Some Liouville Theorems on Finsler Manifolds

: We give some Liouville type theorems of L p harmonic (resp. subharmonic, superharmonic) functions on a complete noncompact Finsler manifold. Using the geometric relationship between a Finsler metric and its reverse metric, we remove some restrictions on the reversibility. These improve the recent literature (Zhang and Xia, 2014).


Introduction
As is well known, Liouville theorems play an important role in analyzing the underlying manifolds. The classical Liouville theorem states that any nonnegative (or bounded) harmonic function on R n must be constant. Up to now, there are many generalizations studied on complete Riemannian manifolds. Yau [1,2] proved that any positive (or bounded) harmonic function on a complete Riemannian manifold with nonnegative Ricci curvature must be constant and there are no nonnegative L p subharmonic functions on such a Riemannian manifold for p ∈ (1, ∞). Yau's results were then generalized by Sturm and Schoen, etc. See [3][4][5] and references therein. For general symmetric diffusion operators, Li [6] extended various Liouville theorems as above.
Recently, Zhang-Xia [7], Yin-He [8] and Yin-Zhang [9] extended the above Liouville theorems in the Finsler setting. Notice that, in [7,8], the Finsler manifolds discussed must have finite reversibility. In this paper, we show that this restriction can be removed. Specifically, we obtain the following results.
where M u = {x ∈ M|du(x) = 0}, then u is a constant. In particular, if u ∈ L 1 (M) and ∆ log u ≥ 0 on M u , then u is a constant.
Theorem 2. Let (M, F, dµ) be an n-dimensional complete noncompact Finsler manifold. Assume that is a nonnegative superharmonic function on M, then u is a constant.
Here, V p (r), p ∈ R is defined in (2) below, and some important concepts such as Finsler metric, Finsler Laplacian and harmonic (resp. subharmonic, superharmonic) functions will be given in Section 2, respectively.

Remark 1.
If the Finsler manifold is compact, then, by the divergence theorem, we can prove all harmonic (resp. subharmonic, superharmonic) functions are constant. Theorem 1 can be regarded as a generalization of Theorem 1 in [2] when p = 1. If (M, F, dµ) is a Riemannian metric measure space, then Theorem 2 is exactly Theorem 1 in [5] or Theorem 13.1 in [10].

Remark 2.
In comparison with [7], the condition on the reversibility is deleted in theorems above. There are many Finsler manifolds with infinity reversibility. Consider the Randers metric in B 3 (1) where | · | denotes the standard Euclid norm. Then the reversibility The geometric quantities between F and its reverse metric ← − F have many important relationships. For example, A forward (backward) distance function w.r.t. F is a backward (forward) distance function w.r.t. ← − F , and vice versa.
To give a more elaborate improvement, we use these relations and thus avoid employing the reversibility. The remainder of the approaches adopted are similar to Zhang-Xia's paper [7]. See also in [2,5,10] for the Riemannian case.
The contents of the paper are arranged as follows. In Section 2, some fundamental concepts which are necessary for the present paper are given, and some lemmas are contained. In Section 3, we prove the main theorems and give some corollaries.

Preliminaries
Let M be an n−dimensional smooth manifold and π : TM → M be the natural projection from the tangent bundle TM. Let (x, y) be a point of TM with x ∈ M, y ∈ T x M, and let (x i , y i ) be the local coordinates on TM with y = y i ∂/∂x i . A Finsler metric on M is a function F : TM → [0, +∞) satisfying the following properties: (i) Regularity: F(x, y) is smooth in TM \ 0; (ii) Positive homogeneity: F(x, λy) = λF(x, y) for λ > 0; (iii) Strong convexity: The fundamental quadratic form where Γ i jk denote the coefficients of the Chern connection. For a smooth function u, the gradient vector of u is The divergence of V with respect to an arbitrary smooth volume form dµ is defined by Then, the Finsler Laplacian of u can be defined by ∆u := div(∇u).
Since ∆u is undefined at x where du(x) = 0, the definition can be viewed in distributional sense. That is, for u ∈ W 1,2 (M), We note here that since the gradient operator ∇ is not linear operator in general, the Finsler Laplacian is quite a bit different from the Riemannian Laplacian. Given a vector field V such that It follows that ∇ ∇u u = ∇u, ∆ ∇u u = ∆u. Let u be a positive harmonic function on M, ∆u = 0. It was proved that u ∈ W 2,2 [11]). We say that u ∈ W 2,2 is a subharmonic (resp. superharmonic) function on M if ∆u ≥ 0 (resp. ∆u ≤ 0). In a weak sense, u is a subharmonic (resp. superharmonic) function in M if, for any positive function ϕ ∈ C ∞ 0 (M), it holds M ϕ∆udµ ≥ (resp. ≤)0.
Let (M, F) be a Finsler n-manifold. Fix a point x 0 ∈ M. We denote a forward (resp. backward) geodesic ball of radius r with center at x 0 by B + x 0 (r)(resp.B − x 0 (r)). Lemma 1. Let (M, F) be a Finsler n-manifold and x 0 ∈ M. Then, there exists a function defined by where C is a positive constant.

