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 low-regularity Sobolev spaces for the skew mean curvature flow in dimensions $d\geq 4$. This extends the local well-posedness result in [7].
Bi-Halfspace and Convex Hull Theorems for Translating Solitons
While it is well known from examples that no interesting “halfspace theorem” holds for properly immersed $n$-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” (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 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 ${\mathbb{R}}^{n}$.
more » « less- NSF-PAR ID:
- 10124704
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract The skew mean curvature flow is an evolution equation for
d dimensional manifolds embedded in$${{\mathbb {R}}}^{d+2}$$ $$d\ge 4$$
