Classification of Stable Minimal Surfaces Bounded by Jordan Curves in Close Planes

1999/05/08 by Rosanna Pearlstein, Pearlstein, Rosanna
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/9905047

38 pages, 2 figures, latex

arxiv created 1999/05/08 · openalex publication_date 1999/05/08 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study compact stable embedded minimal surfaces whose boundary is given by two collections of closed smooth Jordan curves in close planes of Euclidean 3-space. Our main result is a classification of these minimal surfaces, under certain natural geometric asymptotic constraints, in terms of certain associated varifolds which can be enumerated explicitely. One consequence of this result is the uniqueness of the area minimizing examples. Another is the asymptotic nonexistence of stable compact embedded minimal surfaces of positive genus bounded by two convex curves in parallel planes.

Related

• Minimal Surfaces in H²xR
• An asymptotic theorem for minimal surfaces and existence results for minimal graphs in H² × R
• A real-analytic Jordan curve cannot bound infinitely many relative minima of area
• On the asymptotic behavior of minimal surfaces in H²×R
• The geometry of stable minimal surfaces in metric Lie groups