A combinatorial curvature flow for compact 3-manifolds with boundary

Luo, Feng

doi:10.1090/S1079-6762-05-00142-3

A combinatorial curvature flow for compact 3-manifolds with boundary

Author: Feng Luo
Journal: Electron. Res. Announc. Amer. Math. Soc. 11 (2005), 12-20
MSC (2000): Primary 53C44, 52A55
DOI: https://doi.org/10.1090/S1079-6762-05-00142-3
Published electronically: January 28, 2005
MathSciNet review: 2122445
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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.

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