Compact complete minimal immersions in R³

2007/11/15 by Antonio Alarcón, Alarcon, Antonio
Mathematics · #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Analytic and geometric function theory

paper · pdf · doi:10.48550/arxiv.0711.2394

Abstract

In this paper we find, for any arbitrary finite topological type, a compact Riemann surface M, an open domain M\subsetM with the fixed topological type, and a conformal complete minimal immersion X:M→\R³ which can be extended to a continuous map X:M→\R³, such that X_{|∂ M} is an embedding and the Hausdorff dimension of X(∂ M) is 1. We also prove that complete minimal surfaces are dense in the space of minimal surfaces spanning a finite set of closed curves in \R³, endowed with the topology of the Hausdorff distance.

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