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On finiteness of the number of boundary slopes of immersed surfaces in 3-manifolds

Authors: Joel Hass, Shicheng Wang and Qing Zhou
Journal: Proc. Amer. Math. Soc. 130 (2002), 1851-1857
MSC (1991): Primary 57N10; Secondary 57M50, 53A10
Published electronically: October 23, 2001
MathSciNet review: 1887034
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Abstract: For any hyperbolic 3-manifold $M$ with totally geodesic boundary, there are finitely many boundary slopes for essential immersed surfaces of a given genus. There is a uniform bound for the number of such boundary slopes if the genus of $\partial M$ is bounded from above.

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  • 1. C. Adams, Volumes of n-cusped hyperbolic 3-manifolds, J. London Math. Soc. 1988, 38, 2, 555-565. MR 89k:22020
  • 2. I. Agol, Topology of Hyperbolic 3-manifolds, Ph.D. thesis, UCSD, 1998.
  • 3. M. Baker, On the boundary slopes of immersed incompressible surfaces, Ann. Inst Fourier (Grenoble) 46, 1443-1449, (1996). MR 98a:57023
  • 4. A. Basmajian, Tubular neighborhoods of totally geodesic hypersurfaces in hyperbolic manifolds, Invent. Math. 117 (1994), 207-225. MR 95c:57020
  • 5. P. Buser, The collar theorem and examples, Manuscripta Math. 25 (1978), 349-357. MR 80h:53046
  • 6. M. Berger and B. Gostiaux, Differential Geometry: Manifolds, curves and surfaces, GTM 115, Springer Verlag, Berlin, New York, 1988. MR 88h:53001
  • 7. C. Gordon, Dehn surgery on knots, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 631-642, Math. Soc. Japan, Tokyo, 1991. MR 93e:57006
  • 8. J. Hass, H. Rubinstein and S.C.Wang, Immersed surfaces in 3-manifolds, J. Differential Geom. 52 (1999), 303-325. CMP 2000:12
  • 9. A. Hatcher, On the boundary curves of incompressible surfaces, Pacific J. Math. 99 (1982), 373-377. MR 83h:57016
  • 10. S. Kojima and Y. Miyamoto, The smallest hyperbolic $3$-manifolds with totally geodesic boundary, J. Differential Geom. 34 (1991), 175-192. MR 92f:57019
  • 11. J. Luecke, Dehn surgery on knots in $S^3$, Proc. ICM Vol 2 (Zurich, 1994), 585-594. MR 97h:57019
  • 12. W. Meeks III and S.T. Yau, Topology of three-dimensional manifolds and the embedding problems in minimal surface theory, Ann. of Math. (2) 112 (1980), no. 3, 441-484. MR 83d:53045
  • 13. C. McMullen, Complex dynamics and renormalization, Ann. Math. Studies, No. 135, Princeton University Press, 1994. MR 96b:58097
  • 14. U. Oertel, Boundaries of $\pi_1$-injective surfaces, Topology Appl. 78 (1997), 215-234. MR 98f:57029
  • 15. M. Scharlemann and Y. Wu, Hyperbolic manifolds and degenerating handle additions, J. Aust. Math. Soc. (Series A) 55 (1993), 72-89. MR 94e:57019
  • 16. P. Shalen, Representations of 3-manifold groups and its application to topology, Proc. ICM Berkeley, (1986), 607-614. MR 89d:57019
  • 17. M. Spivak, A comprehensive Introduction to Differential Geometry, Vol. 4, Publish or Perish, Inc., Berkeley, 1979. MR 82g:53003d
  • 18. R. Schoen and S.T. Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), 127-142. MR 81k:58029
  • 19. W. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. AMS 6 (1982), 357-388. MR 83h:57019
  • 20. W. Thurston, Geometry and Topology of 3-manifolds, Princeton University Lecture Notes, 1978.

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Additional Information

Joel Hass
Affiliation: Department of Mathematics, University of California, Davis, California 95616

Shicheng Wang
Affiliation: Department of Mathematics, Peking University, Beijing 100871, People’s Republic of China

Qing Zhou
Affiliation: Department of Mathematics, East China Normal University, Shanghai, 200062, People’s Republic of China

Keywords: Boundary slopes, three-dimensional topology, essential surfaces
Received by editor(s): September 2, 1999
Received by editor(s) in revised form: December 28, 2000
Published electronically: October 23, 2001
Additional Notes: The first author was partially supported by NSF grant DMS-9704286.
The second and third authors were partially supported by MSTC and Outstanding Youth Fellowships of NSFC
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2001 American Mathematical Society

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