Finite topology self-translating surfaces for the mean curvature flow in R3

Abstract

Finite topology self-translating surfaces for the mean curva-ture flow constitute a key element in the analysis of TypeII singularities from a compact surface because they arise as lim-its after suitable blow-up scalings around the singularity. We prove the existence of such a surface M⊂R3that is ori-entable, embedded, complete, and with three ends asymptot-ically paraboloidal. The fact that Mis self-translating means that the moving surface S(t) =M+tezevolves by mean curvature flow, or equivalently, that Msatisfies the equation HM=ν·ezwhere HMdenotes mean curvature, νis a choice of unit normal to M, and ezis a unit vector along the z-axis. This surface Mis in correspondence with the classical three-end Costa–Hoffman–Meeks minimal surface with large genus, which has two asymptotically catenoidal ends and one pla-nar 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.