Contracting an axially symmetric torus by its harmonic mean curvature

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We consider the harmonic mean curvature flow of an axially symmetric torus whose axis is a closed geodesic, where the ambient space is a hyperbolic three-manifold. Assuming the initial surface is strictly convex and its harmonic mean curvature is less than 1/2, we show that the evolving surface satisfies a curvature condition comparable to that of a perfectly symmetric torus evolving under harmonic mean curvature flow. In other words, we prove that λ1 ≈ e-t, λ2 ≈ et and λ1λ2 ≈ 1, where λ1 and λ2 are the principal curvatures of the evolving torus.

Original languageEnglish (US)
Pages (from-to)117-133
Number of pages17
JournalPacific Journal of Mathematics
Issue number1
StatePublished - 2014


  • Closed geodesic
  • Harmonic mean curvature flow
  • Hyperbolic manifold


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