Finite topology self-translating surfaces for the mean curvature flow constitute a key element in the analysis of Type II singularities from a compact surface because they arise as limits after suitable blow-up scalings around the singularity. We prove the existence of such a surface that is orientable, embedded, complete, and with three ends asymptotically paraboloidal. The fact that M is self-translating means that the moving surface evolves by mean curvature flow, or equivalently, that M satisfies the equation where denotes mean curvature, ν is a choice of unit normal to M, and is a unit vector along the z-axis. This surface M is in correspondence with the classical three-end Costa–Hoffman–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.
|Number of pages||56|
|Journal||Advances in Mathematics|
|Early online date||14 Sept 2017|
|Publication status||Published - 7 Nov 2017|
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Manuel Del Pino
- Department of Mathematical Sciences - Professor
Person: Research & Teaching