Remarks on Liouville-type theorems on complete noncompact Finsler manifolds

Songting Yin,Pan Zhang

doi:10.33044/revuma.v59n2a03

Abstract

We give a gradient estimate of the positive solution to the equation ∆u = −λ 2u, λ ≥ 0 on a complete noncompact Finsler manifold. Then we obtain the corresponding Liouville-type theorem and Harnack inequality for the solution. Moreover, on a complete noncompact Finsler manifold we also prove a Liouville-type theorem for a C2 -nonnegative function f satisfying ∆f ≥ cfd, c > 0, d > 1, which improves a result obtained by Yin and He.

Highlights

A Finsler space (M, F, dμ) is a differential manifold equipped with a Finsler metric F and a volume form dμ
In [7], [10], [11], [14], [16] and [18], 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

Summary

Introduction

A Finsler space (M, F, dμ) is a differential manifold equipped with a Finsler metric F and a volume form dμ. Liouville-type theorems; Finsler manifold; weighted Ricci curvature. Zhang ([15]) generalized it to Finsler manifolds if the weighted Ricci curvature RicN ≥ −c (c > 0). Assume that the weighted Ricci curvature satisfies RicN (x, y) ≥ −G2(r(x)), ∀y ∈ TxM, N ∈ [n, ∞), where G is a smooth function satisfying. Some definitions —such as those of Finsler manifold, the weighted Ricci curvature, gradient and Finsler Laplacian— will be given in Section 2 below. Given a Finsler manifold (M, F ), the dual Finsler metric F ∗ on M is defined by It is well-known that for any x ∈ M , the Legendre transformation is a smooth diffeomorphism from TxM \ 0 onto Tx∗M \ 0, and it is norm-preserving, namely, F (Y ) = F ∗(L(Y )), ∀Y ∈ T M.

Let V

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