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 | References | Similar Articles | Additional Information

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.

**[BB]**X. Bao and F. Bonahon,*Hyperideal polyhedra in hyperbolic 3-space.*Bull. Soc. Math. France**130**(2002), no. 3, 457-491. MR**1943885 (2003k:52007)****[CL]**B. Chow and F. Luo,*Combinatorial Ricci flows on surfaces.*J. Differential Geom.**63**(2003), no. 1, 97-129. MR**2015261 (2005a:53106)****[CV]**Y. Colin de Verdière,*Un principe variationnel pour les empilements de cercles.*Invent. Math.**104**(1991), no. 3, 655-669. MR**1106755 (92h:57020)****[FP]**R. Frigerio and C. Petronio,*Construction and recognition of hyperbolic 3-manifolds with geodesic boundary,*Trans. AMS.**356**(2004), no. 8, 3243-3282. MR**2052949****[Ha]**R. S. Hamilton,*Three-manifolds with positive Ricci curvature.*J. Differential Geom.**17**(1982), no. 2, 255-306. MR**0664497 (84a:53050)****[HK]**C. D. Hodgson and S. P. Kerckhoff,*Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery.*J. Differential Geom.**48**(1998), no. 1, 1-59. MR**1622600 (99b:57030)****[KR]**E. Kang and J. H. Rubinstein,*Ideal triangulations of 3-manifolds I*, preprint.**[La1]**M. Lackenby,*Word hyperbolic Dehn surgery*, Invent. Math.**140**(2000), no. 2, 243-282. MR**1756996 (2001m:57003)****[La2]**M. Lackenby,*Taut ideal triangulations of 3-manifolds*, Geom. Topol.**4**(2000), 369-395 (electronic). MR**1790190 (2002a:57026)****[Le]**G. Leibon,*Characterizing the Delaunay decompositions of compact hyperbolic surfaces.*Geom. Topol.**6**(2002), 361-391 (electronic). MR**1914573 (2003c:52034)****[Lu]**F. Luo,*Continuity of the volume of simplices in classical geometry*, preprint,`http://front.math.ucdavis.edu/math.GT/0412208`.**[Mo]**E. Moise,*Affine structures in 3-manifolds V*. Ann. Math. (2)**56**(1952), 96-114. MR**0048805 (14:72d)****[Ri1]**I. Rivin,*Combinatorial optimization in geometry.*Adv. in Appl. Math.**31**(2003), no. 1, 242-271. MR**1985831 (2004i:52005)****[Ri2]**I. Rivin,*Euclidean structures on simplicial surfaces and hyperbolic volume.*Ann. of Math. (2)**139**(1994), no. 3, 553-580. MR**1283870 (96h:57010)****[Ri3]**I. Rivin, private communication.**[Sc]**J. Schlenker,*Hyperideal polyhedra in hyperbolic manifolds*, preprint,`http://front.math.ucdavis.edu/math.GT/0212355`.**[Th]**W. Thurston,*Topology and geometry of 3-manifolds*, Lecture notes, Princeton University, 1978.**[Us]**A. Ushijima,*A volume formula for generalized hyperbolic tetrahedra*, preprint,`http://front.math.ucdavis.edu/math.GT/0309216`.

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Additional Information

**Feng Luo**

Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ 07059

Email:
fluo@math.rutgers.edu

DOI:
https://doi.org/10.1090/S1079-6762-05-00142-3

Received by editor(s):
May 14, 2004

Published electronically:
January 28, 2005

Communicated by:
Tobias Colding

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.