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.

Final version, to appear in CCM

References in corpus (2)

• Vortex dynamics in $R^4$
• A Compactness Theorem for the Second Fundamental Form

Cited by in corpus (1)

• Uniqueness of Schrödinger flow on manifolds