A sharp lower bound for the first eigenvalue on Finsler manifolds

Ann. I. H. Poincaré – AN 30 (2013) 983–996

www.elsevier.com/locate/anihpc

A sharp lower bound for the first eigenvalue on Finsler manifolds

Guofang Wang

, Chao Xia

aAlbert-Ludwigs-Universität Freiburg, Mathematisches Institut, Eckerstr. 1, 79104 Freiburg, Germany bMax-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstr. 22, 04103, Leipzig, Germany

Received 22 November 2011; accepted 7 December 2012 Available online 11 January 2013

This paper is dedicated to Professor S.S. Chern on the occasion of his 100th birthday

Abstract

In this paper, we give a sharp lower bound for the first (nonzero) Neumann eigenvalue of Finsler-Laplacian in Finsler manifolds in terms of diameter, dimension, weighted Ricci curvature.

©2013 Elsevier Masson SAS. All rights reserved.

MSC:35P15; 53C60; 35A23

Keywords:Finsler-Laplacian; First eigenvalue; Weighted Ricci curvature; Poincaré inequality

1. Introduction

The study of the first (nonzero) eigenvalue of Laplacian in Riemannian manifolds plays an important role in differ- ential geometry. The first result on this subject, due to Lichnerowicz [11], says that for an n-dimensional smooth compact manifold without boundary, the first eigenvalue λ1 can be estimated below by _n−ⁿ₁K, provided that its RicK >0. In this case, Obata[13] established a rigidity result, asserting the optimality of Lichnerowicz’ esti- mate. Namely, λ₁=K if and only ifM is isometric to then-dimensional sphere with constant curvature _n−¹₁K.

WhenK=0, Li–Yau[9,10]developed a method, which depends on the gradient estimate of the eigenfunctions, to give the lower bound of the first eigenvalue via diameterd, precisely,λ_12d^π22. Their method had been improved by Zhong–Yang[24]to obtainλ1^π_d2², which is optimal in the sense that equality can be attained for one-dimensional circle. Very recently, Hang and Wang showed thatλ₁>^π2

d² in[7], if the dimensionn >1. These results also hold true whenMis a manifold with convex boundary. WhenMis a convex domain inRⁿ, this is a classical result of Payne–

Weinberger[17]. Later Chen–Wang[5]and Bakry–Qian[3]combined these results into a same framework, and gave estimates for the first eigenvalue of very general elliptic symmetric operators, via diameter and Ricci curvature. This sharp estimate on Riemannian manifolds has been also generalized to Alexandrov spaces by Qian–Zhang–Zhu[18].

✩ GW is partly supported by SFB/TR71 “Geometric partial differential equations” of DFG. CX is supported by China Scholarship Council.

* Corresponding author.

E-mail addresses:guofang.wang@math.uni-freiburg.de(G. Wang),chao.xia@mis.mpg.de(C. Xia).

0294-1449/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2012.12.008

Finsler geometry attracts many attentions in recent years, since it has broader applications in nature science. Simul- taneously Finsler manifold is one of the most natural metric measure spaces, which plays an important role in many aspects in mathematics. There exists a natural Laplacian on Finsler manifolds, which we call hereFinsler-Laplacian.

Unlike the usual Laplacian, the Finsler-Laplacian is a nonlinear operator. The objective of this paper is to study the lower bound for the first (nonzero) eigenvalue of this Finsler-Laplacian on Finsler manifolds. In[14]Ohta introduced the weighted Ricci curvature RicN for N ∈ [n,∞] of Finsler manifolds, following the work of Lott–Villani [12]

and Sturm[20]on metric measure space. He proved the equivalence of the lower boundedness of theRicN and the curvature-dimension conditionsCD(K, N )in[12,20]. As a byproduct, he obtained a Lichnerowicz type estimate on the first eigenvalue of Finsler-Laplacian under the assumptionRicNK >0. Another interesting type of eigenvalue estimates was obtained by Ge–Shen in[6], namely the Faber–Krahn type inequality for the first Dirichlet eigenvalue of the Finsler-Laplacian holds. See also[4]and[22]. Recently, we proved in[21]the Li–Yau–Zhong–Yang type sharp estimate for a so-called anisotropic Laplacian on a Minkowski space, which could be viewed as the simplest, but interesting and important case of non-Riemannian Finsler manifolds. In this paper, we shall generalize the results in [21]to general Finsler manifolds. Moreover, as in [5]and[3], we shall put the Li–Yau–Zhong–Yang type and the Lichnerowicz type sharp estimates into a uniform framework.

