Volume-Preserving flow by powers of the mth mean curvature in the hyperbolic space

TL;DR

This paper studies a volume-preserving geometric flow of hypersurfaces in hyperbolic space driven by powers of the mth mean curvature, proving long-term existence and exponential convergence to geodesic spheres under certain curvature conditions.

Contribution

It introduces a new class of volume-preserving flows based on powers of the mth mean curvature in hyperbolic space, establishing existence, uniqueness, and convergence results.

Findings

Flow exists for all time and remains smooth.

Hypersurfaces converge exponentially to geodesic spheres.

Initial curvature ratio close to 1 is preserved during flow.

Abstract

This paper concerns closed hypersurfaces of dimension $n(\geq 2)$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature $\kappa$ evolving in direction of its normal vector, where the speed is given by a power $\beta (\geq 1/m)$ of the $m$th mean curvature plus a volume preserving term, including the case of powers of the mean curvature and of the $\mbox{Gau\ss}$ curvature. The main result is that if the initial hypersurface satisfies that the ratio of the biggest and smallest principal curvature is close enough to 1 everywhere, depending only on $n$, $m$, $\beta$ and $\kappa$, then under the flow this is maintained, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces exponentially converge to a geodesic sphere of ${\mathbb{H}}_{\kappa}^{n+1}$, enclosing the same volume as the initial hypersurface.