  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}.