Our main result of this paper is

Theorem 1.1.Let(Mⁿ, F, m)be ann-dimensional compact Finsler measure space, equipped with a Finsler structure F and a smooth measure m, without boundary or with a convex boundary. Assume that RicN K for some real numbersN ∈ [n,+∞]and K∈R. Letλ1be the first(nonzero)Neumann eigenvalue of the Finsler-Laplacian_m, i.e.,

−mu=λ1u, inM, (1)

with a Neumann boundary condition

∇u(x)∈T_x(∂M), (2)

if∂Mis not empty. Then

λ₁λ₁(K, N, d), (3)

whered is the diameter ofM,λ1(K, N, d)represents the first(nonzero)eigenvalue of the1-dimensional problem v−T (t )v= −λ₁(K, N, d)v in

−d 2,d

2

, v

−d 2

=v

−d 2

=0, (4)

withT (t )varying according to different values ofKandN.T is explicitly defined by

T (t )=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

√(N−1)Ktan(

K

N−1t ), forK >0, 1< N <∞,

−√

−(N−1)Ktanh(

−_N^K₋₁t ), forK <0, 1< N <∞,

0, forK=0, 1< N <∞,

Kt, forN= ∞.

(5)

The precise definition of the Finsler measure space, convex boundary, diameterd, weighted Ricci curvatureRicN, gradient vector field∇, Finsler-Laplacian_mwill be given in Section 2 below.

Equivalently, Theorem1.1gives an optimal Poincaré inequality in Finsler manifolds.

Theorem 1.2.Under the same assumptions as in Theorem1.1, we have

M

F²(∇u) dmλ₁(K, N, d)

M

(u− ¯u)²dm, (6)

whereu¯is the average ofu.

In the case ofK >0 andN=n, Theorem1.1sharpens the Lichnerowicz type estimate given by Ohta[14], since by Meyer theorem, d ^√_(n^π_−1)K, and λ1(K, N, d)λ1(K, n,√ ^π

(n−1)K)=_n^nK₋₁. In the case ofK=0 andN=n,

Theorem1.1gives the Li–Yau–Zhong–Yang type sharp estimate for the Finsler-Laplacian, sinceλ₁(0, n, d)=^π_d2². We remark that the Minkowski space(Rⁿ, F )equipped with then-dimensional Lebesgue measure satisfies thatRicNK withK=0 andN =n(see e.g.[23]theorem on page 908 and[14, Theorem 1.2]), hence Theorem1.1covers the estimate in[21].

Our proof goes along the line of Bakry–Qian[3]. The technique is based on a comparison theorem on the gradient of the first eigenfunction with that of a one-dimensional (1-D) model function (Theorem3.1), which was developed by Kröger[8]and improved by Chen–Wang[5]and Bakry–Qian[3]. By using a Bochner–Weitzenböck formula es- tablished recently by Ohta–Sturm[16], we find that the one-dimensional model coincides with that in the Riemannian case, as presented in Theorem1.1. It should be not so surprising, because when we considerF inR, it can only be two pieces of linear functions. Since the 1-D model has been extensively studied in[3], it also eases our situation, although we deal with a nonlinear operator. One difficulty arises when we deal with the Neumann boundary prob- lem, since the convexity of boundary could not be directly applied due to the difference between the metric induced from the boundary itself and the metric induced from the gradient of the first eigenfunction. We will establish some equivalence between them (see Lemmas3.1and3.2) to overcome this difficulty. Another ingredient is a comparison theorem on the maxima of eigenfunction with that of the 1-D model function (Theorem3.2). Everything in[3]works except the boundedness of the Hessian of eigenfunctions around a critical point (since the eigenfunction is onlyC^1,α amongM), which was used to prove (25). Here we avoid the use of the Hessian of eigenfunctions by using the com- parison theorem on the gradient. For the rest we follow step by step the work of Bakry–Qian[3]to get Theorem1.1.

This paper is organized as follows. In Section 2, the fundamentals in Finsler geometry is briefly introduced and the recent work of Ohta–Sturm is reviewed. We shall first prove the comparison theorem on the gradient and on the maxima of the eigenfunction and then Theorem1.1in Section 3.

2. Preliminaries on Finsler geometry

In this section we briefly recall the fundamentals of Finsler geometry, as well as the recent developments on the analysis of Finsler geometry by Ohta–Sturm[14–16]. For Finsler geometry, we refer to[1]and[19].

2.1. Finsler structure and Chern connection

LetMⁿbe a smooth, connectedn-dimensional manifold. A functionF:T M→ [0,∞)is called aFinsler structure if it satisfies the following properties:

(i) F isC^∞onT M\ {0};

(ii) F (x, t V )=t F (x, V )for all(x, y)∈T M and allt >0;

(iii) for every(x, V )∈T M\ {0}, the matrix g_ij(V ):= ∂²

∂Vi∂Vj

1 2F²

(x, V ) is positive definite.

