Skew Mean Curvature Flow

Abstract

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.