Geometriae Dedicata

, Volume 158, Issue 1, pp 397–411 | Cite as

Minimal immersions of closed surfaces in hyperbolic three-manifolds

Original Paper

Abstract

We study minimal immersions of closed surfaces (of genus g ≥ 2) in hyperbolic three-manifolds, with prescribed data (σ, t α), where σ is a conformal structure on a topological surface S, and α dz 2 is a holomorphic quadratic differential on the surface (S, σ). We show that, for each \({t \in (0,\tau_0)}\) for some τ 0 > 0, depending only on (σ, α), there are at least two minimal immersions of closed surface of prescribed second fundamental form Re(t α) in the conformal structure σ. Moreover, for t sufficiently large, there exists no such minimal immersion. Asymptotically, as t → 0, the principal curvatures of one minimal immersion tend to zero, while the intrinsic curvatures of the other blow up in magnitude.

Keywords

Minimal immersion Second fundamental form Mountain pass solution Hyperbolic three-manifolds 

Mathematics Subject Classification (2000)

53C21 53A10 35J62 

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References

  1. 1.
    Ambrosetti A., Rabinowitz P.H.: Dual variational methods in critical points theory and applications. J. Funct. Anal. 14, 349–381 (1973)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Colding T.H., Minicozzi W.P. II: The space of embedded minimal surfaces of fixed genus in a 3-manifold. I. Estimates off the axis for disks. Ann. Math. (2) 160(1), 27–68 (2004)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Colding T.H., Minicozzi W.P. II: The space of embedded minimal surfaces of fixed genus in a 3-manifold. II. Multi-valued graphs in disks. Ann. Math. (2) 160(1), 69–92 (2004)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Colding, T.H., Minicozzi, W.P. II.: The space of embedded minimal surfaces of fixed genus in a 3-manifold. III. Planar domains.. Ann. Math. (2) 160(2), 523–572 (2004)Google Scholar
  5. 5.
    Colding T.H., Minicozzi W.P. II: The space of embedded minimal surfaces of fixed genus in a 3-manifold. IV. Locally simply connected. Ann. Math. (2) 160(2), 573–615 (2004)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Freedman M., Hass J., Scott P.: Least area incompressible surfaces in 3-manifolds. Invent. Math. 71(3), 609–642 (1983)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Guo R., Huang Z., Wang B.: Quasi-Fuchsian 3-manifolds and metrics on Teichmüller space. Asian J. Math. 14(2), 243–256 (2010)MathSciNetMATHGoogle Scholar
  8. 8.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. 2nd ed., Grundlehren der Mathematischen Wissenschaften. Fundamental principles of mathematical sciences, vol. 224, Springer, Berlin (1983)Google Scholar
  9. 9.
    Hass, J.: Minimal surfaces and the topology of 3-manifolds. Global theory of minimal surfaces, Clay Math. Proc. Am. Math. Soc., vol. 2, Providence, RI, pp. 705–724 (2005)Google Scholar
  10. 10.
    Hopf, H.: Differential geometry in the large. Lecture Notes in Mathematics, vol. 1000, Springer, Berlin (1989)Google Scholar
  11. 11.
    Huang, Z., Wang, B.: On almost Fuchsian manifolds. preprint (2011)Google Scholar
  12. 12.
    Krasnov K., Schlenker J.-M.: Minimal surfaces and particles in 3-manifolds. Geom. Dedicata 126, 187–254 (2007)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Krasnov K., Schlenker J.-M.: On the renormalized volume of hyperbolic 3-manifolds. Com. Math. Phy. 279(3), 637–668 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Kazdan J.L., Warner F.W.: Curvature functions for compact 2-manifolds. Ann. Math. (2) 99, 14–47 (1974)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Kazdan J.L., Warner F.W.: Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures. Ann. Math. (2) 101, 317–331 (1975)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Lawson H.B. Jr: Complete minimal surfaces in S 3. Ann. Math. 92, 335–374 (1970)MATHCrossRefGoogle Scholar
  17. 17.
    McMullen C.T.: The moduli space of Riemann surfaces is Kähler hyperbolic. Ann. Math. (2) 151(1), 327–357 (2000)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Meeks W.H. III, Yau S.-T.: The classical plateau problem and the topology of 3-dimensional manifolds. Topology 21(4), 409–442 (1982)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Rubinstein, J.H.: Minimal surfaces in geometric 3-manifolds. Global theory of minimal surfaces, Clay Math. Proc. Am. Math. Soc., vol. 2, Providence, RI, pp. 725–746Google Scholar
  20. 20.
    Struwe, M.: Variational methods. 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Applications to nonlinear partial differential equations and Hamiltonian systems, vol. 34, Springer, Berlin (2000)Google Scholar
  21. 21.
    Sacks J., Uhlenbeck K.: Minimal immersions of closed Riemann surfaces. Trans. Am. Math. Soc. 271(2), 639–652 (1982)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Schoen R., Yau S.-T.: Existence of incompressible minimal surfaces and the topology of 3-dimensional manifolds with nonnegative scalar curvature. Ann. Math. (2) 110(1), 127–142 (1979)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Taubes, C.H.: Minimal surfaces in germs of hyperbolic 3-manifolds. In: Proceedings of the Casson Fest, Geom. Topol. Monogr. pp. 69–100, Geom. Topol. Publ., Coventry. vol. 7 (2004) (electronic)Google Scholar
  24. 24.
    Uhlenbeck, K.K.: Closed minimal surfaces in hyperbolic 3-manifolds. Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, pp. 147–168, Princeton University Press, NJ (1983)Google Scholar
  25. 25.
    Wolpert S.A.: A generalization of the Ahlfors-Schwarz lemma. Proc. Am. Math. Soc. 84(3), 377–378 (1982)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.College of Staten Island, The City University of New YorkStaten IslandUSA

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