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# Communications in Analysis and Geometry

## Volume 25 (2017)

### Number 2

### Volume-preserving flow by powers of the $m\textrm{-th}$ mean curvature in the hyperbolic space

Pages: 321 – 372

DOI: https://dx.doi.org/10.4310/CAG.2017.v25.n2.a3

#### Authors

#### Abstract

This paper concerns closed hypersurfaces of dimension $n(\geq 2)$ in the hyperbolic space $\mathbb{H}^{n+1}_{\kappa}$ 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 $m\textrm{-th}$ mean curvature plus a volume preserving term, including the case of powers of the mean curvature and of the Gauß curvature. The main result is that if the initial hypersurface satisfies that the ratio of the biggest and smallest principal curvatures 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 converge exponentially to a geodesic sphere of $\mathbb{H}^{n+1}_{\kappa}$, enclosing the same volume as the initial hypersurface.

#### Keywords

powers of the $m\textrm{-th}$ mean curvature, horosphere, convex, hypersurface, hyperbolic space

#### 2010 Mathematics Subject Classification

Primary 35K55, 53C44. Secondary 35B40, 58J35.

Received 8 September 2013

Published 4 August 2017