Local well-posedness of skew mean curvature flow for small data in $d\geq 4$ dimensions

TL;DR

This paper proves local well-posedness for the skew mean curvature flow in dimensions four and higher, establishing existence and uniqueness of solutions for small initial data in low-regularity Sobolev spaces.

Contribution

It introduces the first small data well-posedness results for the skew mean curvature flow in high dimensions, extending the understanding of this Schr"odinger-type geometric evolution.

Findings

Well-posedness established for $d \\geq 4$

Solutions exist and are unique for small initial data

Works in low-regularity Sobolev spaces

Abstract

The skew mean curvature flow is an evolution equation for $d$ dimensional manifolds embedded in $\mathbb{R}^{d+2}$ (or more generally, in a Riemannian manifold). It can be viewed as a Schr\"odinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schr\"odinger Map equation. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension $d\geq 4$.