The classification of doubly periodic minimal tori with parallel ends

2005/01/28 by Joaquin Perez, Joaquín Pérez, M. Magdalena Rodriguez +5 · 2 citations
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #math.DG #msc:49Q05 #msc:53A10

paper · pdf · doi:10.48550/arxiv.math/0501507

55 pages, 8 figures

arxiv created 2005/01/28 · arxiv updated 2009/12/01

Abstract

Let K be the space of properly embedded minimal tori in quotients of \R³ by two independent translations, with any fixed (even) number of parallel ends. After an appropriate normalization, we prove that K is a 3-dimensional real analytic manifold that reduces to the finite coverings of the examples defined by Karcher, Meeks and Rosenberg in \citeka4,ka6,mr3. The degenerate limits of surfaces in K are the catenoid, the helicoid and three 1-parameter families of surfaces: the simply and doubly periodic Scherk minimal surfaces and the Riemann minimal examples.

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• Minimal Surfaces of Finite Genus: Classification, Dynamics and Laminations
• Gluing doubly periodic Scherk surfaces into minimal surfaces

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