  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 $\Sigma(t) = M +
te_z$ 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 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.
