Abstract
AbstractThe skew mean curvature flow is an evolution equation for d dimensional manifolds embedded in $${\mathbb {R}}^{d+2}$$
R
d
+
2
(or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In an earlier paper, the authors introduced a harmonic/Coulomb gauge formulation of the problem, and used it to prove small data local well-posedness in dimensions $$d \geqq 4$$
d
≧
4
. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension $$d\geqq 2$$
d
≧
2
. This is achieved by introducing a new, heat gauge formulation of the equations, which turns out to be more robust in low dimensions.
Funder
Division of Mathematical Sciences
Simons Foundation
NSFC
Publisher
Springer Science and Business Media LLC
Subject
Mechanical Engineering,Mathematics (miscellaneous),Analysis