Contraction of Convex Hypersurfaces in $\mathbb R^3$ by Powers of Principal Curvatures

TL;DR

This paper investigates how strictly convex, axially symmetric hypersurfaces in three-dimensional space contract under a fully nonlinear curvature flow, demonstrating convergence to a point and rescaled convex shapes.

Contribution

It introduces a new analysis of curvature-driven contraction for non-symmetric, non-homogeneous functions on convex hypersurfaces, extending previous symmetry assumptions.

Findings

Hypersurfaces contract to a point in finite time.

Rescaled hypersurfaces converge to convex shapes.

The flow preserves convexity and symmetry.

Abstract

We study the contraction of strictly convex, axially symmetric hypersurfaces by a non-symmetric, non-homogeneous, fully nonlinear function of curvature. Starting from axially symmetric hypersurfaces with even profile curves, we show evolving hypersurfaces converge to a point in a finite time, and under proper rescaling, solutions will converge to a convex hypersurface.