Abstract
The skew mean curvature flow is an evolution equation for d dimensional manifolds embedded in {{mathbb {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 this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension dge 4.
Highlights
For each t ∈ I, we denote the submanifold by t = F(t, ), its tangent bundle by T t, and its normal bundle by N t respectively
The normal bundle N t is a rank two vector bundle with a naturally induced complex structure J (F) which rotates a vector in the normal space by π/2 positively
We describe the gauge choices, so that by the end we obtain (a) a nonlinear Schrödinger equation for ψ, see (2.35). (b) An elliptic fixed time system (2.36) for the dependent variables S = (g, λ, V, A, B), together with suitable compatibility conditions
Summary
0 > 0 sufficiently small such that, for all initial data 0 with metric ∂x (g0 − Id ) Hs ≤. There the tangential component of ∂t F in (1.1) is omitted, and the coordinates on the manifold t are those transported from the initial time The difficulty with such a choice is that the regularity of the map F is no longer determined by the regularity of the second fundamental form, and instead there is a loss of derivatives which may only be avoided if the initial data is assumed to have extra regularity. The advection vector field V , associated to the time dependence of our choice of coordinates These additional variables will be viewed as uniquely determined by our independent variable ψ, provided that a suitable gauge choice was made.
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