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

Abstract

A translating soliton is a hypersurface M in R n+1 such that the family M t = M − t 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 H = e ⊥ n+1 . In this paper we obtain a characterization of hyperplanes which are parallel to the velocity and the family of tilted grim reaper cylinders as the only translating solitons in R n+1 which are C 1 -asymptotic to two half-hyperplanes outside a nonvertical cylinder. This result was proven for translators in 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.

Bi-Halfspace and Convex Hull Theorems for Translating Solitons

“…, w 2,n ). Note that ∇g L does not depend on L. Now take L large enough so that there exists a tilted grim reaper cylinder Σ which is the graph of a function defined on S L and such that it grows linearly in the direction of ∇g L and with the same slope of g L (for a detailed description of tilted grim reaper cylinders, see [GM18] and [BLT18]). Then, since Σ is the graph of a function which is strictly convex w.r.t.…”