Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 


Counting minimal surfaces in quasi-Fuchsian three-manifolds

Authors: Zheng Huang and Biao Wang
Journal: Trans. Amer. Math. Soc. 367 (2015), 6063-6083
MSC (2010): Primary 53A10; Secondary 57M05
Published electronically: April 9, 2015
MathSciNet review: 3356929
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is well known that every quasi-Fuchsian manifold admits at least one closed incompressible minimal surface, and at most finitely many stable ones. In this paper, for any prescribed integer $ N > 0$, we construct a quasi-Fuchsian manifold which contains at least $ 2^N$ such minimal surfaces. As a consequence, there exists some simple closed Jordan curve on $ S_{\infty }^2$ such that there are at least $ 2^N$ disk-type complete minimal surfaces in $ \mathbb{H}^3$ sharing this Jordan curve as the asymptotic boundary.

References [Enhancements On Off] (What's this?)

  • [And83] Michael T. Anderson, Complete minimal hypersurfaces in hyperbolic $ n$-manifolds, Comment. Math. Helv. 58 (1983), no. 2, 264-290. MR 705537 (85e:53076),
  • [AS79] Frederick J. Almgren Jr. and Leon Simon, Existence of embedded solutions of Plateau's problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6 (1979), no. 3, 447-495. MR 553794 (81d:49025)
  • [Bea95] Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1995. Corrected reprint of the 1983 original. MR 1393195 (97d:22011)
  • [Ber72] Lipman Bers, Uniformization, moduli, and Kleinian groups, Bull. London Math. Soc. 4 (1972), 257-300. MR 0348097 (50 #595)
  • [BSE10] Pierre Bérard and Ricardo Sa Earp, Lindelöf's theorem for hyperbolic catenoids, Proc. Amer. Math. Soc. 138 (2010), no. 10, 3657-3669. MR 2661564 (2011e:53087),
  • [Cos09] Baris Coskunuzer, Asymptotic Plateau problem, ArXiv e-print, arXiv:0907.0552 (2009).
  • [FHS83] Michael Freedman, Joel Hass, and Peter Scott, Least area incompressible surfaces in $ 3$-manifolds, Invent. Math. 71 (1983), no. 3, 609-642. MR 695910 (85e:57012),
  • [FK65] Robert Fricke and Felix Klein, Vorlesungen über die Theorie der automorphen Funktionen. Band 1: Die gruppentheoretischen Grundlagen. Band II: Die funktionentheoretischen Ausführungen und die Andwendungen, Bibliotheca Mathematica Teubneriana, Bände 3, vol. 4, Johnson Reprint Corp., New York, 1965 (German). MR 0183872 (32 #1348)
  • [GHW10] Ren Guo, Zheng Huang, and Biao Wang, Quasi-Fuchsian 3-manifolds and metrics on Teichmüller space, Asian J. Math. 14 (2010), no. 2, 243-256. MR 2746123 (2011k:32015)
  • [Gom87] Jonas de Miranda Gomes, Spherical surfaces with constant mean curvature in hyperbolic space, Bol. Soc. Brasil. Mat. 18 (1987), no. 2, 49-73. MR 1018445 (90h:53009),
  • [GS00] Bo Guan and Joel Spruck, Hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity, Amer. J. Math. 122 (2000), no. 5, 1039-1060. MR 1781931 (2001j:53069)
  • [GW07] William M. Goldman and Richard A. Wentworth, Energy of twisted harmonic maps of Riemann surfaces, In the tradition of Ahlfors-Bers. IV, Contemp. Math., vol. 432, Amer. Math. Soc., Providence, RI, 2007, pp. 45-61. MR 2342806 (2008m:58029),
  • [Has05] Joel Hass, Minimal surfaces and the topology of three-manifolds, Global theory of minimal surfaces, Clay Math. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 2005, pp. 705-724. MR 2167285 (2006d:53004)
  • [HL87] Robert Hardt and Fang-Hua Lin, Regularity at infinity for area-minimizing hypersurfaces in hyperbolic space, Invent. Math. 88 (1987), no. 1, 217-224. MR 877013 (88m:49033),
  • [HL12] Zheng Huang and Marcello Lucia, Minimal immersions of closed surfaces in hyperbolic three-manifolds, Geom. Dedicata 158 (2012), 397-411. MR 2922723,
  • [HW13] Zheng Huang and Biao Wang, On almost-Fuchsian manifolds, Trans. Amer. Math. Soc. 365 (2013), no. 9, 4679-4698. MR 3066768,
  • [KS07] Kirill Krasnov and Jean-Marc Schlenker, Minimal surfaces and particles in 3-manifolds, Geom. Dedicata 126 (2007), 187-254. MR 2328927 (2009c:53076),
  • [Leh64] Joseph Lehner, Discontinuous groups and automorphic functions, Mathematical Surveys, No. VIII, American Mathematical Society, Providence, R.I., 1964. MR 0164033 (29 #1332)
