On the geometric and topological rigidity

of hyperbolic 3-manifolds

Author:
David Gabai

Journal:
J. Amer. Math. Soc. **10** (1997), 37-74

MSC (1991):
Primary 57M50

DOI:
https://doi.org/10.1090/S0894-0347-97-00206-3

MathSciNet review:
1354958

Full-text PDF

References | Similar Articles | Additional Information

**[A]**M. Anderson,*Complete Minimal Varieties in Hyperbolic Space*, Invent. Math.**69**(1982), 477-494. MR**84c:53005****[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*, LMS Lect. Notes**111**(1984), 113-255. MR**89c:52014****[F]**W. Fenchel,*Elementary Geometry in Hyperbolic Space*, de Gruyter Stud. in Math.**11**(1989). MR**91a:51009****[FH]**M. H. Freedman and He, personal communication.**[FJ]**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****[G1]**D. Gabai,*Foliations and the Topology of 3-manifolds*, J. Diff. Geom.**18**(1983), 445-503. MR**86a:57009****[G2]**-,*Foliations and 3-manifolds*, Proc. ICM Kyoto-1990**1**(1991), 609-619. MR**93d:57013****[G3]**-,*Homotopy Hyperbolic 3-manifolds are Virtually Hyperbolic*, JAMS**7**(1994), 193-198. MR**94b:57016****[GO]**D. Gabai and U. Oertel,*Essential Laminations in 3-manifolds*, Ann. of Math. (2)**130**(1989), 41-73. MR**90h:57012****[Gr]**M. Gromov,*Hyperbolic Manifolds According to Thurston and Jorgensen*, Sem. Bourbaki**32**(1979), 40-52. MR**84b:53046****[GS]**R. Gulliver and P. Scott,*Least Area Surfaces Can Have Excess Triple Points*, Topology**26**(1987), 345-359. MR**88k:57018****[HS]**J. Hass and P. Scott,*The Existence of Least Area Surfaces in 3-manifolds*, Trans. AMS**310**(1988), 87-114. MR**90c:53022****[Ki]**J. M. Kister,*Isotopies in 3-manifolds*, Trans. AMS**97**(1960), 213-224. MR**22:11378****[L1]**U. Lang,*Quasi-minimizing Surfaces in Hyperbolic Space*, Math. Zeit.**210**(1992), 581-592. MR**93e:53008****[L2]**-,*The Existence of Complete Minimizing Hypersurfaces in Hyperbolic Manifolds*, Int. J. Math.**6**(1995), 45-58. MR**95i:58053****[MSY]**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****[Me]**R. Meyerhoff,*A Lower Bound for the Volume of Hyperbolic 3-manifolds*, Can. J. Math.**39**(1987), 1038-1056. MR**88k:57049****[Mo]**G. D. Mostow,*Quasiconformal Mappings in n-Space and the Rigidity of Hyperbolic Space Forms*, Pub. IHES**34**(1968), 53-104. MR**38:4679****[Mor]**C. B. Morrey,*The Problem of Plateau in a Riemannian Manifold*, Ann. Math (2)**49**(1948), 807-851. MR**10:259f****[Mu]**J. R. Munkres,*Obstructions to Smoothing Piecewise Differentiable Homeomorphisms*, Ann. Math (2)**72**(1960), 521-554. MR**22:12534****[Ne]**M. H. A. Neumann, Quart. J. Math.**2**(1931), 1-8.**[S]**R. Schoen,*Estimates for Stable Minimal Surfaces in Three Dimensional Manifolds*, Ann. of Math. Stud.**103**(1983), 111-126. MR**86j:53094****[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**83h:57019****[W]**F. Waldhausen,*On Irreducible 3-manifolds which are Sufficiently Large*, Annals of Math.**87**(1968), 56-88. MR**36:7146****[We]**J. Weeks,*SnapPea*, undistributed version.

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Additional Information

**David Gabai**

Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, California 91125

Email:
Gabai@cco.caltech.edu

DOI:
https://doi.org/10.1090/S0894-0347-97-00206-3

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