Songting YIN Qun HE

1 Introduction

In recent years,Finsler geometry has developed rapidly in its global and analytic aspects.The present main work is to generalize and improve some famous theorems of Riemann geometry to the Finsler setting.Among these issues,Finsler-Laplacian is one of the most important and interesting projects.As is well-known,there are several definitions of Finsler-Laplacian in Finsler geometry including nonlinear Laplacian,mean-value Laplacian and so on.With regard to nonlinear Finsler-Laplacian,some Laplacian comparison theorems,volume comparison theorems and various estimations on the first eigenvalue have been established(see[9,12–13,15]).

For a Riemanniann-manifold with Ric≥(n−1)k,Cheng[5]gave an upper bound estimation of the first Dirichlet eigenvalue of a geodesic ball.By using the weighted Ricci curvature condition,we extend this result to Finsler manifolds in this article(see Theorem 3.1).Besides,we also obtain an upper bound estimation of the first Neumann eigenvalue on a compact Finsler manifold in terms of the reversibility,the diameter and the lower bound of the weighted Ricci curvature(see Theorem 3.2).It is worth mentioning that in[12]and[4],Shen and Chen discussed these problems under the condition of Ricci curvature andS-curvature.However,in[12],the upper bound dose not have an explicit expression,and in[4]the result only focuses on the manifolds with Ricci curvature Ric≥(n−1)k,k≤0,while in this paper we will give an explicit expression of the upper bound on Finsler manifolds with weighted Ricci curvature RicN≥(n−1)k,∀k.Here RicNis defined in Definition 2.1 below.

It is well-known that Cheeger’s constant is estimated from below by a positive constant which depends only on the diameter and the lower bound of Ricci curvature ofM.By defining Cheeger’s constant on Finsler manifolds,we generalize the Cheeger’s inequality into the Finsler setting(see Theorem 4.1).As to the Faber-Krahn inequality,Shen extended it in[6]to the domains in a Minkowski space.In the present paper,we will consider the Randers manifolds with constant fl ag curvature and obtain another Finsler version of Faber-Krahn inequality(see Theorem 4.2).

In the Riemannian case,by using the gradient estimate,Li and Yau[7]gave the lower bound of the first eigenvalue of manifolds with Ricci curvature bounded from below by a negative constant.Recently,Wang and Xia[14]generalized it to Finsler manifolds by the method of one-dimensional model.To give an explicit expression of the lower bound,we should use some weighted-linear operators.The technique is based on a comparison theorem on the gradient of the eigenfunction.Then we follow step by step the work of Li and Yau to get the result(see Theorem 4.3).

The contents of this paper are arranged as follows.In Section 2,some fundamental formulas which are necessary for the present paper are given,where some lemmas are contained.In Section 3,the upper bounds of the first eigenvalue of Finsler-Laplacian such as Cheng’s type of inequalities are obtained.In Section 4,the lower bounds of the first eigenvalue of Finsler-Laplacian such as Cheeger type inequality,Faber-Krahn type inequality and Li-Yau’s estimation are shown.

2 Preliminaries

LetMbe ann-dimensional smooth manifold andbe the natural projection from the tangent bundleTM.Let(x,y)be a point ofTMwith,and let(xi,yi)be the local coordinates onTMwithA Finsler metric onMis a functionsatisfying the following properties:

(i)(Regularity)F(x,y)is smooth in

(ii)(Positive homogeneity)forλ>0.

(iii)(Strong convexity)The fundamental quadratic form

is positively definite.Letbe a vector field.Then the covariant derivative ofXbyv∈TxMwith reference vectoris defined by(see[2])

wheredenote the coefficients of the Chern connection.

Given two linearly independent vectorsV,W∈TxM{0},the fl ag curvature is defined by

whereRVis the Chern curvature:

Then the Ricci curvature for(M,F)is defined as

whereform an orthonormal basis ofTxMwith respect togV.

