The skew mean curvature flow is an evolution equation for a $d$ dimensional manifold immersed into $\mathbb {R}^{d+2}$, and which moves along the binormal direction with a speed proportional to its mean curvature. In this article, we prove small data global regularity in lowregularity Sobolev spaces for the skew mean curvature flow in dimensions $d\geq 4$. This extends the local wellposedness result in [7].
While it is well known from examples that no interesting “halfspace theorem” holds for properly immersed $n$dimensional selftranslating mean curvature flow solitons in Euclidean space $\mathbb{R}^{n+1}$, we show that they must all obey a general “bihalfspace theorem” (aka “wedge theorem”): two transverse vertical halfspaces can never contain the same such hypersurface. The same holds for any infinite end. The proofs avoid the typical methods of nonlinear barrier construction for the approach via distance functions and the Omori–Yau maximum principle. As an application we classify the closed convex hulls of all properly immersed (possibly with compact boundary) $n$dimensional mean curvature flow selftranslating 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 ${\mathbb{R}}^{n}$.
more » « less NSFPAR ID:
 10124704
 Publisher / Repository:
 Oxford University Press
 Date Published:
 Journal Name:
 International Mathematics Research Notices
 ISSN:
 10737928
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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