Finite topology self-translating surfaces for the mean curvature flow in R-3

Abstract

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 M subset of R-3 that is orientable, embedded, complete, and with three ends asymptotically paraboloidal. The fact that M is self-translating means that the moving surface S(t) = M + te(z) evolves by mean curvature flow, or equivalently, that M satisfies the equation H-M = V.e(z) where H-M denotes mean curvature, v is a choice of unit normal to M, and e(z) 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. (C) 2017 Elsevier Inc. All rights reserved. Keywords. Author Keywords:Mean curvature flow

Self-translating

Solitons

KeyWords Plus:MINIMAL-SURFACES

SINGULARITIES