (n − 1)π2

Myers’ theorem:

n − 1c

CONTRADICTION!

Assume (i) holds

c =(n − 1)λ2π2

λ(λ− c)≥ (n − 1)π2

Myers’ theorem:

n − 1c

CONTRADICTION!

Assume (i) holds

c =(n − 1)λ2π2

λ(λ− c)≥ (n − 1)π2

Myers’ theorem:

n − 1c

CONTRADICTION!

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 ≤(

vol (M,g)

∫M|∇f |2.

|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 |.

Rmax − nλ ≤(

vol (M,g)

∫M|∇f |2.

|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 |.

(i)∫

M(∆f )2 =

((n + 2)λ− R) |∇f |2

(ii) |∇f |2 ≤ Rmax − R∫M

(∆f )2 = (n + 2)λ

∫M|∇f |2 −

R|∇f |2

≥ (n + 2)λ

∫M|∇f |2 − nλRmaxvol (M,g) +

= (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

(i)∫

M(∆f )2 =

((n + 2)λ− R) |∇f |2

(ii) |∇f |2 ≤ Rmax − R∫M

(∆f )2 = (n + 2)λ

∫M|∇f |2 −

R|∇f |2

≥ (n + 2)λ

= (n + 2)λ

M(∆f )2

1vol (M,g)

∫M|∇f |2

(i)∫

M(∆f )2 =

((n + 2)λ− R) |∇f |2

(ii) |∇f |2 ≤ Rmax − R∫M

(∆f )2 = (n + 2)λ

∫M|∇f |2 −

R|∇f |2

≥ (n + 2)λ

= (n + 2)λ

M(∆f )2

1vol (M,g)

∫M|∇f |2

(i)∫

M(∆f )2 =

((n + 2)λ− R) |∇f |2

(ii) |∇f |2 ≤ Rmax − R∫M

(∆f )2 = (n + 2)λ

∫M|∇f |2 −

R|∇f |2

≥ (n + 2)λ

= (n + 2)λ

M(∆f )2

1vol (M,g)

∫M|∇f |2

Outline

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.

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.

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

2λf = R + |∇f |2

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

For r ≥ r1

for r(x) ≥ r2 ≥ r1

For r ≥ r1

for r(x) ≥ r2 ≥ r1

For r ≥ r1

for r(x) ≥ r2 ≥ r1

We have that

(R γ)′(t) = g(∇R(γ(t)), γ′(t)) ≥ 2εk2R2(γ(t)),

along γ ∫ t

(R γ)′(t)(R γ)2(t)

ds ≥∫ t

02εk2dt

1R(γ(0))

− 1R(γ(t))

≥ 2εk2t

Contradiction for t going to infinite.

We have that

(R γ)′(t) = g(∇R(γ(t)), γ′(t)) ≥ 2εk2R2(γ(t)),

along γ ∫ t

(R γ)′(t)(R γ)2(t)

ds ≥∫ t

02εk2dt

1R(γ(0))

− 1R(γ(t))

≥ 2εk2t

We have that

(R γ)′(t) = g(∇R(γ(t)), γ′(t)) ≥ 2εk2R2(γ(t)),

along γ ∫ t

(R γ)′(t)(R γ)2(t)

ds ≥∫ t

02εk2dt

1R(γ(0))

− 1R(γ(t))

≥ 2εk2t

We have that

(R γ)′(t) = g(∇R(γ(t)), γ′(t)) ≥ 2εk2R2(γ(t)),

along γ ∫ t

(R γ)′(t)(R γ)2(t)

ds ≥∫ t

02εk2dt

1R(γ(0))

− 1R(γ(t))

≥ 2εk2t

Outline

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 ) <

for each k ∈ N. If, instead of iii) we assume that

Hu(xk ) <1k

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 , ·)

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 ) <

for each k ∈ N. If, instead of iii) we assume that

Hu(xk ) <1k

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 , ·)

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

Outline

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.

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.

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)∫ ∞

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.

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)∫ ∞

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.

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 ) <

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.

i) u(xk ) > u∗ − 1k, ii) (∆f u)(xk ) <

for each k ∈ N.

i) u(xk ) > u∗ − 1k, ii) (∆f u)(xk ) <

for each k ∈ N.

