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## A combinatorial curvature flow for compact 3-manifolds

with boundary

### Feng Luo

**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.

*Copyright 2005 American Mathematical Society
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#### Article Info

- ERA Amer. Math. Soc.
**11** (2005), pp. 12-20
- Publisher Identifier: S 1079-6762(05)00142-3
- 2000
*Mathematics Subject Classification*. Primary 53C44, 52A55
- Received by editors May 14, 2004
- Posted on January 28, 2005
- Communicated by Tobias Colding
- Comments (When Available)

**Feng Luo**

Department of Mathematics, Rutgers University, Piscataway, NJ 07059

*E-mail address:* `fluo@math.rutgers.edu`

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