Proof.
Let ω(t) be a smooth function on the real line with 0 ≤ ω(t) ≤ 1 and ω (t) ≤ 0 such that Clearly, |ω (t)| ≤ C, where C is some positive constant. Define a.e. on M.

Proof of the Main Theorems
For any nonnegative function u, set which gives v∆v ≥ F(∇v) 2 ≥ 0.
Let ϕ be the function defined in Lemma 1. Then, it is differentiable almost everywhere on (B + x 0 (2R)) with a bounded differential. Note that ∆v = 0 a.e. on M\M v from Lemma 3.5 in [12]. Thus, by the divergence theorem, we have Therefore, By using Lemma 1 and the definition of ϕ, we deduce Letting R → ∞, it follows from lim sup r→∞ r 2 V 1 (r) = ∞ that F(∇v) = 0 everywhere. Since M is connected, v is a constant on M and so is u.
Proof of Theorem 2. Without loss of generality, we might as well assume u > 0. Otherwise, we can replace it byũ = u + ε > 0 for some positive number ε. We first prove (2) in Theorem 2. Let x 0 be a fixed point in M and r 0 be a number with 0 < r 0 < R. Define Using the divergence theorem and similar arguments above, we have Set v = u p 2 . Then, (4) becomes From the conditions in Theorem 2 and (3), it follows that Then, by similar arguments as in [7], we can also reach For fixed r 0 , taking R k = 2 k r 0 , k ∈ N + , we have Letting n → ∞, we have 1 which means that Therefore, by arbitrariness of r 0 , we conclude that v must be constant on M and so is u. Now, we are to prove (1) according to the cases 0 < p < 1, p < 0 and p = 0, respectively.
Case I: 0 < p < 1. Let x 0 and r 0 be as above. Define By similar arguments, we also obtain (5) for the backward geodesic ball. The remainder of the proof is the same as above.
Let ψ be a function as in Case I. We can also obtain 0 ≤(p − 1) Therefore, we have Then, by the same argument as above, one obtains that u is constant.
Case III: p = 0. For every k ∈ R + , set Then, u k is a nonnegative superharmonic function in a weak sense. We will prove this by following the arguments in ( [10], p. 178). Let β be a symmetric, concave and bounded smooth function with |β | ≤ 1 and β ≥ 0 (think of a smooth approximation of x −→ |x|). Definẽ (think of a smooth approximation of u k ). Then, dũ k = 1 2 (1 − β )du. Notice that 1 − β ≥ 0. By Legendre transformation, we have ∇ũ k = 1 2 (1 − β )∇u. Since ∆ũ k = 0 a.e. on M\Mũ k , for ψ defined in Case I, we have The last step holds because (1 − β )ψ is differentiable almost everywhere on B − x 0 (R) with bounded differential and u is superharmonic. Thus, the claim follows by approximation.
It is shown that ∇u k = 0 a.e. on {u ≥ k}, ∇u k = ∇u on {u < k} and µ({x ∈ M|u(x) = k, du(x) = 0}) = 0 w.r.t. to the measure dµ (see [7]). Thus, ∆u k = 0 a.e. on {u ≥ k} and ∆u k = ∆u on {u < k}. Notice that ψ is differentiable almost everywhere on B − x 0 (R) with a bounded differential. Therefore, by similar arguments, we can also obtain (4) for u k on B − x 0 (R). Set v k = u q 2 k for any q ∈ (0, 1). Then, we have (5) as follows: On the other hand, note that which implies that ∞ 1 r V q (r) dr = ∞.
Then, by the same discussion in the proof of (2) and Case I of (1), we show that this u k is constant. Since k is arbitrary, the function u is also constant.
Using Theorem 2, we can reach the following corollaries which extend Theorem 3 in [2] and Corollary 1 in [13], respectively. Corollary 1. Let (M, F, dµ) and u be as in Theorem 2.