Examples of hypersurfaces flowing by curvature in a Riemannian manifold

Robert D Gulliver, Guoyi Xu

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

This paper gives some examples of hypersurfaces φ(Mn) 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 M2.

Original languageEnglish (US)
Pages (from-to)701-719
Number of pages19
JournalCommunications in Analysis and Geometry
Volume17
Issue number4
DOIs
StatePublished - Oct 2009

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