Minimal surfaces in minimally convex domains

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...

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Published in:Transactions of the American Mathematical Society 2019-02, Vol.371 (3), p.1735-1770
Main Authors: Antonio Alarcón, Barbara Drinovec Drnovšek, Franc Forstnerič, Francisco J. López
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Barbara Drinovec Drnovšek
Franc Forstnerič
Francisco J. López
description 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.
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