Compact Embedded Minimal Surfaces of Positive Genus Without Area Bounds

2003/08/22 by Brian Dean, Brian C. Dean, Dean, Brian
Mathematics · #53A10 (Primary) 53C42 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.DG #msc:53A10 #msc:53C42

paper · pdf · doi:10.48550/arxiv.math/0308215

8 pages, 2 figures, to appear in Geomitrae Dedicata

arxiv created 2003/08/22 · openalex publication_date 2003/08/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M be a 3-manifold (possibly with boundary). We show that, for any positive integer g, there exists an open nonempty set of metrics on M for each of which there are stable compact embedded minimal surfaces of genus g with arbitrarily large area. This extends the result of Colding and Minicozzi for g=1.

Citations

• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature

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