On the geometric and topological rigidity
of hyperbolic 3-manifolds
- [A] Michael T. Anderson, Complete minimal varieties in hyperbolic space, Invent. Math. 69 (1982), no. 3, 477–494. MR 679768, https://doi.org/10.1007/BF01389365
- [BS] F. Bonahon and L. Siebenmann, to appear.
- [EM] D. B. A. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 113–253. MR 903852
- [F] Werner Fenchel, Elementary geometry in hyperbolic space, De Gruyter Studies in Mathematics, vol. 11, Walter de Gruyter & Co., Berlin, 1989. With an editorial by Heinz Bauer. MR 1004006
- [FH] M. H. Freedman and He, personal communication.
T. Farrell and L.
E. Jones, A topological analogue of
Mostow’s rigidity theorem, J. Amer. Math.
Soc. 2 (1989), no. 2, 257–370. MR
F. T. Farrell and L. E. Jones, Compact negatively curved manifolds (of dim ≠3,4) are topologically rigid, Proc. Nat. Acad. Sci. U.S.A. 86 (1989), no. 10, 3461–3463. MR 997635, https://doi.org/10.1073/pnas.86.10.3461
F. T. Farrell and L. E. Jones, Rigidity and other topological aspects of compact nonpositively curved manifolds, Bull. Amer. Math. Soc. (N.S.) 22 (1990), no. 1, 59–64. MR 1001606, https://doi.org/10.1090/S0273-0979-1990-15839-2
- [G1] David Gabai, Foliations and the topology of 3-manifolds, J. Differential Geom. 18 (1983), no. 3, 445–503. MR 723813
- [G2] David Gabai, Foliations and 3-manifolds, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 609–619. MR 1159248
- [G3] David Gabai, Homotopy hyperbolic 3-manifolds are virtually hyperbolic, J. Amer. Math. Soc. 7 (1994), no. 1, 193–198. MR 1205445, https://doi.org/10.1090/S0894-0347-1994-1205445-3
- [GO] David Gabai and Ulrich Oertel, Essential laminations in 3-manifolds, Ann. of Math. (2) 130 (1989), no. 1, 41–73. MR 1005607, https://doi.org/10.2307/1971476
- [Gr] Michael Gromov, Hyperbolic manifolds (according to Thurston and Jørgensen), Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., vol. 842, Springer, Berlin-New York, 1981, pp. 40–53. MR 636516
- [GS] Robert Gulliver and Peter Scott, Least area surfaces can have excess triple points, Topology 26 (1987), no. 3, 345–359. MR 899054, https://doi.org/10.1016/0040-9383(87)90006-1
- [HS] Joel Hass and Peter Scott, The existence of least area surfaces in 3-manifolds, Trans. Amer. Math. Soc. 310 (1988), no. 1, 87–114. MR 965747, https://doi.org/10.1090/S0002-9947-1988-0965747-6
- [Ki] J. M. Kister, Isotopies in 3-manifolds, Trans. Amer. Math. Soc. 97 (1960), 213–224. MR 0120628, https://doi.org/10.1090/S0002-9947-1960-0120628-5
- [L1] Urs Lang, Quasi-minimizing surfaces in hyperbolic space, Math. Z. 210 (1992), no. 4, 581–592. MR 1175723, https://doi.org/10.1007/BF02571815
- [L2] Urs Lang, The existence of complete minimizing hypersurfaces in hyperbolic manifolds, Internat. J. Math. 6 (1995), no. 1, 45–58. MR 1307303, https://doi.org/10.1142/S0129167X95000055
- [MSY] William Meeks III, Leon Simon, and Shing Tung Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621–659. MR 678484, https://doi.org/10.2307/2007026
- [Me] Robert Meyerhoff, A lower bound for the volume of hyperbolic 3-manifolds, Canad. J. Math. 39 (1987), no. 5, 1038–1056. MR 918586, https://doi.org/10.4153/CJM-1987-053-6
- [Mo] G. D. Mostow, Quasi-conformal mappings in 𝑛-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53–104. MR 0236383
- [Mor] C. B. Morrey, The Problem of Plateau in a Riemannian Manifold, Ann. Math (2) 49 (1948), 807-851. MR 10:259f
- [Mu] James Munkres, Obstructions to the smoothing of piecewise-differentiable homeomorphisms, Ann. of Math. (2) 72 (1960), 521–554. MR 0121804, https://doi.org/10.2307/1970228
- [Ne] M. H. A. Neumann, Quart. J. Math. 2 (1931), 1-8.
