Nonlinear partial differential systems on Riemannian manifolds with their geometric applications

Abstract

We make the first study of how the existence of (essential) positive supersolutions of nonlinear degenerate partial differential equations on a manifold affects the topology, geometry, and analysis of the manifold. For example, for surfaces in R3 we prove a Bernstein-type theorem that generalizes and unifies three distinct theorems. In higher dimensions, we provide topological obstructions for a minimal hypersurface in Rn+1 to admit an essential positive supersolution. This immediately yields information about the Gauss map of complete minimal hypersurfaces in Rn+1. By coping with a wider class of nonlinear partial differential equations that are involved with (p)-harmonic maps and (p)-superstrongly unstable manifolds, we derive information on the regularity of minimizers, homotopy groups, and solutions to Dirichlet problems, from the existence of essential positive supersolutions.