Tangential limits of stable minimal capillary surfaces

TL;DR

This paper characterizes stable minimal capillary surfaces near angles 0 or π supported on specific minimal surfaces, using curvature estimates to analyze their limits.

Contribution

It provides a detailed characterization of stable minimal capillary surfaces with angles close to 0 or π, supported on certain minimal or mean convex surfaces, advancing understanding of their geometric limits.

Findings

Characterization of stable minimal capillary surfaces near angles 0 or π.

Curvature estimates for sequences of weakly stable surfaces.

Analysis of tangential limits at suitable scales.

Abstract

We characterize all compact embedded stable minimal capillary surfaces with capillary angle close to either $0$ or $\pi$ that are supported on a complete embedded minimal surface with finite total curvature that is not an affine plane. Moreover, we characterize all compact embedded weakly stable minimal capillary surfaces with capillary angle close to either $0$ or $\pi$ that are supported on a closed surface whose mean curvature is positive and has no degenerate maxima. An important ingredient in our work are curvature estimates for sequences of weakly stable minimal capillary surfaces with capillary angles tending to $0$ or $\pi$ that enable us to analyze the tangential limits of such sequences at suitable scales.