Contraction of hypersurfaces with positive sectional curvature in hyperbolic space

TL;DR

This paper investigates curvature flows of compact hypersurfaces with positive sectional curvature in hyperbolic space, demonstrating preservation of curvature and contraction to a round point in finite time.

Contribution

It establishes that certain curvature flows preserve positive sectional curvature and lead to hypersurface contraction to a round point in hyperbolic space.

Findings

Positive sectional curvature is preserved along the flow.

Hypersurfaces contract to a round point in finite time.

Includes the $k$th mean curvature flow as a special case.

Abstract

We study contracting curvature flows of compact hypersurfaces with positive sectional curvature in hyperbolic space $\mathbb{H}^{n+1}$. The speed is assumed to be homogeneous of degree one in the principal curvatures and to satisfy certain conditions. This class of flows includes the $k$th mean curvature flow as a special case. We show that if the initial hypersurface has positive sectional curvature, then this property is preserved along the flow, and the evolving hypersurface contracts to a round point in finite time.