Translating Solitons of the Mean Curvature Flow Asymptotic to Hyperplanes in ℝn+1

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, that is, 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 that are parallel to the velocity and the family of tilted grim reaper cylinders as the only translating solitons in $\mathbb{R}^{n+1}$ that 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 2nd 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.

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