For a given volume form dμ=σ(x)dxand a vector,the distortion of(M,F,dμ)is defined by

To measure the rate of changes of the distortion along geodesics,we define

wherec(t)is the geodesic with˙c(0)=y.Sis called theS-curvature.

Now we introduce the weighted Ricci curvature on Finsler manifolds,which was defined by Ohta.In the present paper,we reform it as follows.

Definition 2.1(see[8])Let(M,F,dμ)be a Finsler n-manifold with volume formdμ.Given a vector V∈TxM,let γ:be a geodesic with γ(0)=x andDefine

where S(V)denotes the S-curvature at(x,V).

The weighted Ricci curvature of(M,F,dμ)is defined by

For a smooth function,the gradient vector ofuatxis defined as

Setfor a vector fieldVonM,andFor a smooth vector fieldVonMandx∈MV,we defineby using the covariant derivative as

For a smooth functionu:andx∈Mu,we setLetbe a local orthonormal basis with respect tog∇uonMuand putThen we have

Letbe aC∞vector field onM.The divergence ofVwith respect to an arbitrary volume form dμis defined by

whereThen the Finsler-Laplacian ofucan be defined by

Given a vector fieldVsuch thatV?=0 onMu,the weighted gradient vector and the weighted Laplacian on the weighted Riemannian manifold(M,gV)are defined by

We note that,and

Let(M,F)be a Finsler manifold.Define the distance function by

where the in fi mum is taken over all differentiable curvesγ:[0,1]→Mwithγ(0)=pandγ(1)=q.Define reversibilityη=η(M,F)as

Obviously,η∈[1,∞),andη=1 if and only if(M,F)is reversible.

Lemma 2.1(see[9])Let(M,F,dμ)be a Finsler manifold with volume formdμ.If its weighted Ricci curvature satisfiesRicN≥K,K∈R,N∈(n,∞),then the Laplacian of the distance function ρ(x)=d(p,x)from any given point p∈M can be estimated as follows:pointwise on M({p}∪Cut(p))and in the sense of distributions on M{z},where

Lemma 2.2(see[11])Let(M,F,dμ)be a Finsler metric measure space with volume formdμ.Let ϕ be a piecewise C1function on M such that every ϕ−1(t)is compact.Then for any continuous function f on M,

Lemma 2.3(see[8])Let(M,F,dμ)be a Finsler n-manifold with volume formdμ.Given u∈C∞(M),we have

pointwise on Mu,wherestands for the Hilbert-Schmidt norm with respect to g∇u.

Lemma 2.4(see[15])Let(M,F,dμ)be a Finsler n-manifold with volume formdμand u:M→R be a smooth function.Then on Muwe have

whereandis a local g∇u-orthonormal basis on Mu.

3 The Upper Bound Estimation of the First Eigenvalue

For a positive numberN,we denote by[N]the integer part ofN.ThenNis defined by

Theorem 3.1Let(M,F,dμ)be a complete Finsler n-manifold with volume formdμ.If the weighted Ricci curvature satisfiesRicN≥(N−1)k,N∈(n,∞),then the first Dirichlet eigenvalue

wheredenotes a geodesic ball with radius r in the N-dim simply connected space form with sectional curvature k,and defined in(3.1).

ProofLetϕbe the nonpositive first eigenfunction ofSince all simply connected space forms are two-point homogenous,ϕis a radial function.Namely,wherex0is the center ofMoreover,we have(see[5])

Letρ(x)=dF(p,x)be the distance function ofSince du=ϕ?dρandϕ?>0,we findUsing Lemma 2.1 and noting thatF(∇ρ)=1,we have

Note thatandIt follows that

Corollary 3.1Let(M,F,dμ)be a complete Finsler n-manifold volume formdμ.If the Ricci curvature satisfiesRic≥(n−1)k and S-curvature S=0,then the first Dirichlet eigenvalue

where Vn(k,r)denotes a geodesic ball with radius r in the n-dim simply connected space form with sectional curvature k.

