2015/01/16 by Juan Dávila, Manuel del Pino, Dávila, Juan +3 · 1 citation
Mathematics · #53C44 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals #math.AP #msc:53C44
paper · pdf · doi:10.48550/arxiv.1501.03867
arxiv created 2015/01/16 · openalex publication_date 2015/01/16 · arxiv updated 2015/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Finite topology self translating surfaces to mean curvature flow of surfaces constitute a key element for the analysis of Type II singularities from a compact surface, since they arise in a limit after suitable blow-up scalings around the singularity. We find in \mathbb R3 a surface M orientable, embedded and complete with finite topology (and large genus) with three ends asymptotically paraboloidal, such that the moving surface Σ(t) = M + tez evolves by mean curvature flow. This amounts to the equation HM = ν⋅ ez where HM denotes mean curvature, ν is a choice of unit normal to M, and ez is a unit vector along the z-axis. The surface M is in correspondence with the classical 3-end Costa-Hoffmann-Meeks minimal surface with large genus, which has two asymptotically catenoidal ends and one planar end, and a long array of small tunnels in the intersection region resembling a periodic Scherk surface. This example is the first non-trivial one of its kind, and it suggests a strong connection between this problem and the theory of embedded, complete minimal surfaces with finite total curvature.