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# Communications in Analysis and Geometry

## Volume 17 (2009)

### Number 4

### Examples of hypersurfaces flowing by curvature in a Riemannian manifold

Pages: 701 – 719

DOI: http://dx.doi.org/10.4310/CAG.2009.v17.n4.a6

#### Authors

#### Abstract

This paper gives some examples of hypersurfaces $\phi_t(M^n)$evolving in time with speed determined by functions of the normalcurvatures in an $(n+1)$-dimensional hyperbolic manifold; weemphasize the case of flow by harmonic mean curvature. Theexamplesconverge to a totally geodesic submanifold of any dimension from$1$ to $n$, and include cases which exist for infinite time.Convergence to a point was studied by Andrews, and onlyoccurs in finite time. For dimension $n=2,$ the destiny ofany harmonic mean curvature flow is strongly influenced by thegenus of the surface $M^2$.