For a complete simply connected Finslern-manifold with finite reversibilityη,the authors obtained in[16]

under the condition thatS≤0 andOn the other hand,from Corollary 3.1 and Cheng’s estimation onVn(k,r),we have

fork=−a2.By the discussion above,we can state the following result.

Corollary 3.2Let(M,F,dμ)be a complete and simply connected reversible Finsler nmanifold with volume formdμ.If S=0and the flag curvature K=−a2,then

Remark 3.1The result of Corollary 3.2 was also obtained by Chen[4].

Let(M,F,dμ)be a compact Finslern-manifold with volume form dμand boundary∂M.If there exists a functionu∈W1,2(M)satisfying

with a boundary condition

we callμ1the first Neumann eigenvalue of the Finsler-Laplacian(see also[14]).

Theorem 3.2Let(M,F,dμ)be a compact Finsler n-manifold with volume formdμ and thefinite reversibility η.Suppose that the weighted Ricci curvature satisfies(n,∞).Then the first Neumann or closed eigenvalue

where d denotes the diameter of M.

ProofLetp1,p2∈Mso that.ChoosingthenIn fact,if there exists a pointthen

which is a contradiction.Thus from(3.2)we conclude

whereuiare defined as in the proof of Theorem 3.1.Extenduito be zero outsideand takeThena1<0,a2>0 and

This completes the proof.

In[5],Cheng obtained the upper bound estimation of the first eigenvalue of a geodesic ball in the space form(see also in[10]).Thus,we give the following proposition.

Proposition 3.1Let(M,F,dμ)be a compact Finsler n-manifold with volume formdμ and the finite reversibility η.If the weighted Ricci curvature satisfiesRicN≥(N−1)k,N∈(n,∞),then the first Neumann eigenvalue

4 The Lower Bound Estimation of the First Eigenvalue

Let(M,F,dμ)be a compact Finslern-manifold with volume form dμand boundary∂M.For anyx∈∂M,there exist exactly two unit norm vectorsncharacterized by

We remark here that ifnis a norm vector,then−nmay not be a norm vector unlessFis reversible.Denote byn+(resp.n−)the normal vector that points outwards(resp.inwards)∂M.Then the induced volume forms of∂Mwith respect ton+andn−are

Consequently,the volumes of∂Mwith respect ton+andn−are

Note thatin general.In fact,the ratiomay be very large(see[4]for details).

In the Riemannian case,Cheeger[3]defined a constant and gave an original estimation on the lower bound of the first eigenvalue.Now we define Cheeger constanth(Ω)of Ω on Finsler manifolds by

where Ω?rangesover all open submanifolds of Ω with compact closure in Ω and smooth boundary∂Ω?,vol(Ω?)denotes the volume of Ω?,and vol+(∂Ω?)and vol−(∂Ω?)denote the volumes of∂Ω?with respect to outwards normal vectors and inwards normal vectors,respectively.

Theorem 4.1Let(M,F,dμ)be a complete Finsler n-manifold with volume formdμ.For any bounded domainΩwith a piecewise smooth boundary in M,the first Dirichlet eigenvalue satisfies

ProofBy using Co-Area formula in Lemma 2.2,we have∀ϕ∈C1(Ω),provided thatϕ|Ω>0,ϕ|∂Ω=0,

Letfbe the first eigenfunction with the Dirichlet boundary condition.Then from[12],we know thatf∈C1,α(Ω).Iff>0 in Ω,then letϕ=f2and we have

On the other hand,it follows from Legendre transform that

Using H¨older’s inequality,one gets

Substituting(4.2)into(4.1),one has

Iff<0 in Ω,we chooseϕ=−f2.Then by a similar argument as above,we have

In this case,dϕ=−2fdf,which yields

Since,a similar argument gives

Iffchanges its sign in Ω,then we can choose a domain?Ω⊂Ω such thatf>0 in?Ω andf|∂?Ω=0.Define

