Wen WANG Rulong XIE Pan ZHANG

Abstract In the paper,the authors provide a new proof and derive some new elliptic type(Hamilton type)gradient estimates for fast diffusion equations on a complete noncompact Riemannian manifold with a fixed metric and along the Ricci flow by constructing a new auxiliary function.These results generalize earlier results in the literature.And some parabolic type Liouville theorems for ancient solutions are obtained.

Keywords Gradient estimate,Fast diffusion equation,Ricci flow,Liouville theorem

1 Introduction and Main Results

In this paper,we continue to consider the fast diffusion equation(FDE for short)

on a family of Riemannian manifolds(M,g(t))for two cases:The one is that g(t)is some fixed metric,and the other one is g(t)deformed by the Ricci flow:

Li and Yau[16]established a famous space-time gradient estimate for positive solutions to the heat equation.In 1993,Hamilton[9]proved the space-only gradient estimate for closed manifolds,which was extended by Souplet and Zhang in[21]to the complete noncompact manifolds.Bailesteanu,Cao and Pulemotov[1]generalized the Hamilton’s gradient estimates to the Ricci flow.For the developments,see[4,10,12,17–18,20,22–24,27].In 2009,Lu,Ni,Vzquez and Villani[19]studied the FDE(1.1)on Riemannian manifolds,and derived a local space-time gradient estimates.Later in[28],Zhu studied the FDE(1.1)on complete noncompact Riemannian manifolds,and derived the following space-only gradient estimate(Hamilton type gradient estimate)and Liouville type theorem.

Recently,Xu[26],Huang and Ma[11]improved the result of Zhu[28].Huang and Ma[10]proved a gradient estimate and Liouville theorem for FDE(1.1)withXu[26]derived the gradient estimate for the FDE(1.1)withCao and Zhu[3]proved Li-Yau-Hamilton type differential Harnack estimates for positive solutions of the FDE(1.1).Li,Bai and Zhang[13]proved Hamilton type gradient estimates for the fast diffusion equations under the Ricci flow.

Our results of this paper are encouraged by the work in[1,3,8,10–11,13,21–22,25,28–29].We consider the FDE(1.1),and derive some elliptic type(Hamilton-Souplet-Zhang type)gradient estimates.

To prove the propertity of the positive solution of the FDE(1.1),we will use the following transformation:LetThen

Our paper is organized as follows:We show our main results in Section 1.We will give some lemmas and the proof of the main results on Riemannian manifolds with a fixed metric in Section 2.The proof of the main results on Riemannian manifolds along the Ricci flow will be given in Section 3.

Theorem 1.1Let(Mn,g)be an n-dimensional complete Riemannian manifold with Ric(Mn)≥−K for some K≥0 in Bx0,R,which is a geodesic ball centered at some fixed point x0in Mnwith radius R.Assume that v is any positive solution to(1.3)in QR,T=Bx0,R×[t0−T,t0]⊂Mn×(−∞,∞)with 0<δ≤v≤A for some constants δ and A.

By applying Theorem 1.1,we deduce the following Liouville type theorem.

Theorem 1.2Let(Mn,g)be an n-dimensional complete,noncompact manifold with nonnegative Ricci curvature.Let u be a positive solution to(1.1)and d(x)be the geodesic distance of g.

Remark 1.1(1)When n≥2,we have

that is

(2)When n≥4,we have

that is

(3)When n≥7,we have

that is

Hence,Theorems 1.1–1.2 generalize some known results in[11,26,28].

Remark 1.2Our proof is a little different from the proofs of Zhu[28],Huang,Ma[11]and Xu[26].We derive the evolution equation of quantity logwith v≤A.

Remark 1.3When n≥4,we have

When 3≤n≤29,we have

that is

So,(1.5)and Theorem 1.2 generalize the results of Zhu[28],Huang and Ma[11].

Remark 1.4The upper bound of the gradient estimate(1.5)does not contain the upper bound of v.

Theorem 1.3Let(Mn,g)be an n-dimensional complete Riemannian manifold with Ric(Mn)≥−K for some K≥0 in Bx0,R,which is a geodesic ball centered at some fixed point x0in Mnwith radius R.Assume that v is any positive solution to(1.3)in QR,T=Bx0,R×[t0−T,t0]⊂Mn×(−∞,∞)with v≤A.LetThen there exist a constant C=C(n,α)such that

Moreover,if(Mn,g)has nonnegative Ricci curvature and u is any positive solution to(1.1)on Mn×(0,∞),then there exists a constant C=C(n,α)such that

By applying Theorem 1.3,we deduce the following Liouville type theorem.

Theorem 1.4Let(Mn,g)be an n-dimensional complete,noncompact manifold with nonnegative Ricci curvature.Let u be a positive solution to(1.1)withsuch thatnear infinity,where d(x)is the geodesic distance of g.Then u is a constant.

