Potential Theory on Minimal Hypersurfaces II: Hardy Structures and Schr\"odinger Operators

Abstract

We extend the potential theory on almost minimzers from Part 1. We introduce so-called Hardy structures to study many classical operators using the tools from part 1. Furthermore, we show that for a naturally defined operator L, minimal growth of positive solutions of Lw = 0 towards the singular set is a stable property. It persists under perturbations or blow-ups of the underlying spaces. This is the key result to develop a dimensional induction scheme for the asymptotic analysis of these operators near the singular set.