Improvement of flatness in annuli

TL;DR

This paper introduces a PDE-based method to improve flatness in annuli for exterior domains, providing new insights into minimal hypersurfaces and adaptable to other PDE problems.

Contribution

It develops a novel flatness improvement argument tailored for annuli in exterior domains, with applications to minimal surfaces and other PDE contexts.

Findings

Provides an alternative proof of end-structure for minimal hypersurfaces.

Develops a flexible flatness improvement technique for annuli.

Method applicable to Bernoulli and Allen–Cahn problems.

Abstract

We present a short and flexible improvement-of-flatness argument adapted to the setting of exterior domains, where one is naturally led to work with annuli instead of balls. As a model application in the classical setting of minimal surfaces, we give an alternative proof of the end-structure and asymptotics for finite Morse index minimal hypersurfaces with Euclidean area growth in low dimensions. The method is largely PDE-based and general in its application. Suitable variants have been employed in Bernoulli and Allen--Cahn settings.