Such a pair(Mⁿ, F )is called aFinsler manifold. A Finsler structure is said to bereversibleif, in addition,F is even.

OtherwiseF is nonreversible. By aFinsler measure spacewe mean a triple(Mⁿ, F, m)constituted with a smooth, connectedn-dimensional manifoldM, a Finsler structureF onMand a measuremonM.

Forx₁, x₂∈M, thedistance functionfromx₁tox₂is defined by

d(x1, x2):=inf

γ 1

0

F

˙ γ (t)

dt,

where the infimum is taken over all C¹-curvesγ:[0,1] →M such that γ (0)=x₁ and γ (1)=x₂. Note that the distance function may not be symmetric unlessF is reversible. AC^∞-curveγ:[0,1] →Mis called ageodesicif F (γ )˙ is constant and it is locally minimizing. ThediameterofMis defined by

d:= sup

x,y∈M

d(x, y).

Theforwardandbackward open ballsare defined by B⁺(x, r):=

y∈M:d(x, y) < r

, B⁻(x, r):=

y∈M: d(y, x) < r . We denoteB^±(x, r):=B⁺(x, r)∪B⁻(x, r).

For every nonvanishing vector fieldV,g_ij(V )induces a Riemannian structureg_V ofT_xMvia g_V(X, Y )=

n i,j=1

g_ij(V )XⁱY^j, forX, Y∈T_xM.

In particular,g_V(V , V )=F²(V ).

Letπ:T M\ {0} →Mthe projection map. The pull-back bundleπ^∗T Madmits a unique linear connection, which is theChern connection. The Chern connection is determined by the following structure equations, which characterize

“torsion freeness”:

D_X^VY−D_Y^VX= [X, Y] (7)

and “almostg-compatibility”

Z

gV(X, Y )

=gV

D_Z^VX, Y +gV

X, D_Z^VY +CV

D^V_ZV , X, Y

(8) forV ∈T M\ {0},X, Y, Z∈T M. Here

C_V(X, Y, Z):=C_{ij k}(V )XⁱY^jZ^k=1 4

∂³F²

∂VⁱV^jV^k(V )XⁱY^jZ^k

denotes the Cartan tensor andD_X^VY the covariant derivative with respect to reference vectorV ∈T M\ {0}. We mention here that CV(V , X, Y )=0 due to the homogeneity ofF. In terms of the Chern connection, a geodesicγ satisfiesD^γ_γ˙^˙γ˙=0. For local computations in Finsler geometry, we refer to[19].

2.2. Hessian and Finsler-Laplacian

We shall introduce the Finsler-Laplacian on Finsler manifolds. First of all, we recall the notion of the Legendre transform.

Given a Finsler structureF onM, there is a natural dual normF^∗on the cotangent spaceT^∗M, which is defined by

F^∗(x, ξ ):= sup

F (x,V )1

ξ(V ) for anyξ∈T_x^∗M.

One can show thatF^∗is also a Minkowski norm onT^∗Mand g_ij^∗(ξ ):= ∂²

∂ξ_i∂ξ_j 1

2F^∗2

(x, ξ )

is positive definite for every(x, ξ )∈T^∗M\ {0}.

TheLegendre transformis defined by the mapl:TxM→T_x^∗M:

l(V ):=gV(V ,·) forV ∈TxM\ {0},

0 forV =0.

One can verify thatF (V )=F^∗(l(V ))for anyV ∈T M andg_ij^∗(x, l(V ))is the inverse matrix ofg_ij(x, V ).

Let u:M→R be a smooth function onM andDu be its differential 1-form. Thegradient of u is defined as

∇u(x):=l⁻¹(Du(x))∈TxM. DenoteMu:= {Du=0}. Locally we can write in coordinates

∇u= n i,j=1

g^ij(x,∇u)∂u

∂x_i

∂

∂x_j inM_u.

TheHessianofuis defined by using Chern connection as

∇²u(X, Y )=g_∇u

D_X^∇u∇u, Y

. (9)

One can show that ∇²u(X, Y ) is symmetric, see [22] and [16]. Indeed, using (7) and (8) and noticing that C_∇u(∇u, X, Y )=0, we have

g_∇u

D_X^∇u∇u, Y

=X

g_∇u(∇u, Y )

−g_∇u

∇u, D^∇_X^uY

=XY (u)−g_∇u

∇u, D_Y^∇uX+ [X, Y]

=Y X(u)+ [X, Y](u)−g_∇u

∇u, D_Y^∇uX

− [X, Y](u)

=Y

g_∇u(∇u, X)

−g_∇u

∇u, D^∇_Y^uX

=g_∇u

D_Y^∇u∇u, X .

