Density theorems for complete minimal surfaces in R³

2006/03/31 by A. Alarcon, Antonio Alarcón, L. Ferrer +6
Mathematics · #49Q05 #53A10 #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #math.DG #msc:49Q05 #msc:53A10

paper · pdf · doi:10.48550/arxiv.math/0603737

32 pages, 8 figures. To appear in G.A.F.A

openalex publication_date 2006/03/31 · arxiv created 2006/11/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we have proved several approximation theorems for the family of minimal surfaces in R³ that imply, among other things, that complete minimal surfaces are dense in the space of all minimal surfaces endowed with the topology of C^k convergence on compact sets, for any k. As a consequence of the above density result, we have been able to produce the first example of a complete proper minimal surface in R³ with uncountably many ends.

Citations

• On the existence of a proper minimal surface in R³ with the conformal type of a disk

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