Rough isometry and Dirichlet finite harmonic functions on Riemannian manifolds
Let \(M\) be a Riemannian manifold. We say that \(M\) satisfies a \(D\)-Liouville property if every harmonic function with finite Dirichlet integral on \(M\) is constant. In the present paper, the author considers a generalized \(D\)-Liouville property. To be precise, the author proves that the dimension of harmonic functions with finite Dirichlet integral is invariant under rough isometries between Riemannian manifolds satisfying some local conditions. One should remark that the finiteness of the Dirichlet integral is necessary in proving the bi-Lipschitz invariance of the Liouville property. For example, see \textit{A. Grigor'yan} [Math. USSR, Sb. 60, No. 2, 485-504 (1988); translation from Mat. Sb., Nov. Ser. 132(174), No. 4, 496-516 (1987; Zbl 0646.31009)] and \textit{T. Lyons} [J. Differ. Geom. 26, 33-66 (1987; Zbl 0599.60011)]. The main result of the present paper directly generalizes those of \textit{M. Kanai} [J. Math. Soc. Japan 38, 227-238 (1986; Zbl 0577.53031)], \textit{A. Grigor'yan} (loc. cit.) and \textit{I. Holopainen} [Rev. Mat. Iberoam. 10, No. 1, 143-176 (1994; Zbl 0797.31008)]. The author also proves that the dimension of harmonic functions with finite Dirichlet integral is preserved under the rough isometries between a Riemannian manifold satisfying the same local conditions and a graph of bounded degree; and between graphs of bounded degree. These results generalize those of \textit{I. Holopainen} and \textit{P. M. Soardi} [Manuscr. Math. 94, No. 1, 95-110 (1997; Zbl 0898.31007)], and of \textit{P. M. Soardi} [Proc. Am. Math. Soc. 119, 1239-1248 (1993; Zbl 0801.31002)], respectively.
scientific article; zbMATH DE number 1319557