In order to define a Laplacian on Finsler manifolds, we need a measurem(or a volume formdm) onM. From now on, we consider the Finsler measure space(M, F, m)equipped with a fixed smooth measurem. LetV ∈T Mbe a smooth vector field onM. ThedivergenceofV with respect tomis defined by

divmV dm=d(Vdm),

whereVdmdenotes the inner product ofV with the volume formdm. In a local coordinate(xⁱ), expressingdm= e^Φdx¹dx²· · ·dxⁿ, we can write divmV as

divmV = n i=1

∂Vⁱ

∂xⁱ +Vⁱ∂Φ

∂xⁱ

.

A Laplacian, which is called theFinsler-Laplacian, can now be defined by mu:=divm(∇u).

We remark that the Finsler-Laplacian is better to be viewed in a weak sense that foru∈W^1,2(M),

M

φmu dm= −

M

Dφ(∇u) dm forφ∈C_c^∞(M).

The relationship between_muand∇²uis that mu+DΨ (∇u)=trg_∇u

∇²u

= n i=1

∇²u(ei, ei),

whereΨ is defined bydm=e^{−Ψ (V )}dVolgV and{e_i}is an orthonormal basis ofT_xM with respect tog_∇u. See e.g.

[22, Lemma 3.3].

Given a vector fieldV, theweighted Laplacianis defined on the weighted Riemannian manifold(M, g_V, m)by ^V_mu:=divm

∇^Vu , where

∇^Vu:= ^ni,j=1g^ij(x, V )_∂x^∂u

i

∂

∂x_j forx∈M_u,

0 forx∈M\M_u.

Similarly, the weighted Laplacian can be viewed in a weak sense that foru∈W^1,2(M). We note that^∇_m^uu=_mu.

2.3. Finsler manifolds with boundary

Assume that(M, F, m)is a Finsler measure space with boundary∂M, then we shall view∂M as a hypersurface embedded inM.∂Mis also a Finsler manifold with a Finsler structureF∂M induced byF. For anyx∈∂M, there exists exactly two unitnormal vectorsν, which are characterized by

T_x(∂M)=

V ∈T_xM: g_ν(ν, V )=0, gν(ν, ν)=1 .

Throughout this paper, we choose the normal vector that points outwardM. Note that, ifνis a normal vector,−νmay be not a normal vector unlessF is reversible.

The normal vectorνinduces a volume formdm_νon∂Mfromdmby Vdm=g_ν(ν, V ) dm_ν, for allV ∈T (∂M).

One can check that Stokes theorem holds (see[19, Theorem 2.4.2])

M

divm(V ) dm=

∂M

g_ν(ν, V ) dm_ν. We recall the convexity of the boundary ofM.

Thenormal curvatureΛ_ν(V )atx∈∂Min a directionV ∈T_x(∂M)is defined by Λ_ν(V )=g_ν

ν, D_γ^γ_˙^˙γ (0)˙

, (10)

whereγis the unique local geodesic for the Finsler structureF_∂M on∂Minduced byF with the initial dataγ (0)=x andγ (0)˙ =V.

Mis said to hasconvex boundaryif for anyx∈∂M, the normal curvatureΛν atxis nonpositive in any directions V ∈Tx(∂M). We remark that the convexity ofMmeans thatD^γ_γ˙^˙γ (0)˙ lies at the same side ofTxMasM. Hence the choice of normal is not essential for the definition of convexity. (See Lemma 3.2 below.) There are several equivalent definitions of convexity, see for example[2]and[19].

2.4. Weighted Ricci curvature

The Ricci curvature of Finsler manifolds is defined as the trace of the flag curvature. Explicitly, given two linearly independent vectorsV , W∈T_xM\ {0}, theflag curvatureis defined by

K^V(V , W )= g_V(R^V(V , W )W, V ) gV(V , V )gV(W, W )−gV(V , W )², whereR^V is theChern curvature(orRiemannian curvature):

R^V(X, Y )Z=D^V_XD_Y^VZ−D^V_YD_X^VZ−D^V_[X,Y]Z.

Then theRicci curvatureis defined by Ric(V ):=

n−1

i=1

K^V(V , ei),

wheree₁, . . . , e_n−1,_{F (V )}^V form an orthonormal basis ofT_xMwith respect tog_V.

We recall the definition of the weighted Ricci curvature on Finsler manifolds, which was introduced by Ohta in[14], motivated by the work of Lott–Villani[12]and Sturm[20]on metric measure spaces.

