Curvature flows in the sphere

TL;DR

This paper studies curvature flows in the sphere, showing how convex hypersurfaces contract or expand, relate via the Gauss map, and converge to spheres or the equator with exponential speed.

Contribution

It establishes a relation between contracting and expanding convex hypersurfaces via the Gauss map and describes their asymptotic behavior.

Findings

Contracting hypersurfaces shrink to a point.

Expanding hypersurfaces converge to the equator.

Rescaled hypersurfaces approach unit spheres exponentially fast.

Abstract

We consider contracting and expanding curvature flows in $\Ss$. When the flow hypersurfaces are strictly convex we establish a relation between the contracting hypersurfaces and the expanding hypersurfaces which is given by the Gau{\ss} map. The contracting hypersurfaces shrink to a point $x_0$ while the expanding hypersurfaces converge to the equator of the hemisphere $\mc H(-x_0)$. After rescaling, by the same scale factor, the rescaled hypersurfaces converge to the unit spheres with centers $x_0$ \resp $-x_0$ exponentially fast in $C^\un(\Ss[n])$.