Communications in Analysis and Geometry

Volume 17 (2009)

Number 4

Examples of hypersurfaces flowing by curvature in a Riemannian manifold

Pages: 701 – 719



Robert Gulliver (School of Mathematics, University of Minnesota)

Guoyi Xu (School of Mathematics, University of Minnesota)


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 $M^2$.

Published 1 January 2009