Minimal surfaces in minimally convex domains

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.

Trans. Amer. Math. Soc

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