Rough isometry and \(p\)-harmonic boundaries of complete Riemannian manifolds
The \(p\)-harmonic boundary of a complete Riemannian manifold \(M\) is introduced. This is a certain subset of \(\widehat{M}\setminus M\), where \(\widehat{M}\) is the so-called Royden \(p\)-compactification of \(M\). The author proves that every bounded, energy-finite \(p\)-harmonic function on \(M\) whose distributional gradient belongs to \(L^p(M)\) is uniquely determined by its limit values on the \(p\)-harmonic boundary. It is also proven that a rough isometry between complete Riemannian manifolds always induces a homeomorphism between their \(p\)-harmonic boundaries and a bijection between their spaces of bounded, energy-finite \(p\)-harmonic functions with distributional gradient in \(L^p\). In particular, in the case of the Laplacian, a rough isometry induces a linear isomorphism between the spaces of bounded harmonic functions with finite Dirichlet integral.
scientific article; zbMATH DE number 2164739