Local well-posedness of the Skew mean curvature flow for small data in $d\geq 2$ dimensions

TL;DR

This paper establishes local well-posedness for the skew mean curvature flow in dimensions two and higher using a novel heat gauge approach, extending previous results to lower dimensions and lower regularity spaces.

Contribution

The authors introduce a heat gauge formulation that enables proving small data local well-posedness for the skew mean curvature flow in low dimensions, improving upon earlier methods.

Findings

Proved local well-posedness in dimensions d ≥ 2.

Developed a heat gauge formulation for the flow.

Extended well-posedness results to lower regularity 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 an earlier paper, the authors introduced a harmonic/Coulomb gauge formulation of the problem, and used it to prove small data local well-posedness in dimensions $d \geq 4$. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension $d\geq 2$. This is achieved by introducing a new, heat gauge formulation of the equations, which turns out to be more robust in low dimensions.