Soap films with gravity and almost-minimal surfaces

TL;DR

This paper establishes a compactness theorem for surfaces with near-zero mean curvature, providing conditions under which almost-minimal surfaces converge to true minimal surfaces, with quantitative estimates on their proximity.

Contribution

It introduces a new compactness result for almost-minimal surfaces with boundary, linking their limits to minimal surfaces and offering quantitative proximity estimates.

Findings

Sufficient geometric conditions for convergence to minimal surfaces

Quantitative estimates on the distance between almost-minimal and minimal surfaces

Characterization of limit surfaces for sequences of almost-minimal surfaces

Abstract

Motivated by the study of the equilibrium equations for a soap film hanging from a wire frame, we prove a compactness theorem for surfaces with asymptotically vanishing mean curvature and fixed or converging boundaries. In particular, we obtain sufficient geometric conditions for the minimal surfaces spanned by a given boundary to represent all the possible limits of sequences of almost-minimal surfaces. Finally, we provide some sharp quantitative estimates on the distance of an almost-minimal surface from its limit minimal surface.