Definition 2.1. (See[14].) Given a unit vector V ∈T_xM, letη:[−ε, ε] →M be the geodesic such thatη(0)˙ =V. Decomposemasm=e^−Ψdvolη_˙alongη, where volη_˙ is the volume form ofg_η˙as a Riemannian metric. Then

Ricn(V ):=

Ric(V )+(Ψ◦η)(0) if(Ψ ◦η)(0)=0,

−∞ otherwise;

RicN(V ):=Ric(V )+(Ψ◦η)(0)−(Ψ◦η)(0)²

N−n , forN∈(n,∞), Ric_∞(V ):=Ric(V )+(Ψ◦η)(0).

Forc0 andN∈ [n,∞], define RicN(cV ):=c²RicN(V ).

Ohta proved in[14]that, forK∈R, the boundRicN(V )KF²(V )is equivalent to Lott–Villani and Sturm’sweak curvature-dimension condition CD(K, N ).

2.5. Bochner–Weitzenböck formula

The following Bochner–Weitzenböck type formula, established by Ohta–Sturm in[16], plays an important role in this paper.

Theorem 2.1.(See[16, Theorem 3.6].) Givenu∈W_loc^2,2(M)∩C¹(M)with_mu∈W_loc^1,2(M), we have

−

M

Dη

∇^∇u

F²(x,∇u) 2

dm=

M

η

D(mu)(∇u)+Ric_∞(∇u)+∇²u²

H S(∇u)

dm

as well as

−

M

Dη

∇^∇u

F²(x,∇u) 2

dm

M

η

D(_mu)(∇u)+RicN(∇u)+(_mu)² N

dm

for anyN ∈ [n,∞]and all nonnegative functionsη∈W_c^1,2(M)∩L^∞(M). Here∇²u²_{H S(∇u)}denotes the Hilbert–

Schmidt norm with respect tog_∇u.

Based on Bochner–Weitzenböck formula, a similar argument as Bakry–Qian[3, Theorem 6], leads to a refined inequality, which was referred to as an extended curvature-dimension inequality there. Another direct proof was also given in[21, Lemma 2.3].

Theorem 2.2.Assume that RicNK for someN ∈ [n,∞] and someK∈R. Givenu∈W_loc^2,2(M)∩C¹(M)with _mu∈W_loc^1,2(M), we have

−

M

Dη

∇^∇u

F²(x,∇u) 2

dm

M

η

D(mu)(∇u)+KF (∇u)²+(_mu)² N + N

N−1 _mu

N −D(F²(x,∇u))(∇u) 2F²(x,∇u)

2

dm (11)

for anyN∈ [n,∞]and all nonnegative functionsη∈W_c^1,2(M)∩L^∞(M).

3. Proof of Theorem1.1

We first remark that a weak eigenfunctionu∈W^1,2(M)of Finsler-Laplacian defined in (1) has regularity that u∈C^1,α(M)∩W^2,2(M)∩C^∞(M_u)(see[6]).

Let us recall the 1-D modelsL_K,Ndescribed in[3]. LetK∈RandN∈(1,∞].

(i) ForK >0 and 1< N <∞,LK,N is defined on(−₂^√_K/(N^π ₋₁₎, ^π

2√

K/(N−1))by L_K,N(v)(t)=v−

K(N−1)tan K

N−1t

v;

(ii) ForK <0 and 1< N <∞,L_K,N is defined on(−∞,0)∪(0,∞)by L_K,N(v)(t)=v−

−K(N−1)coth

− K N−1t

v and on(−∞,∞)by

LK,N(v)(t)=v−

−K(N−1)tanh

− K N−1t

v;

(iii) ForK=0 and 1< N <∞,L_K,Nis defined on(−∞,0)∪(0,∞)by L_K,N(v)(t)=v+N−1

t v and on(−∞,∞)by

LK,N(v)(t)=v;

(iv) ForK=0 andN= ∞,L_K,N is defined on(−∞,∞)by L_K,N(v)(t)=v−Kt v;

(v) ForK=0 andN= ∞,L_K,N is defined on(−∞,∞)by L_K,N(v)(t)=v−cv

for any constantc.

For convenience, we writeLK,N(v)(t)=v−T (t )v. It is easy to check thatT=K+_N^T₋²₁. Denote byμK,N the invariant measure associated withLK,N, that is, a measure satisfying_b

aLK,N(v) dμK,N=0 forv(a)=v(b)=0.

For instance, in the case (i),dμ_K,N=cos^N−1( K

N−1t ) dt.

The following gradient comparison theorem plays the most crucial role in the proof of our main theorem.

