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Harnack inequalities for positive solutions of the heat equation on closed Finsler manifolds

The main goal of this paper is to generalize some Li-Yau type gradient estimates to Finsler geometry in order to derive Harnack type inequalities. Moreover, we obtain, under some curvature assumption, a general gradient estimate for positive solutions of the heat equation when the manifold evolving along the Finsler Ricci flow.


Introduction
A very active research topic in geometric analysis is the study of the heat equation on manifolds because of its several applications in physics and natural sciences. In their famous paper [11], P. Li and S-T. Yau have proved the following Then, for any α > 1, it holds |∇f | 2 − α∂ t f nα t + nα 2 k 2(α − 1) , (1.2) where f = log u. Particularly, when k = 0, letting α → 1, one obtains It is worth to mention that the equation (1.2) is not sharp unless k = 0. An important application of this estimate is that it provides Harnack type inequalities. There is a rich litterature about improvement and generalization of the Li-Yau gradient estimate (1.2), see for example [5,15,14,2] and the references therein. These results do not only concern the heat equation on Riemannian manifolds but also general linear and semi-linear parabolic equations on Kaeler manifolds or Alexandrov spaces. Despite these important efforts done, the question of sharpness of the Li-Yau gradient estimate remain open.
Recently, a general gradient estimate for positive solutions of the heat equation on closed Riemannian manifold which extends many existing estimates was proved [6]. The first purpose of this paper is to generalize this result to Finsler geometry in order to derive Harnack type inequalities. Finsler manifolds are natural generalization of Riemannian manifolds in the sense that each tangent space is endowed with a Minkowskian norm instead of the Euclidean one. Observe that, in this paper, we use the nonlinear Shen's Finsler Laplacian which is an extension among others of the Laplace-Beltrami operator to Finsler geometry. Because of the lack of linearity, method used in Riemannian setting does not work. To overcome this difficulty, we use the weighted gradient and Laplacian which are linear operators (see Section 2 for the definitions). Our first result is stated as follows. Theorem 1.1. Let (M, F ) be a closed Finsler manifold of dimension n, equiped with a smooth measure dµ, such that the weighted Ricci curvature satisfies Ric N K for some N ∈ [n, ∞) and K ∈ R and take a positive global solution u : [0, T ] × M → R to the heat equation. Let us consider the functions λ, β, Ψ ∈ C 1 ((0, T ]) satisfying where K − := min{K, 0}. Then, for any x, y ∈ M and 0 < s < t T , we have where d F is the distance function induced by the Finsler metric F .
The related definitions such as weighted Ricci curvature and the Finsler distance function are given in Section 2 below. Recall that Ohta and Sturm [13] have already proved some Harnack type inequalities in Finsler setting. Theorem 1.1 can be view as a generalisation of their results.
The Li-Yau type gradient estimates are also investigated on manifolds evolving along geometric flows. The case of the Ricci flow is well known and was iniated by Hamilton in [7] on Riemannian manifolds ; see [10,16] and references therein. In [3], Bao has introduced the notion of Finsler Ricci flow and recently, Lakzian [8] has derived Harnack estimates for positive solutions to the heat equation under this flow. This paper gives also a generalization of Lakzian result. Theorem 1.2. Let (M, F ) be a closed Finsler manifold of dimension n endowed with a smooth measure dµ. Let (F t ) t∈[0,T ] be a solution of the Ricci flow on M with F 0 = F . Assume that there exists some positive constants K i , i = 1, 2, 3, 4, such that for all t ∈ [0, T ], F t has isotropic S-curvature, S = σF t + dϕ for some time dependent functions σ and ϕ with −K 3 F t ≤ dσ and −K 4 F 2 t ≤ Hess ϕ, and its Ricci curvature satisfies −K 1 ≤ Ric ij ≤ K 2 . Let u : [0, T ] × M → R be a positive solution of the heat equation under the Finsler Ricci flow (F t ). Let β, λ ∈ C 1 ((0, T ]) satisfying Then we have, for any t ∈ (0, T ],
We obtain an improvement of [8, Theorem 1.1] with a suitable choice of functions β and λ. The paper is organised as follows. In section 2, we review some facts about Finsler geometry which we need for the sequel. Thereafter, we prove Li-Yau gradient estimate in section 3. That allows us to show the Theorem 1.1. In the last section, we deal with gradient estimate in time dependent Finsler manifolds where we prove Theorem 1.2.

