Global regularity of Skew mean curvature flow for small data in $d\geq 4$ dimensions

TL;DR

This paper proves that small initial data lead to globally regular solutions for the skew mean curvature flow in dimensions four and higher, extending previous local well-posedness results to global regularity in low-regularity Sobolev spaces.

Contribution

It establishes global regularity for small data in high dimensions, advancing understanding of the skew mean curvature flow beyond local solutions.

Findings

Global regularity for small data in $d extgreater=4$ dimensions.

Extension of local well-posedness to global solutions.

Works in low-regularity Sobolev spaces.

Abstract

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 \cite{HT}.