Theorem 3.1.Let(M, F, m)andλ₁be as in Theorem1.1andube the eigenfunction. Letvbe a solution of the1-D model problem on some interval(a, b):

L_K,N(v)= −λ₁v, v(a)=v(b)=0, v>0. (12)

Assume that[minu,maxu] ⊂ [minv,maxv], then F

x,∇u(x) v

v⁻¹ u(x)

. (13)

Proof. First, since

Mu=0, minu <0 while maxu >0. We may assume that[minu,maxu] ⊂(minv,maxv)by multiplyingu by a constant 0< c <1. If we prove the result for this u, then letting c→1 implies the original statement.

Under the condition[minu,maxu] ⊂(minv,maxv),v⁻¹is smooth on a neighborhoodUof[minu,maxu].

ConsiderP (x)=ψ (u)(¹₂F²(x,∇u)−φ(u)), whereψ, φ∈C^∞(U )are two positive smooth functions to be deter- mined later. We first consider the case thatP attains its maximum atx₀∈M, then study the case thatx₀∈∂Mif∂M is not empty.

Case1.P attains its maximum atx₀∈M.

Due to the lack of regularity ofu, we shall compute in the distributional sense. Letηbe any nonnegative function inWc^1,2(M)∩L^∞(M). We first compute−

MDη(∇^∇uP ) dm.

−

M

Dη

∇^∇uP

dm= −

M

ψ

ψP−ψ φ

Dη(∇u)+ψ Dη

∇^∇u 1

2F²(x,∇u)

dm

=

M

−D ψ

ψP −ψ φ

η

(∇u)+ηD ψ

ψP −ψ φ

(∇u)

−D(ψ η)

∇^∇u 1

2F²(x,∇u)

+ηDψ

∇^∇u 1

2F²(x,∇u)

dm :=I+II+III+IV.

By usingDu(∇u)=F²(x,∇u)=2(^P_ψ +φ)and_mu= −λ₁uin weak sense, we compute I=

M

−λ₁u ψ

ψP −ψ φ

η dm,

II=

M

η ψ

ψ −ψ² ψ²

P −ψ φ−ψφ

Du(∇u)+ψ

ψDP (∇u)

dm

=

M

η

2 ψ

ψ −ψ² ψ²

P−ψ φ−ψφ P

ψ +φ

+ψ

ψDP (∇u)

dm,

IV=

M

ηψ 1

ψDu

∇^∇uP +

−ψ ψ²P +φ

Du(∇u)

dm

=

M

2ηψ

−ψ ψ²P+φ

P ψ +φ

+terms ofDP (∇u)

dm.

For the termIII, we apply the refined integral Bochner–Weitzenböck formula (11) to derive III

M

ψ η

D(_mu)(∇u)+KF²+(mu)²

N + N

N−1 mu

N −D(F²(x,∇u))(∇u) 2F²(x,∇u)

2 dm

=

M

ψ η

2(K−λ₁) P

ψ+φ

+λ²₁u²

N + N

N−1 −λ1u

N −

−ψ ψ²P+φ

− 1

ψ F²DP (∇u) 2

dm

=

M

ψ η

2(K−λ₁) P

ψ+φ

+ λ²₁u² N−1+ N

N−1

−ψ ψ²P +φ

2

+ 2 N−1λ₁u

−ψ ψ²P+φ

+terms ofDP (∇u)

dm.

Combining all we obtain

−

M

Dη

∇^∇uP

dm

M

η 1

ψ

2ψ ψ −

4− N

N−1 ψ²

ψ²

P²

+

2φ ψ

ψ −2ψ² ψ²

−N+1 N−1

ψ

ψλu− 2N N−1

ψ

ψφ+2(K−λ₁)−2φ

P +ψ

1

N−1λ²₁u²+N+1

N−1λ₁uφ+ N

N−1φ²+2(K−λ₁)φ−2φφ

+terms ofDP (∇u)

dm := −

M

a₁P²+a₂P +a₃+terms ofDP (∇u)

dm. (14)

Therefore,

^∇_m^uP+terms ofDP (∇u)=a₁P²+a₂P +a₃ (15)

holds in the distributional sense inM.

We claim that at the maximum pointx₀ofP,

a₁P²+a₂P +a₃0. (16)

In fact, if not, then in a neighborhoodUofx₀,a₁P²+a₂P+a₃>0. It follows from (15) that the functionP is a strict subsolution to an elliptic operator inU. By maximum principle,P (x₀) <max∂UP, which contradicts the maximality ofP (x₀).

It is interesting to see that the coefficientsa_i,i=1,2,3, coincide with that appeared in the Riemannian case (see e.g.[3, Lemma 1]). The next step is to choose suitable positive functionsψ andφsuch thata₁, a₂>0 everywhere anda3=0, which has already been done in[3]. For completeness, we sketch the main idea here.

