A combinatorial curvature flow for compact 3-manifolds with boundary

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Abstract

We introduce a combinatorial curvature flow for piecewise constant curvature metrics on compact triangulated 3-manifolds with boundary consisting of surfaces of negative Euler characteristic. The flow tends to find the complete hyperbolic metric with totally geodesic boundary on a manifold. Some of the basic properties of the combinatorial flow are established. The most important one is that the evolution of the combinatorial curvature satisfies a combinatorial heat equation. It implies that the total curvature decreases along the flow. The local convergence of the flow to the hyperbolic metric is also established if the triangulation is isotopic to a totally geodesic triangulation.

Original languageEnglish (US)
Pages (from-to)12-20
Number of pages9
JournalElectronic Research Announcements of the American Mathematical Society
Volume11
Issue number2
DOIs
StatePublished - Jan 28 2005

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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