Translating solitons of the mean curvature flow asymptotic to hyperplanes in ${\mathbb{R}}^{n+1}$

Abstract

A translating soliton is a hypersurface $M$ in ${\mathbb{R}}^{n+1}$ such that the family $M_t=M-t{\mathbf{e}}_{n+1}$ is a mean curvature flow, i.e., such that normal component of the velocity at each point is equal to the mean curvature at that point $\mathbf{H}={\mathbf{e}}_{n+1}^{\perp }.$ In this paper we obtain a characterization of hyperplanes which are parallel to ${\mathbf{e}}_{n+1}$ and the family of tilted grim reaper cylinders as the only translating solitons in ${\mathbb{R}}^{n+1}$ which are $C^1$-asymptotic to two half-hyperplanes outside a non-vertical cylinder. This result was proven for translators in ${\mathbb{R}}^3$ by the second author, Perez-Garcia, Savas-Halilaj and Smoczyk under the additional hypotheses that the genus of the surface was locally bounded and the cylinder was perpendicular to the translating velocity.

37 pages, 12 figures. This revised version will appear in IMRN