Chooseφ(u)=¹₂v(v⁻¹(u))², wherevis a solution of 1-D problem (12). One can compute that φ(u)=v

v⁻¹(u)

, φ(u)=v v

v⁻¹(u) . Sett=v⁻¹(u)andu=v(t )then

a3(t)

ψ = 1

N−1λ²₁v²+N+1

N−1λ₁vv+ N

N−1v²+(K−λ₁)v²−vv

= −v

v−T v+λ₁v + 1

N−1

v−T v+λ₁v

N v+T v+λ₁v

=0.

Here we have used thatT satisfiesT=K+_N^T₋²₁. Fora₁, a₂, we introduce X(t )=λ₁v(t )

v(t), ψ (u)=exp h v(t )

, f (t )= −h v(t )

v(t).

With these notations, we have f= −hv²+f (T −X), v²

v⁻¹a₁ψ=2f (T −X)−N−2

N−1f²−2f:=2

Q₁(f )−f , a₂=f

3N−1 N−1 T −2X

−2T N

N−1T −X

−f²−f:=Q₂(f )−f.

We may now use Corollary 3 in [3], which says that there exists a bounded function f on [minu,maxu] ⊂ (minv,maxv)such thatf<min{Q₁(f ), Q₂(f )}.

In view of (16), we know that by our choice ofψandφ,P (x₀)0, and henceP (x)0 for everyx∈M, which leads to (13).

Case2.∂M= ∅andx₀∈∂M.

To handle this case, we need to define a new normal vector field on∂M, that is normal with respect to the Rieman- nian metricg_∇u. To be more general, for everyX∈T M, there is a unique normal vector fieldνXsuch that

g_X(ν_X, Y )=0 for anyY∈T (∂M), g_X(ν_X, ν_X)=1, g_ν(ν, ν_X) >0. (17) A simple calculation shows that

g_X(ν, ν_X) >0. (18)

Indeed, let νX=Z+aν for some a∈R andZ∈T (∂M). (17) tells that a >0. HencegX(ν, νX)=gX(¹_a(νX− Z), ν_X)=¹_a>0.

The following lemma follows directly from the definition ofνandν_X. Lemma 3.1.LetX, Y∈T M. Then

g_ν(ν, Y )=0 ⇔ Y ∈T (∂M) ⇔ g_X(ν_X, Y )=0. 2 Define four sets

T_±^νM:=

Y∈T M: g_ν(ν, Y ) >0(<0) and

T_±^νXM:=

Y∈T M: g_X(ν_X, Y ) >0(<0) .

We have the following simple but important observation, which may be familiar to expects.

Lemma 3.2.T₊^νM=T₊^νXM, T₋^νM=T₋^νXM.

Proof. We first claim that eitherT₊^νM⊂T₊^νXM orT₊^νM⊂T₋^νXM. Otherwise, there are two vector fields Y₁, Y₂∈ T₊^νM, such thatg_X(ν_X, Y1) >0 andg_X(ν_X, Y2) <0. Then by the continuity ofg_X(ν_X,·)inT₊^νM, there existsY ∈ T₊^νMwithgX(νX, Y )=0, which meansgν(ν, Y )=0 from Lemma3.1. A contradiction. Taking into consideration thatν∈T₊^νXM, we see thatT₊^νM⊂T₊^νXM. A similar argument implies thatT₊^νXM⊂T₊^νM. The second equivalence follows in a similar way. 2

Return to the case whenP attains its maximum atx₀∈∂M. If∇u(x₀)=0, nothing needs to be proved. Thus we assumex₀∈M_u. Recall thatP∈C^∞(M_u). Sinceν_∇upoints outward due to its definition, by taking normal derivative ofP with respect toν_∇u, we have

DP (ν_∇u)(x₀)0.

On one hand, the Neumann boundary condition∇u∈T (∂M)implies that g_∇u(ν_∇u,∇u)(x)=0,

or equivalently,

Du(ν_∇u)(x)=0 forx∈∂M.

Thus we have

DP (ν_∇u)(x₀)=1 2ψ (u)

D

F²(∇u) (ν_∇u)

(x₀). (19)

On the other hand, using (8) and the symmetry of∇²u, we have D

F²(∇u)

(ν_∇u)=D

g_∇u(∇u,∇u)

(ν_∇u)=2g_∇u

D_ν^∇u

∇u(∇u),∇u

=2g_∇u

D_∇^∇_u^u(∇u), ν_∇u

. (20)

By the convexity of∂M, for anyX∈T (∂M),g_ν(ν, D^X_XX)0. In particular, setX= ∇u, we know that gν

ν, D_∇^∇u_u(∇u)

0. (21)

It follows from Lemmas3.1and3.2that (21) is equivalent to g_∇u

ν_∇u, D_∇^∇_u^u(∇u)

0. (22)

Combining (19), (20) and (22), we conclude thatDP (ν_∇u)(x₀)≤0, and henceDP (ν_∇u)(x₀)=0. The tangent derivatives ofP obviously vanish due to its maximality. Hence we have also

∇P (x₀)=0.

