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Finite topology self translating surfaces to mean curvature flow of surfacesconstitute a key element for the analysis of Type II singularities from acompact surface, since they arise in a limit after suitable blow-up scalingsaround the singularity. We find in $\mathbb R^3$ a surface $M$ orientable,embedded and complete with finite topology (and large genus) with three endsasymptotically 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 unitnormal 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 minimalsurface with large genus, which has two asymptotically catenoidal ends and oneplanar end, and a long array of small tunnels in the intersection regionresembling a periodic Scherk surface. This example is the first non-trivial oneof its kind, and it suggests a strong connection between this problem and thetheory of embedded, complete minimal surfaces with finite total curvature.

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