Some Liouville-type theorems for harmonic functions on Finsler manifolds
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.
scientific article; zbMATH DE number 6380667
scientific article