Arsove–Huber Theorem in Higher Dimensions

Abstract

In this note we briefly present the progress in the research project to extend Huber’s theory of surfaces to general dimensions. The full paper [42] is in progress. We discuss n-Laplace equations and n-subharmonic functions using nonlinear potential theory. Particularly we build the Brezis–Merle type sharp inequality for Wolff potential and establish Taliaferro’s estimates in higher dimensions. We then apply the theory of n-subharmonic functions developed here to study hypersurfaces in hyperbolic space with nonnegative Ricci curvature as well as locally conformal flat manifolds with nonnegative Ricci.