Finite topology selftranslating surfaces for the mean curvature flow in $\mathbb R^3$
Abstract
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 blowup scalings around the singularity. We find in $\mathbb R^3$ a surface $M$ orientable, embedded and complete with finite topology (and large genus) with three ends asymptotically paraboloidal, such that the moving surface $\Sigma(t) = M + te_z$ evolves by mean curvature flow. This amounts to the equation $H_M = \nu\cdot e_z$ where $H_M$ denotes mean curvature, $\nu$ is a choice of unit normal to $M$, and $e_z$ is a unit vector along the $z$axis. The surface $M$ is in correspondence with the classical 3end CostaHoffmannMeeks 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 nontrivial 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2015
 arXiv:
 arXiv:1501.03867
 Bibcode:
 2015arXiv150103867D
 Keywords:

 Mathematics  Analysis of PDEs;
 53C44