Outline

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.

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 .

2(R2−|Ric|2).

Under our assumption one has

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∑

|∇Rii |2 = 0

The Ricci soliton is rigid.

2(R2 − |Ric|2)(R2 − |Ric|2) = 2|Ric|2.

⇒n∑

|∇Rii |2 = 0

2(R2 − |Ric|2)(R2 − |Ric|2) = 2|Ric|2.

⇒n∑

|∇Rii |2 = 0

2(R2 − |Ric|2)(R2 − |Ric|2) = 2|Ric|2.

⇒n∑

|∇Rii |2 = 0

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

Taking the limit when k goes to infinity we get that R∗ = 0.

∆f R = −2|Ric|2.

k and (∆f R)(xk ) > − 1k .

1k≥ 2|Ric(xk )|2 ≥ 2R(xk )2

∆f R = −2|Ric|2.

k and (∆f R)(xk ) > − 1k .

1k≥ 2|Ric(xk )|2 ≥ 2R(xk )2

∆f R = −2|Ric|2.

k and (∆f R)(xk ) > − 1k .

1k≥ 2|Ric(xk )|2 ≥ 2R(xk )2

∆f R = −2|Ric|2.

k and (∆f R)(xk ) > − 1k .

1k≥ 2|Ric(xk )|2 ≥ 2R(xk )2

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

2 ≤ R ≤ −12 .

∆f R = −R − 2|Ric|2 ≥ −R − 2R2 = −R(1 + 2R)

R∗(1 + 2R∗) ≥ 0⇒ R∗ ≤ −12

2 ≤ R ≤ −12 .

∆f R = −R − 2|Ric|2 ≥ −R − 2R2 = −R(1 + 2R)

R∗(1 + 2R∗) ≥ 0⇒ R∗ ≤ −12

Outline

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√

B. Chow, P. Lu and B. Yang; A lower bound for the scalarcurvature of certain steady gradient Ricci solitons

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√

B. Chow, P. Lu and B. Yang; A lower bound for the scalarcurvature of certain steady gradient Ricci solitons

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)

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|

1− R≤ 1

2 . Then

|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|

1− R≤ 1

2 . Then

|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|

1− R≤ 1

2 . Then

|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|

1− R≤ 1

Integrating −(Rγ)′

1−Ralong a minimizing geodesic γ(s)

1 +√

1− R1−√

1− R

0= −

(R γ)′

1− Rds ≤

|∇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 =

sech2 t2

1 +√

1− R1−√

1− R

0= −

(R γ)′

1− Rds ≤

|∇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

R(γ(t)) ≥ 4cc2et + 2c + e−t ≥

4cc2et + 2c2 + c2e−t =

sech2 t2

1 +√

1− R1−√

1− R

0= −

(R γ)′

1− Rds ≤

|∇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

R(γ(t)) ≥ 4cc2et + 2c + e−t ≥

4cc2et + 2c2 + c2e−t =

sech2 t2

1 +√

1− R1−√

1− R

0= −

(R γ)′

1− Rds ≤

|∇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

R(γ(t)) ≥ 4cc2et + 2c + e−t ≥

4cc2et + 2c2 + c2e−t =

sech2 t2

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

dx2 + dy2

1 + x2 + y2

)satisfies

R(x) = 4sech2r(x)

4(dx2 + dy2)

1 + x2 + y2

)satisfies

R(x) = sech2 r(x)

dx2 + dy2

1 + x2 + y2

)satisfies

R(x) = 4sech2r(x)

4(dx2 + dy2)

1 + x2 + y2

)satisfies

R(x) = sech2 r(x)

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),

|∇R|2 ≤ 4|Hf |2|∇f |2

Since |Hf |2 = |Rc|2 ≤ R2 one has

|∇R|R√

1− R≤ 2

|∇R|2 ≤ 4|Hf |2|∇f |2

|∇R|R√

1− R≤ 2

|∇R|2 ≤ 4|Hf |2|∇f |2

|∇R|R√

1− R≤ 2

THANK YOU VERY MUCHFOR YOUR ATTENTION

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