  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
$d\geq 4$.
