Curvature estimates for minimal surfaces with total boundary curvature less than 4\pi

TL;DR

This paper provides curvature estimates for minimal surfaces with boundary curves having total curvature less than 4π, leading to bounds on their genus and topological properties based on boundary geometry.

Contribution

It introduces a new curvature estimate for minimal surfaces with boundary total curvature under 4π and explores topological bounds and openness results in the boundary curve space.

Findings

Curvature estimates depend only on boundary geometry.

Bound on the genus of minimal surfaces based on boundary curvature.

Openness of certain boundary curve classes in the C^{2,α} topology.

Abstract

We establish a curvature estimate for classical minimal surfaces with total boundary curvature less than 4\pi. The main application is a bound on the genus of these surfaces depending solely on the geometry of the boundary curve. We also prove that the set of simple closed curves with total curvature less than 4\pi and which do not bound an orientable compact embedded minimal surface of genus greater than g, for any given g, is open in the C^{2,\alpha} topology.