When u(x,t)is independent on t,by(1.6),we can derive the following Liouville type theorem.

Theorem 1.5Let(Mn,g)be an n-dimensional complete,noncompact manifold with nonnegative Ricci curvature.Let u be a positive solution to the equation

Daskalopoulos et al.[6–7]observed that the metricsatisfies the Yamabe flow(see[2])

on Rn,n≥3,for 0

Theorem 1.6Let(Mn,g)be an n-dimensional complete Riemannian manifold with Ric(Mn)≥−K for some K≥0 in Bx0,R,which is a geodesic ball centered at some fixed point x0in Mnwith radius R.Assume that u is any positive solution to the equation

in QR,T=Bx0,R×[t0−T,t0]⊂Mn×(−∞,∞).Assume also thatwith 0<δ≤v≤A for some constants δ and A.Then there exists a constant C=C(n,α)such that

By using Theorem 1.6,we deduce the following Liouville type theorem.

Next,we state our estimates for the FDE coupled with the Ricci flow(see[5,9]),which are similar to the fixed metric case.

Let(Mn,g(t))t∈[0,T]be a complete solution to the Ricci flow

Theorem 1.8Let(Mn,g(x,t))t∈[0,T]be a complete solution to(1.11).Suppose that|Ric(x,t)|≤K for some K≥0 and all(x,t)∈BR,T=B(x0,R)×(0,T]for some fixed x0∈Mn.Assume that v is any positive solution to the equation

Remark 1.5Since

So,Theorem 1.8 generalizes the one of Li,Bai and Zhang[13].

Theorem 1.9Let(Mn,g(x,t))t∈[0,T]be a complete solution to(1.11).Suppose that|Ric(x,t)|≤K for some K≥0 and all(x,t)∈BR,T=B(x0,R)×(0,T]for some fixed x0∈Mn.Assume that v is any positive solution to the equation

Theorem 1.10Let(Mn,g(x,t))t∈[0,T]be a complete solution to(1.11).Suppose that|Ric(x,t)|≤K for some K≥0 and all(x,t)∈BR,T=B(x0,R)×(0,T]for some fixed x0∈Mn.Assume that u is any positive solution to the equation

in BR,T.Assume also thatwith 0<δ≤v≤A for some constants δ and A.

Then there exists a constant C=C(n,α)such that

2 FED Under the Fixed Metric

2.1 Basic lemmas

Before proving the main theorems,we need some lemmas.Consider the equation

on a complete Riemannian manifold(Mn,g).Let v(x,t)be a solution of(2.1)and 0

here t0∈R and T>0.We first introduce a new smooth function

in QR,T.Then v=A·e−g,

From(2.1),we have

By utilizing the above equation(2.3),we can derive the following lemma.

Lemma 2.1Let ω=|∇g|2.Then for any(x,t)∈QR,T,

ProofBy using the Bochner-Weitzenbck formula

we have

By(2.3),we obtain

By applying(2.2)–(2.3)and the Cauchy inequality,we have

Substituting(2.7)into(2.6)and noting that Ric≥−K,we have

ProofApplying(2.5),we have

By utilizing(2.1)and(2.3),we have

By the Cauchy inequality,we have

Combining(2.9)and(2.10),we have

where we use the fact that

In order to obtain the gradient estimates,we need to require the coefficient f(β)of|∇g|4to be positive.In fact,

Taking β=1 in(2.11),the following lemma is derived.

Lemma 2.3LetThen

We next introduce a smooth cut-offfunction(see[10,17,24]),which will be used in the proof of our main theorems.

Lemma 2.4(see[16,21,28])We use the geodesic polar coordinate here.Assume that a function ϕ=ϕ(x,t)is a smooth cut-offfunction supported in QR,T,satisfying the following properties:

2.2 The proof of theorems

In this section,we will prove our main theorems by Lemma 2.4.

Proof of Theorem 1.1 Part 1:Assume that the maximum of ϕϖ is arrived at a point(x1,t1).By[16],we can suppose,without loss of generality,that x1is not on the cut-locus of Mn.Therefore,at(x1,t1),it yields∆(ϕϖ)≤0,(ϕϖ)t≥0 and∇(ϕϖ)=0.By

then

Hence,by(2.8)and a straightforward calculation,it yields that

This implies

We next estimate upper bounds for each term of the right hand side of(2.14).Applying the Young inequality,we have

and

We substitute(2.15)–(2.19)into(2.14),and have

at(x1,t1).Therefore,for all(x,t)∈QR,T,we obtain

This proves part 1 of the theorem.

