REMARKS ON LIOUVILLE-TYPE THEOREMS ON COMPLETE NONCOMPACT FINSLER MANIFOLDS

In this paper, we give a gradient estimate of positive solution to the equation ∆u = −λu, λ ≥ 0 on a complete non-compact Finsler manifold. Then we obtain the corresponding Liouville-type theorem and Harnack inequality for the solution. Moreover, on a complete non-compact Finsler manifold we also prove a Liouville-type theorem for a C2-nonegative function f satisfying ∆f ≥ cf, c > 0, d > 1, which improves a result obtained by Yin and He.


Introduction
A Finsler space (M, F, dµ) is a differential manifold equipped with a Finsler metric F and a volume form dµ. The class of Finsler spaces is one of the most important metric measure spaces.Up to now, Finsler geometry has developed rapidly in its global and analytic aspects.In [7][9] [11][14] [16] [17], the study was well implemented on Laplacian comparison theorem, Bishop-Gromov volume comparison theorem and Liouville-type theorem, and so on.
In [13], Yau derived a gradient estimate for harmonic functions on complete, noncompact Riemannian manifolds with the Ricci curvature bounded below by negative constant and proved that complete Riemannian manifolds with nonnegative Ricci curvature must have Liouville property.Recently, the result was extended by Xia ([12]) to the Finsler manifolds under the condition that the weighted Ricci curvature has a lower bound, and by ) to the Alexandrov spaces.Let ∆u = −λ 2 u, λ ≥ 0 (1.1) be an equation on the Finsler manifold (M, F, dµ).Using the gradient estimate obtained in [12], we can give a gradient estimate of the positive solution to (1.1).This is inspired by the work by Ma ([3]) on similar result in Riemannian geometry.Specifically, we prove Theorem 1.1.
Let (M, F, dµ) be a complete noncompact Finsler n−manifold, equipped with a uniformly smooth and uniformly convex Finsler structure F. Assume that Ric N ≥ −K for some real numbers N ∈ [n, +∞) and K ≥ 0. Let u be a positive solution to (1.1) in a forward geodesic ball B + 2R (p) ⊂ M .Then there exists some constant C = C(N, Λ 1 , Λ 2 ), depending on N, the uniform constants Λ 1 and Λ 2 , such that, in Using (1.2) we obtain the corresponding Liouville-type theorem and a Harnack inequality for the solution in Section 3 below.We remark that if λ = 0 in (1.1), Theorem 1.1 becomes the main result in [12].
On the other hand, Nishikawa ([4]) proved that if a C 2 -nonnegative function f satisfies ∆f ≥ 2f 2 on a complete Riemannian manifold with Ricci curvature bounded from below, then f vanishes identically.The result was extended by Choi, Kwon and Suh ( [2]) to the general case ∆f ≥ cf d for c > 0, d > 1.Recently, Zhang ([15]) generalized it to Finsler manifolds if the weighted Ricci curvature Ric N ≥ − c( c > 0).Jointed with He, the first author further generalized it under the condition that the weighted Ricci curvature is bounded from below by some function, and F is reversible (see Corollary 4.7 in [14]).Now we show that the last condition is redundant.
Theorem 1.2.Let (M, F, dµ) be a complete noncompact Finsler n-manifold, and r(x) = d F (p, x) be the distance function from a fixed point p ∈ M .Assume that the weighted Ricci curvature satisfies where G is a smooth function satisfying then f vanishes identically.
Some definitions such as Finsler manifold, the weighted Ricci curvature, gradient and Finsler Laplacian will be given in Section 2 below.We remark that the Finsler gradient and Laplacian are nonlinear operators, which are much different from those on Riemannian manifolds.Besides, the results obtained do not coincide with those on weighted Riemannian manifolds, since the two kinds of weighted Ricci curvature Ric N (x, y) and Ric ∇u N are not the same.To prove the theorems, we borrow some methods from the related literatures (see [2][3] [10]) and give them some adjustments.Precisely, let M = M × R be a Finsler (n + 1)-manifold equipped with the product metric F = √ F 2 + t 2 , Then its weighted Ricci curvature are also not less than −K.By setting f (x, t) = e λt u(x) we find a harmonic function f (x, t) on M , and the gradient estimate is obtained from [12].Then (1.2) follows, as required.As to the proof of Theorem 1.2, we make full use of the relationship between the gradient and the reverse gradient, as well as the Finsler-Laplacian and the reverse Finsler-Laplacian of a function.Then the arguments can be followed step by step as in [2] (see also in [14]).

