Mathematische Annalen

, Volume 354, Issue 3, pp 955–966

Minimal surfaces in quasi-Fuchsian 3-manifolds



In this paper, we prove that if a quasi-Fuchsian 3-manifold M contains a closed geodesic with complex length \({\fancyscript {L} = l + i\theta}\) such that \({|\theta|/l \gg 1}\) , where l > 0 and −π ≤ θπ, then it contains at least two incompressible minimal surfaces near the geodesic.

Mathematics Subject Classification (2000)

Primary 53A10 Secondary 57M05 


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  1. 1.
    Anderson M.T.: Complete minimal hypersurfaces in hyperbolic n-manifolds. Comment. Math. Helvetici 58(2), 264–290 (1983)MATHCrossRefGoogle Scholar
  2. 2.
    Calegari D., Gabai D.: Shrinkwrapping and the taming of hyperbolic 3-manifolds. J. Am. Math. Soc. 19(2), 385–446 (2006)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Meeks W. III, Simon L., Yau S.T.: Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature. Ann. Math. 116(3), 621–659 (1982)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Meeks W. III, Yau S.T.: 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(4), 409–442 (1982)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Meeks W. III, Yau S.T.: The existence of embedded minimal surfaces and the problem of uniqueness. Math. Z. 179(2), 151–168 (1982)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Meyerhoff R.: A lower bound for the volume of hyperbolic 3-manifolds. Canad. J. Math. 39(5), 1038–1056 (1987)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Mori H.: On surfaces of right helicoid type in H 3. Bol. Soc. Brasil. Mat. 13(2), 57–62 (1982)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Morrey C.B. Jr: The problem of plateau on a Riemannian manifold. Ann. Math. 49, 807–851 (1948)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Otal, J.-P.: Les géodésiques fermées d’une variété hyperbolique en tant que nœuds. In: Kleinian Groups and Hyperbolic 3-manifolds (Warwick, 2001), London Mathematical Society Lecture Note Series, vol. 299, pp. 95–104. Cambridge University Press, Cambridge (2003)Google Scholar
  10. 10.
    Schoen R., Yau S.T.: Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature. Ann. Math. 110(1), 127–142 (1979)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Schoen R., Yau S.T.: Lectures on Differential Geometry. International Press, Cambridge (1994)MATHGoogle Scholar
  12. 12.
    Tuzhilin, A.A.: Global properties of minimal surfaces in \({{\mathbb R}^3}\) and \({{\mathbb H}^3}\) and their Morse type indices. In: Minimal Surfaces, Advances in Soviet Mathematics, vol. 15, pp. 193–233. American Mathematical Society, Providence (1993)Google Scholar
  13. 13.
    Uhlenbeck, K.K.: Closed minimal surfaces in hyperbolic 3-manifolds. In: Seminar on Minimal Submanifolds, Annals of Mathematics Studies, vol. 103, pp. 147–168. Princeton University Press, PrincetonGoogle Scholar

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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsCentral Connecticut State UniversityNew BritainUSA

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