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

TL;DR

This paper extends potential theory on minimal hypersurfaces by introducing Hardy structures, demonstrating stability of minimal growth of solutions, and developing a dimensional induction scheme for analyzing operators near singularities.

Contribution

It introduces Hardy structures for classical operators and proves stability of minimal growth of solutions, enabling a new asymptotic analysis framework.

Findings

Hardy structures facilitate studying classical operators on minimal hypersurfaces.

Minimal growth of solutions is stable under perturbations and blow-ups.

A dimensional induction scheme is developed for asymptotic analysis near singular sets.

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.