Preliminaries
To meet the requirements in the next section, here, some fundamentals of Finsler geometry are briefly presented.
Let M be an n-dimensional smooth manifold and π : T M → M be the natural projection from the tangent bundle T M .A Finsler metric on M is a function F : T M → [0, +∞) satisfying the following properties: (i) Regularity: F is smooth in T M \ 0; (ii) Positive homogeneity: F (x, λy) = λF (x, y) for all (x, y) ∈ T M and all λ > 0; (iii) Strong convexity: for every (x, y) ∈ T M \ 0, the matrix Such a pair (M, F ) is called a Finsler manifold.We say that F is uniformly smooth and uniformly convex if there exist two uniform constants 0 It is proved that a large class of Finsler manifolds satisfies the above property (see [6]).
The reversibility η of (M, F ) is defined by [8] For every non-vanishing vector There exists a unique linear connection, which is called the Chern connection, on Finsler manifolds.The Chern connection is determined by the following structure equations, which characterize torsion freeness: Given two linearly independent vectors V, W ∈ T x M \0, the flag curvature is defined by where R V is the Chern curvature: where e 1 , • • • , e n−1 , V F (V ) form an orthonormal basis of T x M with respect to g V .Given a Finsler manifold (M, F ), the dual Finsler metric F * on M is defined by , ∀ξ ∈ T * M, and the corresponding fundamental tensor is defined by The Legendre transformation L : T M → T * M is defined by It is well-known that for any x ∈ M , the Legendre transformation is a smooth diffeomorphism from T x M \ 0 onto T * x M \ 0, and it is norm-preserving, namely, For a smooth function u on M , the gradient vector of u at x is defined by ∇u(x) := L −1 (du).Locally we can write in coordinates where A volume form dµ on (M, F ) is noting but a global nondegenerate n-form on M .In local coordinates we can express dµ as dµ = σ(x)dx 1 be a smooth vector field on M .Then the divergence of V with respect to dµ and the Finsler-Laplacian of u are defined by The Finsler-Laplacian is better to be viewed in a weak sense due to the lack of regularity, that is, for u ∈ W 1,2 (M ), Let (M, F, dµ) be a Finsler n-manifold.
τ is called the distortion of (M, F, dµ).To measure the rate of distortion along geodesics, we define where γ : (−ε, ε) → M is a geodesic with γ(0) = x, γ(0) = V .S is called the S-curvature (see [9]).Define Then the weighted Ricci curvature of (M, F, dµ) is defined by (see [5]) Then M = M × R have the product metric F = √ F 2 + t 2 and the volume form dμ = dµdt = σ(x)dxdt.It is easy to check that F is a Finsler metric on M .Moreover, we have Denote by ∇, ∆ the gradient and the Laplacian on M .Let f (x, t) be a smooth function defined on M .Then Recall that the Christoffel symbol with respect to the Chern connection on ( M , F ) (see [1]) Therefore, one obtains By a direct computation, we further have and ˙ S = Ṡ, where G α = 1 2 Γ α βγ y β y γ .Thus we still have the lower bound for the weighted Ricci curvature of M .That is Ric Then, from (3.2), f (x, t) is a positive harmonic function on M .Namely, ∆f = 0.By using the gradient estimate in [12], we have max{F (x, ∇ log u(x)), F (x, ∇(− log u(x)))} As applications of Theorem 1.1, we give a Liouville property and a Harnack inequality in the following.
Proof.Assume that u > 0. Letting K = 0 and R → +∞ in (1.2), we have F (x, ∇ log u(x)) = F (x, ∇(− log u(x))) = 0, which implies that u is constant.Then from Equation (1.1) we get u ≡ 0 on M .This contradicts the assumption.Corollary 3.2.Let (M, F, dµ) be as in Theorem 1.1 and u be a positive solution of (1.1) in forward geodesic ball B + 2R (p) ⊂ M .Then there exists some constant C = C(N, Λ 1 , Λ 2 ), depending on N, the uniform constants Λ 1 and Λ 2 , such that sup , where η is the reversibility of F .Since F is uniformly smooth and uniformly convex, η < +∞ and depends on Λ 1 and Λ 2 .Therefore, Taking the differential in both sides, we have For a positive number λ and any smooth function u, we have Thus, by a straight calculation we obtain which can be rewritten as ∆f (f + a) q = − Observe that −G is bounded from above, we can apply the Omori-Yau maximum principle (Theorem 0. From the definition of G, we have f (p k ) → sup M f when −G(p k ) → sup M (−G).Since ∆f ≥ cf d , we obtain c(sup M f ) d (sup M f + a) q ≤ 0 for d > 1 and any q > 1.We claim that sup M f < +∞.If not, then we choose q < d, the left side of the above inequality is +∞, which is a contradiction.Thus sup M f < +∞.Using the inequality above again, we find sup M f = 0.This means f ≡ 0.
If f has an upper bound, the restriction on d > 1 in (1.3) can be improved to d > 0.