Examples of hypersurfaces flowing by curvature in a Riemannian manifold

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

2009-04-08

arxiv.org/abs/0904.1255v2

Mathematics

Differential Geometry

some changes, 18 pages, no figures, accepted by Comm. Anal. Geom

Scientific paper

This paper gives some examples of hypersurfaces $\phi_t(M^n)$ evolving in time with speed determined by functions of the normal curvatures in an $(n+1)$-dimensional hyperbolic manifold; we emphasize the case of flow by harmonic mean curvature. The examples converge 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 only occurs in finite time. For dimension $n=2,$ the destiny of any harmonic mean curvature flow is strongly influenced by the genus of the surface $M^2$.

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