The relationship between some Liouville theorems on Riemannian manifolds of a special type
type: Item
instance of: scholarly article
title: The relationship between some Liouville theorems on Riemannian manifolds of a special type
zbMATH Open document ID: 0927.58021
author: K. Appert
published in: Russian Mathematics
publication date: 1999-04-27T00:00:00Z
review text: Let \(M\) be a complete Riemannian manifold and let \(\Delta = \sum_{ij} g^{ij} ({\partial \over \partial x^i} {\partial \over \partial x^j} - \sum_k\Gamma_{ij}^k {\partial \over \partial x^k})\) be the Laplace-Beltrami operator acting on functions. We say that \(M\) satisfies the Liouville theorem (L1) if every bounded solution \(u\) of \(\Delta u = 0\) is constant. Moreover, we say that \(M\) satisfies (L2) if the equation \(\Delta u = \mu u\) has no nontrivial bounded solutions for all \(\mu > 0\). Euclidean space \(M={\mathbb R}^n\) satisfies both (L1) and (L2). Now let \(M\) be the union \(M = B \cup D\) where \(B\) is a compact set and \(D\) is diffeomorphic to \((0,\infty)\times S\), \(S\) compact, and \(D\) carries a warped product metric. The main result of the paper states that for such \(M\) (L1) implies (L2). The converse is not true as one can see from the example of hyperbolic space.
Mathematics Subject Classification ID: 58J99
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 1256815
zbMATH Keywords: Liouville theorem
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Laplace equation
reviewed by: Christian Bär
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1256815