Volume preserving flow by powers of symmetric polynomials in the principal curvatures

TL;DR

This paper investigates a volume-preserving curvature flow driven by powers of symmetric polynomials in principal curvatures, proving long-time existence, bounded speed, and convergence to a sphere without initial pinching assumptions.

Contribution

It introduces a new volume-preserving flow based on symmetric polynomials and establishes convergence results without curvature pinching conditions.

Findings

Flow exists for all time

Speed remains bounded and converges to a constant

Hypersurfaces converge exponentially to a sphere

Abstract

We study a volume preserving curvature flow of convex hypersurfaces, driven by a power of the $k$-th elementary symmetric polynomial in the principal curvatures. Unlike most of the previous works on related problems, we do not require assumptions on the curvature pinching of the initial datum. We prove that the solution exists for all times and that the speed remains bounded and converges to a constant in an integral norm. In the case of the volume preserving scalar curvature flow, we can prove that the hypersurfaces converge smoothly and exponentially fast to a round sphere.