Part 2Assume that the maximum of ϕω is arrived at a point(x1,t1).By[16],we can suppose,without loss of generality,that x1is not on the cut-locus of Mn.Therefore,at(x1,t1),it yields∆(ϕω)≤0,(ϕω)t≥0 and∇(ϕω)=0.By 0=∇(ϕω)=ω∇ϕ+ϕ∇ω,thenHence,by(2.4)and a straightforward calculation,it yields that

This implies

We next estimate upper bounds for each term of the right hand side of(2.23).Applying the Young inequality,we have

and

We substitute(2.24)–(2.28)into(2.23),and have

at(x1,t1).Therefore,for all(x,t)∈QR,T,we obtain

The proof is completed.

This implies

We next estimate upper bounds for each term of the right hand side of(2.32).Applying the Young inequality,we have

and

We substitute(2.33)–(2.37)into(2.32),and have

at(x1,t1).Therefore,for all(x,t)∈QR,T,we obtain

The proof is completed.

3 FDE along the Ricci Flow

The Ricci flow(1.11)was first introduced by Hamilton[9],and was an important tool of analyzing the structure of manifolds.In 2010,Bailesteanu,Cao and Pulemotov[1]generalized the Hamiltons gradient estimates for the heat equation on Riemannian manifolds with a fixed metric to the Ricci flow,and proved the theorem below.

Theorem B(see[1])Let(Mn,g(x,t))t∈(0,T]be a complete solution along the Ricci flow.Let|Ric(x,t)|≤K for some K>0 and all(x,t)∈BR,T:≡B(x0,R)×[0,T].Suppose that u is a smooth positive solution to the heat equation

If u≤A for some A>0 and all(x,t)∈BR,T,then there exists a constant C=C(n)such that

In this section,we will derive some Hamilton type gradient estimates for fast diffusion equations(1.1)on a Riemannian manifold evolved by the Ricci flow.

3.1 Basic lemmas

Before the proof of the main theorems,we need some lemmas.Consider the equation

on a complete Riemannian manifold(Mn,g)along the Ricci flow.Let v(x,t)be a solution of(3.2)and 0

here T>0.

Now,in order to simplify writing,we all set∆=∆g(t)and∇=∇g(t).

We first introduce a new smooth function

in BR,T.From(3.2),we have

By utilizing the above equation(3.3),we can derive the following lemma.

Lemma 3.1Let ω=|∇g|2.Then for any(x,t)∈BR,T,

ProofThe Ricci flow equation(3.1)implies

By further using the Bochner-Weitzenbck formula(2.5),we have

By(3.3),we obtain

Substituting(2.7)into(3.6)and noting that|Ric|≤K,we have

ProofApplying(2.5),we have

By utilizing(2.1),(2.3)and(3.5),we have

Therefore,by(2.10)we have

where we use the fact that

In order to obtain the gradient estimates,we need to require the coefficient f(β)of|∇g|4to be positive.In fact,

Taking β=1 in(3.9),the following lemma is derived.

Lemma 3.3Let ω1=v|∇g|2.Then

We next introduce a smooth cut-offfunction(see[1,16]),which will be used in the proof of our main theorems.

3.2 The proof of theorems

In this section,we will prove our main theorems by Lemma 3.4.Let dist(x,x0,t)be the distance between x∈Mnand x0with respect to the metric g(x,t).

Proof of Theorem 1.8Part 1:In order to derive the result,we also need a cutofffunction ϕ by Li-Yau[16]on BR,T.Define a smooth function ϕ:Mn×[0,T]→R by ϕ(x,t)=(dist(x,x0,t),t)supported in BR,T,wheresatisfies Lemma 3.4.

Let ω=|∇g|2.Assume that the function ϕω arrives its maximum at a point(x1,t1)and x1is not in the cut-locus of Mnby[15].Therefore,at(x1,t1),it yields∆(ϕω)≤0,(ϕω)t≥0 and∇(ϕω)=0.

By 0=∇(ϕω)=ω∇ϕ+ϕ∇ω,then we haveHence,by(3.4)and a straightforward calculation,it yields that

This implies

We next estimate upper bounds for each term of the right hand side of(3.12).Applying the Young inequality,we have

and

For the last term,by[1],we have

We substitute(3.13)–(3.17)into(3.12),and have

at(x1,t1).Therefore,for all(x,t)∈we obtain

Part 2:Define a smooth function

This implies

We next estimate upper bounds for each term of the right hand side of(3.21).Applying the Young inequality,we have

and

We substitute(3.22)–(3.26)into(3.21),and have

at(x1,t1).Therefore,for all(x,t)∈we obtain

So,we prove Theorem 1.8.

This implies

We next estimate upper bounds for each term of the right hand side of(3.30).Applying the Young inequality,we have

and

We substitute(3.31)–(3.35)into(3.30),and have

at(x1,t1).Therefore,for all(x,t)∈we obtain

AcknowledgementsWe are grateful to Professors Jiayu Li and Xiaobao Zhu for their support and encouragement.The first author would like to thank Professor Qi S Zhang for introducing this topic in the summer course at USTC.We appreciate the referees for the valuable suggestions and the very careful reading of the original manuscript.