The relationship between some Liouville theorems on Riemannian manifolds of a special type
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.
scientific article; zbMATH DE number 1256815
scientific article