Preliminaries
In this section, we briefly recall some basic concepts of Finsler geometry necessary for further discussions. We refer to [12,3] and references therein for more details.
Let M be an n-dimensional smooth manifold and π : T M → M be the natural projection from the tangent bundle T M . A point (x, y) ∈ T M is such that y = y i ∂ ∂x i in the local coordinates (x i , y i ) on T M . Let F : T M → [0, ∞) be a Finsler metric on M , that is F satisfies (i) Regularity: F is smooth on T M {0}; (ii) Positive homogeneity: F (x, λy) = λF (x, y), ∀ λ > 0; (iii) Strong convexity: The fundamental quadratic form is positive definite for any (x, y) ∈ T M with y = 0.
For x, y ∈ M , we define the distance from x to y by where the infimum is taken over all C 1 -curves γ : [0, 1] → M such that γ(0) = x and γ(1) = y.
Since the Finsler metric is only positively homogeneous, the distance function may be asymmetric. We call geodesic any C ∞ -curve γ on M which is locally minimising and has constant speed. We also define the exponential map by and a covariant derivative by are coefficients of the Chern connection. The flag curvature of the plane spanned by two linearly independent vectors V, W ∈ T x M {0} is defined by where R V is the Chern curvature as follows Then, the Ricci curvature of (M, F ) is given by where {e 1 , . . . , e n−1 , e n := V F (V ) } is an orthonormal basis of T x M with respect to g V and the Ricci tensor is defined as follows : Now, let us consider an arbitrary volume form dµ = σ(x)dx on M . The distortion of (M, F, dµ) is defined, for any y ∈ T x M {0}, by measures the rate of changes of the distortion along geodesics. One also defineṡ The weighted Ricci curvature of the Finsler manifold (M, F, dµ) is defined as follows In analogy with the Ricci flow in Riemannian setting, Bao [3] proposed the following definition of the Finsler Ricci flow: which contrating to y i y j , via Euler theorem, provides where F 0 is the initial Finsler structure. Short time existence and uniqueness of the Finsler Ricci flow has been studied in some particular cases, see e.g [1,4].
For their convenient use, let us recall the definitions of the gradient, the Hessian and the Laplacian operators on (M, F, dµ). Let u be a differentiable function and V a smooth vector field on M . We set where F * stands for the dual norm of F . Then the gradient vector and the Laplacian of u are given by ∆u(x) := div(∇u).
If V does not vanish on M u , one can also define the weighted gradient vector and the weighted Laplacian of u on the Riemannian manifolds (M, g V ) as In particular, on M u , we have In other words, the Laplacian of u can be expressed using the S-curvature as where ∇ 2 u := D ∇u . ∇u is the Hessian of u. Here the trace is taken with respect to an g ∇uorthonormal basis. The point-wise Finslerian version of the Bochner-Weitzenbock formulas are given by We conclude this section by recalling some notions about the heat equation ∂ t u = ∆u associated with the Finsler Laplacian. We consider a Finsler space (M, F, dµ) where (M, F ) is a Finsler space and dµ a smooth measure on M .
A function u : [0, T ] × M → R is said to be a global solution of the heat equation ∂ t u = ∆u if it satifies the following: where u t := u(t, .).

Now, let us define the function H(t, x) = λ(t)G(t, x). From (3.3), we can easily deduce that the following equation
on some neighborhood of (t 0 , x 0 ). Therefore, H is a strict subsolution of the linear parabolic operator div µ (∇ V H) + 2dH(∇f ) − ∂ t H, (3.6) on such a neighborhood. This implies that H(t 0 , x 0 ) is trictly less than the supremum of H on the boundary of any small parabolic cylinder Hence, we have proved the following gradient estimate: where f := ln u.
Following [6], we obtain this immediate consequence of Theorem 3.3: Then, We can now derive our Harnack type inequality.

Gradient estimates with time dependant Finsler metrics
In this section, we consider a smooth manifold evolving along the Finsler Ricci flow. Our aim is to prove a general gradient estimate for positive solutions of the heat equation under the Ricci flow, that is functions where the gradient is with respect to the Finsler metric F t . Let us observe that, here we didn't investigated about existence and regularity of such solutions. Let (M, F ) be a closed n-dimensional Finsler manifold endowed with a smooth mesure dµ. Let where the gradient is taken with respect to the Finsler metric F t and let f = ln u.
We have the following evolution equation Let β, λ ∈ C 1 ((0, T ]) be three functions satisfying the following assumptions: On in the sense of distribution on (0, T ) × M , where Proof. We have Using this relation, we compute and From the Bochner-Weitzenbock formula, we have Therefore, Now, we assume that there exists some positive constant K 1 , K 2 , K 2 , K 4 such that for all t ∈ [0, T ], • the Ricci curvature of (M, F t ) satifies in any g v -orthonormal basis, as a quadratic form on T x M , for all v ∈ T x M \{0}, • (M, F t , dµ) has isotropic S-curvature, S t := σF t + dϕ, where σ and ϕ are time dependent functions which verify with C 1 = K 1 , C 2 = max{K 2 1 , K 2 2 } and C 3 = K 3 + K 4 . Proof. By Young's inequality, we have Choose a nornal coordinate system {x i } with respect to g ∇f such that ∇f (x) = ∂ ∂x 1 and for all i, j, Γ 1 ij (∇f (x)) = 0 . Therefore, one can easily check the following  We have also n i,j=1 Using (4.12)-(4.14) with estimates on hypothesis of the Theorem, we get where we set C 1 = K 1 , C 2 = max{K 2 1 , K 2 2 } and C 3 = K 3 + K 4 .
Corollary 4.6. Let (F t ) [0,t) be a Finsler Ricci flow on a closed n-dimensional manifold M such that each Finsler metric F t has Ricci curvature satisfying 0 ≤ Ric ij ≤ C for some positive constnate C and isotropic S-curvature S t = σF t + dϕ where σ and ϕ satisfy assumption (4.10).
Let b ∈ C 1 ((0, T ]) be a positive increasing function such that lim t→0 + b(t) = 0. Then a positive solution u of the heat equation under the Ricci flow satisfies , (4.21) where f = ln u. Particulary, if all Finsler metrics F t have vanishing S-curvature, then we have Proof. The result follows by choosing β = 1 1+b and λ = b in Theorem 4.4. Here C 1 = 0 and C 2 = C 2 .