Some Liouville-type theorems for harmonic functions on Finsler manifolds
type: Item
instance of: scholarly article
title: Some Liouville-type theorems for harmonic functions on Finsler manifolds
zbMATH Open document ID: 1310.53065
author: Fu-e Zhang
author: Qiaoling Xia
published in: Journal of Mathematical Analysis and Applications
publication date: 2014-12-15T00:00:00Z
review text: The authors give some generalizations of Liouville theorems on Riemannian manifolds to Finsler manifolds. Using the Laplacian introduced by \textit{Z. Shen} [in: The theory of Finslerian Laplacians and applications. Dordrecht: Kluwer Academic Publishers. 187--198 (1998; Zbl 0930.53047)], they study some Liouville properties of \(L^p\) harmonic (respectively, subharmonic and superharmonic) functions, \((p\in\mathbb{R})\), on a Finsler manifold. Also, considering the weighted Ricci curvature \(\mathrm{Ric}_N(x,y)\), for \(N\in(n,\infty)\), defined by \textit{S.-I. Ohta} on Finsler manifolds in [Calc. Var. Partial Differ. Equ. 36, No. 2, 211--249 (2009; Zbl 1175.49044)], the authors derive a gradient estimate for harmonic functions on a closed Finsler manifold and show that any harmonic function on a closed Finsler manifold with non-negative weighted Ricci curvature must be constant.
reviewed by: Nicoleta Aldea
Mathematics Subject Classification ID: 53C60
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 31C05
zbMATH DE Number: 6380667
zbMATH Keywords: Finsler Laplacian
zbMATH Keywords: Liouville-type theorem
zbMATH Keywords: harmonic function
zbMATH Keywords: weighted Ricci curvature
MaRDI profile type: Publication
full work available at URL: j.jmaa.2014.03.078
OpenAlex ID: W2017757294
cites work: Q4891304
cites work: Finsler Laplacians and minimal-energy maps.
cites work: Differential equations on riemannian manifolds and their geometric applications
cites work: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds
cites work: Uniqueness of \(L^ 1\) solutions for the Laplace equation and the heat equation on Riemannian manifolds
cites work: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds
cites work: \(L^ p\) and mean value properties of subharmonic functions on Riemannian manifolds
cites work: Ricci curvature for metric-measure spaces via optimal transport
cites work: The existence of harmonic maps from Finsler manifolds to Riemannian manifolds
cites work: Finsler interpolation inequalities
cites work: Heat flow on Finsler manifolds
cites work: Bochner-Weitzenböck formula and Li-Yau estimates on Finsler manifolds
cites work: Q4257729
cites work: Lectures on Finsler geometry
cites work: Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and Lp-Liouville properties.
cites work: On the geometry of metric measure spaces. I
cites work: On the geometry of metric measure spaces. II
cites work: A natural Finsler-Laplace operator
cites work: Comparison theorems in Finsler geometry and their applications
cites work: Harmonic functions on complete riemannian manifolds
cites work: Q4102442
Wikidata QID: Q125553813
DOI: 10.1016/J.JMAA.2014.03.078
description: scientific article; zbMATH DE number 6380667