Local Well-Posedness of Skew Mean Curvature Flow for Small Data in $$d\ge 4$$ Dimensions

The 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 this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension $$d\ge 4$$ d ≥ 4 .

On the Schrödinger map for regular helical polygons in the hyperbolic space

The main purpose of this article is to understand the evolution of X t = X s ∧− X ss , with X(s, 0) a regular polygonal curve with a nonzero torsion in the three-dimensional Minkowski space. Unlike in the case of the Euclidean space, a nonzero torsion now implies two different helical curves. This generalizes recent works by the author with de la Hoz and Vega on helical polygons in the Euclidean space as well as planar polygons in the Minkowski space. Numerical experiments in this article show that the trajectory of the point X(0, t) exhibits new variants of Riemann’s non-differentiable function whose structure depends on the initial torsion in the problem. As a result, we observe that the smooth solutions (helices, straight line) in the Minkowski space show the same instability as displayed by their Euclidean counterparts and curves with zero-torsion. These numerical observations are in agreement with some recent theoretical results obtained by Banica and Vega.