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The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3171–3182 | Cite as

A Gap Theorem for Free Boundary Minimal Surfaces in Geodesic Balls of Hyperbolic Space and Hemisphere

  • Haizhong Li
  • Changwei XiongEmail author
Article
  • 157 Downloads

Abstract

In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the support function of the surface, and a natural potential function in hyperbolic space and hemisphere.

Keywords

Gap theorem Minimal surface Free boundary Hyperbolic space Hemisphere 

Mathematics Subject Classification

53C42 53C20 

Notes

Acknowledgements

The authors would like to thank the referees for careful reading of the paper and for the valuable suggestions and comments which made this paper better and more readable. They wish to thank Professor Ben Andrews for his interest in this work. The second author is also grateful to Professor Jaigyoung Choe for discussions on minimal surfaces in hyperbolic space at the workshop on Nonlinear and Geometric Partial Differential Equations at Kioloa Campus of ANU, 2016, Australia. The first author was supported by NSFC Grant No. 11671224. The second author was supported by a postdoctoral fellowship funded via ARC Laureate Fellowship FL150100126.

References

  1. 1.
    Almgren Jr., F.J.: Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem. Ann. Math. (2) 84, 277–292 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ambrozio, L., Nunes, I.: A gap theorem for free boundary minimal surfaces in the three-ball. arXiv:1608.05689
  3. 3.
    Andrews, B., Li, H.: Embedded constant mean curvature tori in the three-sphere. J. Differ. Geom. 99(2), 169–189 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brendle, S.: A sharp bound for the area of minimal surfaces in the unit ball. Geom. Funct. Anal. 22(3), 621–626 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brendle, S.: Embedded minimal tori in \(S^3\) and the Lawson conjecture. Acta Math. 211(2), 177–190 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brendle, S.: Constant mean curvature surfaces in warped product manifolds. Publ. Math. Inst. Hautes Études Sci. 117, 247–269 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cheng, S.Y.: Eigenfunctions and nodal sets. Comment. Math. Helv. 51(1), 43–55 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chern, S.S., do Carmo, M., Kobayashi, S.: Minimal submanifolds of a sphere with second fundamental form of constant length. In: 1970 Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968), pp. 59–75. Springer, New YorkGoogle Scholar
  9. 9.
    Devyver, B.: Index of the critical catenoid. arXiv:1609.02315
  10. 10.
    do Carmo, M., Dajczer, M.: Rotation hypersurfaces in spaces of constant curvature. Trans. Am. Math. Soc. 277(2), 685–709 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fraser, A., Li, M.: Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary. J. Differ. Geom. 96(2), 183–200 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fraser, A., Schoen, R.: The first Steklov eigenvalue, conformal geometry, and minimal surfaces. Adv. Math. 226(5), 4011–4030 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fraser, A., Schoen, R.: Uniqueness theorems for free boundary minimal disks in space forms. Int. Math. Res. Not. 2015(17), 8268–8274 (2015)Google Scholar
  14. 14.
    Fraser, A., Schoen, R.: Sharp eigenvalue bounds and minimal surfaces in the ball. Invent. Math. 203(3), 823–890 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lawson Jr., H.B.: Local rigidity theorems for minimal hypersurfaces. Ann. Math. (2) 89, 187–197 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Li, H., Wei, Y., Xiong, C.: A note on Weingarten hypersurfaces in the warped product manifold, Internat. J. Math. 25(14), 1450121 (2014)Google Scholar
  17. 17.
    López, R.: Constant Mean Curvature Surfaces with Boundary. Springer Monographs in Mathematics. Springer, Heidelberg (2013)CrossRefzbMATHGoogle Scholar
  18. 18.
    Marques, F.C., Neves, A.: Min-max theory and the Willmore conjecture. Ann. Math. (2) 179(2), 683–782 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Maximo, D., Nunes, I., Smith, G.: Free boundary minimal annuli in convex three-manifolds. J. Differ. Geom. 106(1), 139–186 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    McGrath, P.: A characterization of the critical catenoid. arXiv:1603.04114
  21. 21.
    Mori, H.: Minimal surfaces of revolution in \(H^3\) and their global stability. Indiana Univ. Math. J. 30(5), 787–794 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nitsche, J.C.: Stationary partitioning of convex bodies. Arch. Rational Mech. Anal. 89(1), 1–19 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ros, A., Souam, R.: On stability of capillary surfaces in a ball. Pac. J. Math. 178(2), 345–361 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ros, A., Vergasta, E.: Stability for hypersurfaces of constant mean curvature with free boundary. Geom. Dedicata 56(1), 19–33 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Simons, J.: Minimal varieties in riemannian manifolds. Ann. Math. (2) 88, 62–105 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Smith, G., Zhou, D.: The Morse index of the critical catenoid. arXiv:1609.01485 (to appear in Geom. Dedicata)
  27. 27.
    Souam, R.: On stability of stationary hypersurfaces for the partitioning problem for balls in space forms. Math. Z. 224(2), 195–208 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Tran, H.: Index characterization for free boundary minimal surfaces. arXiv:1609.01651 (to appear in Comm. Anal. Geom)

Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China
  2. 2.Mathematical Sciences InstituteAustralian National UniversityCanberra, ACTAustralia

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