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Arkiv för Matematik

, Volume 52, Issue 1, pp 61–92 | Cite as

The Chern–Osserman inequality for minimal surfaces in a Cartan–Hadamard manifold with strictly negative sectional curvatures

  • Antonio Esteve
  • Vicente PalmerEmail author
Article
  • 142 Downloads

Abstract

We state and prove a Chern–Osserman-type inequality in terms of the volume growth for minimal surfaces S which have finite total extrinsic curvature and are properly immersed in a Cartan–Hadamard manifold N with sectional curvatures bounded from above by a negative quantity K N b<0 and such that they are not too curved (on average) with respect to the hyperbolic space with constant sectional curvature given by the upper bound b. We also prove the same Chern–Osserman-type inequality for minimal surfaces with finite total extrinsic curvature and properly immersed in an asymptotically hyperbolic Cartan–Hadamard manifold N with sectional curvatures bounded from above by a negative quantity K N b<0.

Keywords

Minimal Surface Sectional Curvature Hyperbolic Space Euler Characteristic Warped Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Anderson, M. T., Complete minimal varieties in hyperbolic space, Invent. Math. 69 (1982), 477–494. CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Anderson, M. T., The compactification of a minimal submanifold in Euclidean space by the Gauss map, I.H.E.S. Preprint, Bures-sur-Yvette, 1984. Google Scholar
  3. 3.
    Chen, Q., On the area growth of minimal surfaces in \(\mathbb{H}^{n}\), Geom. Dedicata 75 (1999), 263–273. CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Chen, Q. and Cheng, Y., Chern–Osserman inequality for minimal surfaces in \(\mathbb{H}^{n}\), Proc. Amer. Math. Soc. 128 (1999), 2445–2450. MathSciNetGoogle Scholar
  5. 5.
    Chern, S. S. and Osserman, R., Complete minimal surface in E n, J. Anal. Math. 19 (1967), 15–34. CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    De Oliveira, G., Compactification of minimal submanifolds of hyperbolic space, Comm. Anal. Geom. 1 (1993), 1–29. zbMATHMathSciNetGoogle Scholar
  7. 7.
    Greene, G. and Wu, S., Function Theory on Manifolds Which Posses a Pole, Lecture Notes in Math. 699, Springer, Berlin–Heidelberg, 1979. Google Scholar
  8. 8.
    Grigor′yan, A., Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. 36 (1999), 135–249. CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Hurtado, A., Markvorsen, S. and Palmer, V., Torsional rigidity of submanifolds with controlled geometry, Math. Ann. 344 (2009), 511–542. CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Hurtado, A. and Palmer, V., A note on the p-parabolicity of submanifolds, Potential Anal. 34 (2011), 101–118. CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Jorge, L. P. and Meeks, W. H., The topology of minimal surfaces of finite total Gaussian curvature, Topology 122 (1983), 203–221. CrossRefMathSciNetGoogle Scholar
  12. 12.
    Markvorsen, S. and Min-Oo, M., Global Riemannian Geometry: Curvature and Topology, Advanced Courses in Mathematics—CRM Barcelona, Birkhäuser, Basel, 2003. CrossRefGoogle Scholar
  13. 13.
    Markvorsen, S. and Palmer, V., The relative volume growth of minimal submanifolds, Arch. Math. (Basel) 79 (2002), 507–514. CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Markvorsen, S. and Palmer, V., Generalized isoperimetric inequalities for extrinsic balls in minimal submanifolds, J. Reine Angew. Math. 551 (2002), 101–121. zbMATHMathSciNetGoogle Scholar
  15. 15.
    Markvorsen, S. and Palmer, V., Torsional rigidity of minimal submanifolds, Proc. Lond. Math. Soc. 93 (2006), 253–272. CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Markvorsen, S. and Palmer, V., Extrinsic isoperimetric analysis on submanifolds with curvatures bounded from below, J. Geom. Anal. 20 (2010), 388–421. CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Milnor, J., Morse Theory, Princeton University Press, Princeton, NJ, 1969. Google Scholar
  18. 18.
    O’Neill, B., Semi-Riemannian Geometry, Academic Press, New York, 1983. zbMATHGoogle Scholar
  19. 19.
    Palmer, V., Isoperimetric inequalities for extrinsic balls in minimal submanifolds and their applications, J. Lond. Math. Soc. 60 (1999), 607–616. CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Palmer, V., On deciding whether a submanifold is parabolic of hyperbolic using its mean curvature, in Simon Stevin Transactions on Geometry 1, pp. 131–159, Simon Stevin Institute for Geometry, Tilburg, 2010. Google Scholar
  21. 21.
    Sakai, T., Riemannian Geometry, Translations of Mathematical Monographs 149, Amer. Math. Soc., Providence, RI, 1996. zbMATHGoogle Scholar
  22. 22.
    Shi, Y. and Tian, G., Rigidity of asymptotically hyperbolic manifolds, Comm. Math. Phys. 259 (2005), 545–559. CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    White, B., Complete surfaces of finite total curvature, J. Differential Geom. 26 (1987), 315–326. zbMATHMathSciNetGoogle Scholar

Copyright information

© Institut Mittag-Leffler 2013

Authors and Affiliations

  1. 1.Instituto de Enseñanza Secundaria Alfonso VIII Departamento de Análisis Económico y FinanzasUniversidad de Castilla-La ManchaCuencaSpain
  2. 2.Department of Mathematics Institute of New Imaging TechnologiesUniversitat Jaume ICastellónSpain

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