A combinatorial Yamabe flow in three dimensions

Topology

Volumn 44, Issue 4, 2005, Pages 791-808

  Glickenstein, David ^a  

^a University of Arizona   (United States)

Author keywords

Curvature flow; Discrete Riemannian geometry; Laplacian; Sphere packing; Yamabe flow

Indexed keywords

EID: 17644373737     PISSN: 00409383     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.top.2005.02.001     Document Type: Article

References (31)

1. N. Altshiller-Court Modern Pure Solid Geometry 1964 Chelsea Publishing Co. New York
2. J. Ambjørn, M. Carfora, A. Marzuoli, The geometry of dynamical triangulations, Lecture Notes in Physics, New Series m: Monographs, vol. 50, Springer, Berlin, 1997.
3. T. Aubin Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire J. Math. Pures Appl. (9) 55 3 1976 269-296
4. A.L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer, Berlin, 1987.
5. B. Chow F. Luo Combinatorial Ricci flows on surfaces J. Differential Geom. 63 2003 97-129
6. F.R.K. Chung Spectral graph theory 1997 Published for the Conference Board of the Mathematical Sciences Washington, DC
7. Y. Colin de Verdière Un principe variationnel pour les empilements de cercles Invent. Math. 104 3 1991 655-669
8. D. Cooper I. Rivin Combinatorial scalar curvature and rigidity of ball packings Math. Res. Lett. 3 1 1996 51-60
9. J. Fröhlich, Regge calculus and discretized gravitational functional integrals, Nonperturbative quantum field theory: Mathematical aspects and applications, Selected papers, Advanced Series in Mathematical Physics, vol. 15, World Scientific Publishing Co., River Edge, NJ, 1992, pp. 523-545.
10. D. Glickenstein, A maximum principle for combinatorial Yamabe flow, topology, in press doi: 10.1016/j.top.2005.02.002
11. H.W. Hamber, Simplicial quantum gravity, Phénomènes Critiques, Systèmes Aléatoires, Théories de Jauge, Part I, II (Les Houches, 1984), North-Holland, Amsterdam, 1986, pp. 375-439.
12. H.W. Hamber R.M. Williams Simplicial quantum gravity in three dimensions: Analytical and numerical results Phys. Rev. D (3) 47 2 1993 510-532
13. R.S. Hamilton, The Ricci flow on surfaces, Mathematics and General Relativity (Santa Cruz, CA, 1986), Contemporary Mathematics, vol. 71, American Mathematical Society, Providence, RI, 1988, pp. 237-262.
14. R.S. Hamilton, The formation of singularities in the Ricci flow, Surveys in Differential Geometry, vol. II (Cambridge, MA, 1993), International Press, Cambridge, MA, 1995, pp. 7-136.
15. R.S. Hamilton Non-singular solutions of the Ricci flow on three-manifolds Comm. Anal. Geom. 7 4 1999 695-729
16. Z.-X. He Rigidity of infinite disk patterns Ann. Math. (2) 149 1 1999 1-33
17. A. N. Hirani, Discrete exterior calculus, Ph.D. Thesis, California Institute of Technology, Pasadena, CA, May 2003.
18. J.C. Lagarias C.L. Mallows A.R. Wilks Beyond the descartes circle theorem Amer. Math. Monthly 109 4 2002 338-361
19. J.M. Lee T.H. Parker The Yamabe problem Bull. Amer. Math. Soc. (N.S.) 17 1 1987 37-91
20. F. Luo, Combinatorial Yamabe flow on surfaces, Commun. Contemp. Math. 6(5) (2004) 765-780.
21. F.H. Lutz, The manifold page, http://www.math.tu-berlin.de/diskregeom/ stellar
22. F.H. Lutz, Triangulated manifolds with few vertices and vertex-transitive group actions, Berichte aus der Mathematik [Reports from Mathematics], Verlag Shaker, Aachen, 1999; Dissertation, Technischen Universität Berlin, Berlin, 1999.
23. A. Marden, B. Rodin, On Thurston's formulation and proof of Andreev's theorem, Computational Methods and Function Theory, Valparaíso, 1989, Springer, Berlin, 1990, pp. 103-115.
24. M. Meyer, M. Desbrun, P. Schröder, A.H. Barr, Discrete differential geometry operators for triangulated 2-manifolds, Visualization and Mathematics III, Math. Vis., Springer, Berlin, 2003, pp. 35-57.
25. J. Milnor, The Schläfli differential equality, Collected Papers, vol. 1, Publish or Perish Inc., Houston, TX, 1994.
26. D. Pedoe Geometry, a Comprehensive, Course second ed. 1988 Dover Publications New York
27. T. Regge General relativity without coordinates Nuovo Cimento (10) 19 1961 558-571
28. R. Schoen Conformal deformation of a Riemannian metric to constant scalar curvature J. Differential Geom. 20 2 1984 479-495
29. W.P. Thurston, The geometry and topology of 3-manifolds, Princeton University Mathematics Department Notes, 1980 (Chapter 13).
30. Y. Wang, X. Gu, T.F. Chan, P.M. Thompson, S.-T. Yau, Intrinsic brain surface conformal mapping using a variational method, Proc. SPIE Int. Soc. Opt. Eng. 5370 (2004) 241-251.
31. R. Ye Global existence and convergence of Yamabe flow J. Differential Geom. 39 1 1994 35-50