## On the geometric and topological rigidity of hyperbolic 3-manifolds

HTML articles powered by AMS MathViewer

- by David Gabai
- J. Amer. Math. Soc.
**10**(1997), 37-74 - DOI: https://doi.org/10.1090/S0894-0347-97-00206-3
- PDF | Request permission

## References

- Michael T. Anderson,
*Complete minimal varieties in hyperbolic space*, Invent. Math.**69**(1982), no. 3, 477–494. MR**679768**, DOI 10.1007/BF01389365 - 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*, 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** - 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**, DOI 10.1515/9783110849455 - M. H. Freedman and He, personal communication.
- F. 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**973309**, DOI 10.1090/S0894-0347-1989-0973309-4 - David Gabai,
*Foliations and the topology of $3$-manifolds*, J. Differential Geom.**18**(1983), no. 3, 445–503. MR**723813** - 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** - David Gabai,
*Homotopy hyperbolic $3$-manifolds are virtually hyperbolic*, J. Amer. Math. Soc.**7**(1994), no. 1, 193–198. MR**1205445**, DOI 10.1090/S0894-0347-1994-1205445-3 - David Gabai and Ulrich Oertel,
*Essential laminations in $3$-manifolds*, Ann. of Math. (2)**130**(1989), no. 1, 41–73. MR**1005607**, DOI 10.2307/1971476 - 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** - Robert Gulliver and Peter Scott,
*Least area surfaces can have excess triple points*, Topology**26**(1987), no. 3, 345–359. MR**899054**, DOI 10.1016/0040-9383(87)90006-1 - 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**, DOI 10.1090/S0002-9947-1988-0965747-6 - J. M. Kister,
*Isotopies in $3$-manifolds*, Trans. Amer. Math. Soc.**97**(1960), 213–224. MR**120628**, DOI 10.1090/S0002-9947-1960-0120628-5 - Urs Lang,
*Quasi-minimizing surfaces in hyperbolic space*, Math. Z.**210**(1992), no. 4, 581–592. MR**1175723**, DOI 10.1007/BF02571815 - Urs Lang,
*The existence of complete minimizing hypersurfaces in hyperbolic manifolds*, Internat. J. Math.**6**(1995), no. 1, 45–58. MR**1307303**, DOI 10.1142/S0129167X95000055 - 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**, DOI 10.2307/2007026 - Robert Meyerhoff,
*A lower bound for the volume of hyperbolic $3$-manifolds*, Canad. J. Math.**39**(1987), no. 5, 1038–1056. MR**918586**, DOI 10.4153/CJM-1987-053-6 - 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**236383**, DOI 10.1007/BF02684590 - Morgan Ward,
*Ring homomorphisms which are also lattice homomorphisms*, Amer. J. Math.**61**(1939), 783–787. MR**10**, DOI 10.2307/2371336 - James Munkres,
*Obstructions to the smoothing of piecewise-differentiable homeomorphisms*, Ann. of Math. (2)**72**(1960), 521–554. MR**121804**, DOI 10.2307/1970228 - M. H. A. Neumann, Quart. J. Math.
**2**(1931), 1–8. - 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** - 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**, DOI 10.1090/S0273-0979-1982-15003-0 - Friedhelm Waldhausen,
*On irreducible $3$-manifolds which are sufficiently large*, Ann. of Math. (2)**87**(1968), 56–88. MR**224099**, DOI 10.2307/1970594 - J. Weeks,
*SnapPea*, undistributed version.

## Bibliographic Information

**David Gabai**- Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
- MR Author ID: 195365
- Email: Gabai@cco.caltech.edu
- 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.
- © Copyright 1997 American Mathematical Society
- 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