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 discs whose boundaries converge to $K$ in the measure theoretic sense.