Local Well-Posedness of the Skew Mean Curvature Flow for Small Data in \(d\geqq 2\) Dimensions

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 Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger 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 \(d \geqq 4\). In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension \(d\geqq 2\). This is achieved by introducing a new, heat gauge formulation of the equations, which turns out to be more robust in low dimensions.