Finite topology self-translating surfaces for the mean curvature flow constitute a key element in the analysis of Type II singularities from a compact surface because they arise as limits after suitable blow-up scalings around the singularity. We prove the existence of such a surface that is orientable, embedded, complete, and with three ends asymptotically paraboloidal. The fact that M is self-translating means that the moving surface evolves by mean curvature flow, or equivalently, that M satisfies the equation where denotes mean curvature, ν is a choice of unit normal to M, and is a unit vector along the z-axis. This surface M is in correspondence with the classical three-end Costa–Hoffman–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.
Dávila, J., Del Pino, M., & Nguyen, X. H. (2017). Finite topology self-translating surfaces for the mean curvature flow in R3. Advances in Mathematics, 320, 674-729. https://doi.org/10.1016/j.aim.2017.09.014