  • [Mag74] Wilhelm Magnus, Noneuclidean tesselations and their groups, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Pure and Applied Mathematics, Vol. 61. MR 0352287 (50 #4774)
  • [Mar74] Albert Marden, The geometry of finitely generated kleinian groups, Ann. of Math. (2) 99 (1974), 383-462. MR 0349992 (50 #2485)
  • [Mor48] Charles B. Morrey Jr., The problem of Plateau on a Riemannian manifold, Ann. of Math. (2) 49 (1948), 807-851. MR 0027137 (10,259f)
  • [Mor81] Hiroshi Mori, Minimal surfaces of revolution in $ H^{3}$ and their global stability, Indiana Univ. Math. J. 30 (1981), no. 5, 787-794. MR 625602 (82k:53082),
  • [MSW02] David Mumford, Caroline Series, and David Wright, Indra's pearls, Cambridge University Press, New York, 2002. The vision of Felix Klein. MR 1913879 (2003f:00005)
  • [MT98] Katsuhiko Matsuzaki and Masahiko Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1998. Oxford Science Publications. MR 1638795 (99g:30055)
  • [MY82a] William H. Meeks III and Shing Tung Yau, The classical Plateau problem and the topology of three-dimensional manifolds. The embedding of the solution given by Douglas-Morrey and an analytic proof of Dehn's lemma, Topology 21 (1982), no. 4, 409-442. MR 670745 (84g:53016),
  • [MY82b] William W. Meeks III and Shing Tung Yau, The existence of embedded minimal surfaces and the problem of uniqueness, Math. Z. 179 (1982), no. 2, 151-168. MR 645492 (83j:53060),
  • [Poi83] Henri Poincaré, Mémoire sur les groupes kleinéens, Acta Math. 3 (1883), no. 1, 49-92.
  • [Poi85] -, Papers on Fuchsian functions, Springer-Verlag, New York, 1985.
  • [Rol90] Dale Rolfsen, Knots and links, Mathematics Lecture Series, vol. 7, Publish or Perish Inc., Houston, TX, 1990. Corrected reprint of the 1976 original. MR 1277811 (95c:57018)
  • [Rub05] J. Hyam Rubinstein, Minimal surfaces in geometric 3-manifolds, Global theory of minimal surfaces, Clay Math. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 2005, pp. 725-746. MR 2167286 (2006g:57038)
  • [Rub07] J. H. Rubinstein, Problems around 3-manifolds, Workshop on Heegaard Splittings, Geom. Topol. Monogr., vol. 12, Geom. Topol. Publ., Coventry, 2007, pp. 285-298. MR 2408251 (2009i:57050),
  • [Sco83] Peter Scott, The geometries of $ 3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401-487. MR 705527 (84m:57009),
  • [Seo11] Keomkyo Seo, Stable minimal hypersurfaces in the hyperbolic space, J. Korean Math. Soc. 48 (2011), no. 2, 253-266. MR 2789454 (2012c:53096),
  • [SU82] J. Sacks and K. Uhlenbeck, Minimal immersions of closed Riemann surfaces, Trans. Amer. Math. Soc. 271 (1982), no. 2, 639-652. MR 654854 (83i:58030),
  • [SY79] R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127-142. MR 541332 (81k:58029),
  • [Tau04] Clifford Henry Taubes, Minimal surfaces in germs of hyperbolic 3-manifolds, Proceedings of the Casson Fest, Geom. Topol. Monogr., vol. 7, Geom. Topol. Publ., Coventry, 2004, pp. 69-100 (electronic). MR 2172479 (2007a:53157),
  • [Uhl83] Karen K. Uhlenbeck, Closed minimal surfaces in hyperbolic $ 3$-manifolds, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 147-168. MR 795233 (87b:53093)
  • [Wan12a] Biao Wang, Least area spherical catenoids in hyperbolic three-dimensional space, Preprint (2012).
  • [Wan12b] Biao Wang, Minimal surfaces in quasi-Fuchsian 3-manifolds, Math. Ann. 354 (2012), no. 3, 955-966. MR 2983075,

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53A10, 57M05

Retrieve articles in all journals with MSC (2010): 53A10, 57M05

Additional Information

Zheng Huang
Affiliation: Department of Mathematics, The City University of New York, Staten Island, New York 10314; The Graduate Center, The City University of New York, 365 Fifth Avenue, New York, New York 10016

Biao Wang
Affiliation: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459
Address at time of publication: Department of Mathematics and Computer Science, Queensborough Community College, City University of New York, 222-05 56th Avenue, Bayside, New York 11364

Received by editor(s): September 10, 2012
Received by editor(s) in revised form: February 24, 2013, and April 10, 2013
Published electronically: April 9, 2015
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society