  The skew mean curvature flow(SMCF), which origins from the study of fluid
dynamics, describes the evolution of a codimension two submanifold along its
binormal direction. We study the basic properties of the SMCF and prove the
existence of a short-time solution to the initial value problem of the SMCF of
compact surfaces in Euclidean space $\mathbb{R}^4$. A Sobolev-type embedding
theorem for the second fundamental forms of two dimensional surfaces is also
proved, which might be of independent interest.