Thus the proof for Case 1 works in this case. This finishes the proof of Theorem3.1. 2 Another ingredient is a comparison theorem for the maxima of the eigenfunctions.

Theorem 3.2.Let(M, F, m), λ₁be as in Theorem1.1and1< N <∞. Letv=v_K,Nbe a solution of the1-Dmodel problem on some interval(a, b) L_K,Nv= −λ₁v, with initial datav(a)= −1, v(a)=0, where

a=

−₂^√_K/(N^π ₋₁₎ forK >0,

0 forK0

and b=b(a) be the first number after a withv(b)=0. Denotem_K,N =v_K,N(b)=max(v). Assume that λ₁>

max{_N^KN₋₁,0}andmin(u)= −1. Thenmaxum_K,N.

Proof. We argue by contradiction. Suppose max(u) < mK,N. Then[minu,maxu] ⊂ [minv,maxv]. The condition λ₁>max{_N^KN₋₁,0}ensures that

b

_π

2√

K/(N−1) forK >0,

∞ forK0,

which in turn ensures thatv>0 in(a, b). Hence we could apply Theorem3.1foruandv.

The same argument as Theorem 12 in[3]implies that the ratio R(c)=

{uc}u dm

{vc}v dμK,N

is increasing on[min(u),0]and decreasing on[0,max(u)]. Therefore, forc−¹₂, we have that m

{uc}

2

{uc}

|u|dm2R(0)

{vc}

|v|dμ_K,N2R(0)μK,N

{vc}

. (23)

Letc= −1+εforε >0 small. A simple calculation gives thatv(a)=^λ_N¹. Hence fort close toa,v(t)has positive lower and upper bound. Together withv(a)=0, we see thatv(t )−v(a)C(t−a)². Thus ift∈ {v−1+ε}, then t∈(a, a+Cε¹²). It follows that

μ_K,N

{v−1+ε}

μ_K,N

a, a+Cε¹²

Cε^N/2. (24)

On the other hand, we shall prove that m

{u−1+ε}

m

B^±

x₀, Cε¹²

. (25)

Letx₀∈Mbe such thatu(x₀)= −1. For anyx∈B^±(x₀, δ)withδ small,u(x)is close to−1 ands:=v⁻¹(u(x))is close toa. Thus we see again from the upper bound ofvandv(a)=0 thatv(s)C(s−a). Therefore, we have from Theorem3.1thatF (x,∇u(x))v(v⁻¹(u(x)))C(s−a)andF (x,∇v⁻¹(u(x)))=(v⁻¹)(u(x))F (x,∇u(x))1.

In turn, we get s−a=v⁻¹

u(x)

−v⁻¹ u(x₀)

F

˜ x,∇v⁻¹

u(x)˜ δδ, and

u(x)u(x0)+Fx,˜˜ ∇u(x)˜˜

δ−1+C(s−a)δ−1+Cδ²,

for somex,˜ x˜˜∈B^±(x₀, δ). Letε=Cδ², we concludeB^±(x₀, δ)⊂ {u−1+ε}, which implies (25).

Combining (23), (24) and (25), we see that there exists some constantC >0 such that m

B^±(x0, r)

Cr^N. (26)

This will lead to a contradiction. In fact, since max(u) < mK,N andmK,N is continuous with respect to(K, N ), we also have that max(u) < mK,N for anyN> Nclose toN. Argued as before, we will obtain (26) withN instead ofN, i.e.

m

B^±(x₀, r)

Cr^N. (27)

However, the volume comparison theorem for Finsler manifolds under the assumption of lower bound forRicN (see [14, Theorem 7.3]), implies thatm(B^±(x₀, r))Cr^Nforr >0 small. A contradiction to (27). The previous argument also works in the casex₀∈∂M. The proof is completed. 2

Besides the comparison theorem on the gradient and maxima, in order to prove Theorem1.1, we also need some properties of the 1-D models, which has been extensively studied in[3]. We refer to[3]for the elementary properties, meanwhile we list two of them, one presents the full range of the maximum functionm_K,N, the other reveals that the central interval has the lowest first Neumann eigenvalue.