Then

From the above formula we find thatf1is the eigenfunction ofandλ1is an eigenvalue ofwhich implies.On the other hand,sinceThusAccording to the discussion above,we have Combining three cases above,we complete the proof.

whereand

Theorem 4.2Let(M,F,dμ)be a Randers manifold with Busemann-Hausdorff volume form and the finite reversibility η,where F is expressed by(4.3)and has constant flag curvature.LetΩ⊂M be a domain and B be a geodesic ball of Riemannian manifold(M,h).Ifvoldμ(Ω)=voldμ(B),then the first Dirichlet eigenvalue satisfies

where the equality holds if and only if F=h andΩ=B.

ProofFirstly we have that if(M,F)has constant flag curvature,then(M,h)has constant sectional curvature(see[1]).Letfbe the first Dirichlet eigenfunction corresponding to the first eigenvalueλ1in Ω,that is,

Whenf>0 in Ω,we setand.Using a symmetrization procedure,we construct the geodesic ballBtin(M,h)such that voldμ(Bt)=voldμ(Ωt)for eacht,andB=B0.We define a functionsuch thatgis a radially decreasing function and

Notice the important fact that dμ=dVhwhere dVhis the Riemaniann volume form of(M,h)(see[1,p.12]).Using Co-Area formula in Lemma 2.2,we have

where dνtand dAtdenote the volume elements on Γtand∂Btrespectively,and∇hgdenotes the gradient ofgwith respect toh.Here we have used the identity

From the discussion in[10,p.28],we find

wheredenote the unit normal vector fields of∂Ωtwith respect toFandhrespectively.Sincefor an arbitrary vectorX,one gets

Using H¨older’s inequality,the isoperimetric inequality and(4.5),we obtain

which yields

Recall that the dual metric of(4.3)is

whereand.Thus for aC1functionf,we have

from which we conclude

From(4.6)–(4.7),we obtain

By using(4.4)and(4.8),it follows that

Here we have used the fact thatIf the equality holds,then the above inequalities become equalities.In particular,

The third formula implieswhich together with the first two formulas yieldsb0=0.In this case,F=hand Ω=B.

Whenf<0 in Ω,we setandThen by a similar argument as the above,we also give the conclusion.

Whenfchanges its sign in Ω,we can choose a domain?Ω⊂Ω such thatf|?Ω>0 andf|∂?Ω=0.Define

Thenf1is the first eigenfunction of?Ω corresponding toλ1.Using the conclusion of the first case,we have

whereis a geodesic ball of Riemannian manifold(M,h)withSincewe obtain

Theorem 4.3Let(M,F,dμ)be a compact Finsler n-manifold with volume formdμ,without boundary.If the weighted Ricci curvatureRicN≥(n−1)(−k)for k≥0and N∈(n,∞),then the first closed eigenvalue

where d denotes the diameter of M.

ProofLetube the first eigenfunction of Finsler-Laplacian.ThenWithout loss of generality,we assume that−m=infu1 be a real number.We de fi ne the function

Suppose thatx0is a maximum point onM.Thenx0∈Muand

From(4.9),we get

Then atx0we have

Using Lemma 2.3,one gets

where

Sinceuis the eigenfunction of Laplacian,we have

A direct calculation on the first formula of(4.10)yields

Choosing ag∇u-orthonormal basis atx0such thatwe have

By using the standard inequality and Lemma 2.4,one has for 0<ε<1 that

Using(4.11)–(4.13)and noting that RicN≥(n−1)(−k),we obtain

SetThen.Thus(4.14)can be changed into

Viewing(4.15)as a quadratic inequality and noting that,we have

Therefore,for an arbitrary pointx∈M,it is concluded that

Letx1,x2be two points such thatu(x1)=0 andu(x2)=1.Consider a regular minimal geodesicγ(s)joiningx1andx2,and then

wheredis the diameter ofM.Hence

AcknowledgementThe authors would like to thank the referees for their useful comments on the manuscript.

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