- [S] Richard Schoen, Estimates for stable minimal surfaces in three-dimensional manifolds, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 111–126. MR 795231
- [Th] William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, https://doi.org/10.1090/S0273-0979-1982-15003-0
- [W] Friedhelm Waldhausen, On irreducible 3-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 0224099, https://doi.org/10.2307/1970594
- [We] J. Weeks, SnapPea, undistributed version.
- M. Anderson, Complete Minimal Varieties in Hyperbolic Space, Invent. Math. 69 (1982), 477-494. MR 84c:53005
- F. Bonahon and L. Siebenmann, to appear.
- D. B. A. Epstein and A. Marden, Convex Hulls in Hyperbolic Space, a Theorem of Sullivan and Measured Pleated Surfaces, LMS Lect. Notes 111 (1984), 113-255. MR 89c:52014
- W. Fenchel, Elementary Geometry in Hyperbolic Space, de Gruyter Stud. in Math. 11 (1989). MR 91a:51009
- M. H. Freedman and He, personal communication.
- F. T. Farrell and L. Jones, A Topological analogue of Mostow's Rigidity Theorem, J. Amer. Math. Soc. 2 (1989), 257-370. MR 90h:57023a
- D. Gabai, Foliations and the Topology of 3-manifolds, J. Diff. Geom. 18 (1983), 445-503. MR 86a:57009
- -, Foliations and 3-manifolds, Proc. ICM Kyoto-1990 1 (1991), 609-619. MR 93d:57013
- -, Homotopy Hyperbolic 3-manifolds are Virtually Hyperbolic, JAMS 7 (1994), 193-198. MR 94b:57016
- D. Gabai and U. Oertel, Essential Laminations in 3-manifolds, Ann. of Math. (2) 130 (1989), 41-73. MR 90h:57012
- M. Gromov, Hyperbolic Manifolds According to Thurston and Jorgensen, Sem. Bourbaki 32 (1979), 40-52. MR 84b:53046
- R. Gulliver and P. Scott, Least Area Surfaces Can Have Excess Triple Points, Topology 26 (1987), 345-359. MR 88k:57018
- J. Hass and P. Scott, The Existence of Least Area Surfaces in 3-manifolds, Trans. AMS 310 (1988), 87-114. MR 90c:53022
- J. M. Kister, Isotopies in 3-manifolds, Trans. AMS 97 (1960), 213-224. MR 22:11378
- U. Lang, Quasi-minimizing Surfaces in Hyperbolic Space, Math. Zeit. 210 (1992), 581-592. MR 93e:53008
- -, The Existence of Complete Minimizing Hypersurfaces in Hyperbolic Manifolds, Int. J. Math. 6 (1995), 45-58. MR 95i:58053
- W. H. Meeks III, L. Simon, S. T. Yau, Embedded Minimal Surfaces, Exotic Spheres, and Manifolds with Positive Ricci Curvature, Ann. of Math (2) 91 (1982), 621-659. MR 84f:53053
- R. Meyerhoff, A Lower Bound for the Volume of Hyperbolic 3-manifolds, Can. J. Math. 39 (1987), 1038-1056. MR 88k:57049
- G. D. Mostow, Quasiconformal Mappings in n-Space and the Rigidity of Hyperbolic Space Forms, Pub. IHES 34 (1968), 53-104. MR 38:4679
- C. B. Morrey, The Problem of Plateau in a Riemannian Manifold, Ann. Math (2) 49 (1948), 807-851. MR 10:259f
- J. R. Munkres, Obstructions to Smoothing Piecewise Differentiable Homeomorphisms, Ann. Math (2) 72 (1960), 521-554. MR 22:12534
- M. H. A. Neumann, Quart. J. Math. 2 (1931), 1-8.
- R. Schoen, Estimates for Stable Minimal Surfaces in Three Dimensional Manifolds, Ann. of Math. Stud. 103 (1983), 111-126. MR 86j:53094
- William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357-381. MR 83h:57019
- F. Waldhausen, On Irreducible 3-manifolds which are Sufficiently Large, Annals of Math. 87 (1968), 56-88. MR 36:7146
- J. Weeks, SnapPea, undistributed version.
Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 57M50
Retrieve articles in all journals with MSC (1991): 57M50
Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
Received by editor(s): October 1, 1993
Received by editor(s) in revised form: September 1, 1995
Additional Notes: Partially supported by NSF Grants DMS-8902343, DMS-9200584, DMS-9505253 and SERC grant GR/H60851.
Article copyright: © Copyright 1997 American Mathematical Society