Francesco Chini

In this work we show that $2$-dimensional, simply connected, translating solitons of the mean curvature flow embedded in a slab of ${\mathbb{R}}^3$ with entropy strictly less than $3$ must be mean convex and thus, thanks to a result by J. Spruck and L. Xiao, are convex. Recently, such $2$-dimensional convex translating solitons have been completely classified by Hoffman, Ilmanen, Martín and White, up to an ambient isometry, as vertical plane, (tilted) grim reaper cylinders, $\Delta$-wings and bowl translater. These are all contained in a slab, except for the rotationally symmetric bowl translater. New examples show that the bound on the entropy is necessary.

We study properly immersed ancient solutions of the codimension one mean curvature flow in $n$-dimensional Euclidean space, and classify the convex hulls of the subsets of space reached by any such flow. In particular, it follows that any compact convex ancient mean curvature flow can only have a slab, a halfspace or all of space as the closure of its set of reach. The proof proceeds via a bi-halfspace theorem (also known as a wedge theorem) for ancient solutions derived from a parabolic Omori-Yau maximum principle for ancient mean curvature flows.

While it is well known from examples that no interesting `halfspace theorem' holds for properly immersed complete $n$-dimensional self-translating mean curvature flow solitons in Euclidean space ${\mathbb{R}}^{n+1}$, we show that they must all obey a general `bi-halfspace theorem': Two transverse vertical halfspaces can never contain the same such hypersurface. The proof avoids the typical methods of nonlinear barrier construction, not readily available here, for the approach via distance functions and the Omori-Yau maximum principle. As an application we classify the convex hulls of all properly immersed complete (possibly with compact boundary) $n$-dimensional mean curvature flow self-translating solitons ${\Sigma }^n$ in ${\mathbb{R}}^{n+1}$, up to an orthogonal projection in the direction of translation. This list is short, coinciding with the one given by Hoffman-Meeks in 1989, for minimal submanifolds: All of ${\mathbb{R}}^n$, halfspaces, slabs, hyperplanes and convex compacts in ${\mathbb{R}}^n$.