Minimal surfaces in minimally convex domains

TL;DR

This paper demonstrates that conformal minimal immersions in minimally convex domains can be approximated by proper complete immersions, introduces a rigidity theorem for minimal surfaces of finite total curvature, and characterizes the minimal surface hull of compact sets.

Contribution

It provides new approximation results for minimal surfaces in minimally convex domains, a rigidity theorem for finite total curvature surfaces, and a characterization of minimal surface hulls in higher dimensions.

Findings

Approximation of conformal minimal immersions by proper complete ones.

Rigidity theorem for minimal surfaces with finite total curvature.

Characterization of minimal surface hulls via conformal minimal disks.

Abstract

In this paper, we prove that every conformal minimal immersion of a compact bordered Riemann surface $M$ into a minimally convex domain $D\subset \mathbb{R}^3$ can be approximated, uniformly on compacts in $\mathring M=M\setminus bM$, by proper complete conformal minimal immersions $\mathring M\to D$. We also obtain a rigidity theorem for complete immersed minimal surfaces of finite total curvature contained in a minimally convex domain in $\mathbb{R}^3$, and we characterize the minimal surface hull of a compact set $K$ in $\mathbb{R}^n$ for any $n\ge 3$ by sequences of conformal minimal disks whose boundaries converge to $K$ in the measure theoretic sense.