Finite topology self-translating surfaces for the mean curvature flow in ${\mathbb{R}}^3$

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 ${\mathbb{R}}^3$ a surface $M$ orientable, embedded and complete with finite topology (and large genus) with three ends asymptotically paraboloidal, such that the moving surface $\Sigma(t) = M + te_z$evolvesbymeancurvatureflow\mathrm{.}Thisamountstotheequation$H_M = \nu\cdot e_z$where$H_M$denotesmeancurvature,$\nu$ 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 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.