Skew Mean Curvature Flow

TL;DR

This paper investigates the skew mean curvature flow (SMCF), proving short-time existence of solutions for compact surfaces in four-dimensional space and establishing a Sobolev embedding theorem for their second fundamental forms.

Contribution

It introduces the SMCF in the context of fluid dynamics, proves short-time existence of solutions, and develops a Sobolev embedding theorem for 2D surface second fundamental forms.

Findings

Proved short-time existence of SMCF solutions for compact surfaces in R^4.

Established a Sobolev-type embedding theorem for second fundamental forms.

Analyzed basic properties of the 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.