Global and local theory of skew mean curvature flows

TL;DR

This paper investigates the skew mean curvature flow, establishing global regularity for small perturbations, analyzing asymptotic behavior through scattering theory, and exploring the Cauchy problem for large data.

Contribution

It provides the first comprehensive analysis of the global behavior and scattering properties of solutions to the skew mean curvature flow.

Findings

Global regularity for small initial perturbations

Existence of wave operators and modified scattering

Analysis of the Cauchy problem for large data

Abstract

In this paper, we study the skew mean curvature flow. The results are threefold. First, we prove the global regularity of solutions with initial data which are small perturbations of planes in Sobolev spaces. Second, we prove the modified scattering and the existence of wave operators for small data, which completely determines the set of asymptotic states. Third, we study the Cauchy problem for arbitrary large data.