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Index estimates for free boundary minimal hypersurfaces

Abstract

We show that the Morse index of a properly embedded free boundary minimal hypersurface in a strictly mean convex domain of the Euclidean space grows linearly with the dimension of its first relative homology group (which is at least as big as the number of its boundary components, minus one). In ambient dimension three, this implies a lower bound for the index of a free boundary minimal surface which is linear both with respect to the genus and the number of boundary components. Thereby, the compactness theorem by Fraser and Li implies a strong compactness theorem for the space of free boundary minimal surfaces with uniformly bounded Morse index inside a convex domain. Our estimates also imply that the examples constructed, in the unit ball, by Fraser–Schoen and Folha–Pacard–Zolotareva have arbitrarily large index. Extensions of our results to more general settings (including various classes of positively curved Riemannian manifolds and other convexity assumptions) are discussed.

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References

  1. 1.

    Ambrozio, L., Carlotto, A., Sharp, B.: Comparing the Morse index and the first Betti number of minimal hypersurfaces. J. Differ. Geom. (to appear)

  2. 2.

    Chen, J., Fraser, A., Pang, C.: Minimal immersions of compact bordered Riemann surfaces with free boundary. Trans. Am. Math. Soc. 367(4), 2487–2507 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Courant, R.: The existence of minimal surfaces of given topological structure under prescribed boundary conditions. Acta Math. 72, 51–98 (1940)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Folha, A., Pacard, F., Zolotareva, T.: Free boundary minimal surfaces in the unit 3-ball. Preprint arXiv:1502.06812

  5. 5.

    Fraser, A.: Index estimates for minimal surfaces and k-convexity. Proc. Am. Math. Soc. 135(11), 3733–3744 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    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)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Fraser, A., Schoen, R.: The first Steklov eigenvalue, conformal geometry, and minimal surfaces. Adv. Math. 226(5), 4011–4030 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Fraser, A., Schoen, R.: Minimal Surfaces and Eigenvalue Problems. Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations. Contemporary Mathematics, vol. 599, pp. 105–121. American Mathematical Society, Providence (2013)

    MATH  Google Scholar 

  9. 9.

    Fraser, A., Schoen, R.: Sharp eigenvalue bounds and minimal surfaces in the ball. Invent. Math. 203(3), 823–890 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Freidin, B., Gulian, M., McGrath, P.: Free boundary minimal surfaces in the unit ball with low cohomgeneity. Proc. Am. Math. Soc. 145(4), 1671–1683 (2017)

    MATH  Google Scholar 

  11. 11.

    Grüter, M., Jost, J.: On embedded minimal disks in convex bodies. Ann. Inst. H. Poincaré Anal. Non Linaire 3(5), 345–390 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Grüter, M., Jost, J.: Allard type regularity results for varifolds with free boundaries. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13(1), 129–169 (1986)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  14. 14.

    Hildebrandt, S., Nitsche, J.C.C.: Minimal surfaces with free boundaries. Acta Math. 143(3–4), 251–272 (1979)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Hildebrandt, S., Nitsche, J.C.C.: Optimal boundary regularity for minimal surfaces with a free boundary. Manuscr. Math. 33(3–4), 357–364 (1980/1981)

  16. 16.

    Jost, J.: On the regularity of minimal surfaces with free boundaries in Riemannian manifolds. Manuscr. Math. 56(3), 279–291 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Jost, J.: On the existence of embedded minimal surfaces of higher genus with free boundaries in Riemannian manifolds. In: Concus, P., Finn, R. (eds.) Variational Methods for Free Surface Interfaces. Proceedings of a Conference Held in Menlo Park, California, September 7–12, 1985, pp. 65–75. Springer, New York (1987)

  18. 18.

    Li, M.: A general existence theorem for embedded minimal surfaces with free boundary. Commun. Pure Appl. Math. 68(2), 286–331 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Máximo, D., Nunes, I., Smith, G.: Free boundary minimal annuli in convex three-manifolds. J. Differ. Geom. 106(1), 139–186 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Meeks III, W.H., Yau, S.-T.: Topology of three-dimensional manifolds and the embedding problems in minimal surface theory. Ann. Math. (2) 112(3), 441–484 (1980)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Ros, A.: One-sided complete stable minimal surfaces. J. Differ. Geom. 74(1), 69–92 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Ros, A.: Stability of minimal and constant mean curvature surfaces with free boundary. Mat. Contemp. 35, 221–240 (2008)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Ros, A., Vergasta, E.: Stability for hypersurfaces of constant mean curvature with free boundary. Geom. Dedicata 56(1), 19–33 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Sargent, P.: Index bounds for free boundary minimal surfaces of convex bodies. Proc. Am. Math. Soc. 145, 2467–2480 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Savo, A.: Index bounds for minimal hypersurfaces of the sphere. Indiana Univ. Math. J. 59(3), 823–837 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Schoen, R.: Minimal submanifolds in higher codimension. Mat. Contemp. 30, 169–199 (2006)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Schwarz, G.: Hodge Decomposition: A Method for Solving Boundary Value Problems. Lecture Notes in Mathematics, vol. 1607. Springer, Berlin (1995)

    Book  Google Scholar 

  28. 28.

    Struwe, M.: On a free boundary problem for minimal surfaces. Invent. Math. 75(3), 547–560 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Taylor, M.: Partial Differential Equations, I: Basic Theory. Texts in Applied Mathematics, vol. 23. Springer, New York (1996)

    Book  Google Scholar 

  30. 30.

    Ye, R.: On the existence of area-minimizing surfaces with free boundary. Math. Z. 206(3), 321–331 (1991)

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Ivaldo Nunes, Fernando Codá Marques and André Neves for their interest in this work. L. A. is supported by the ERC Start Grant PSC and LMCF 278940 and would like to thank the Scuola Normale Superiore where part of this project was completed. This article was done while A. C. was an ETH-ITS fellow: the outstanding support of Dr. Max Rössler, of the Walter Haefner Foundation and of the ETH Zürich Foundation are gratefully acknowledged. B.S. would like to thank the ETH-FIM for their hospitality and excellent working environment during the completion of this project. B.S. was partially supported by the Scuola Normale Superiore (Commissione Ricerca, Progetto Giovani Ricercatori).

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Ambrozio, L., Carlotto, A. & Sharp, B. Index estimates for free boundary minimal hypersurfaces. Math. Ann. 370, 1063–1078 (2018). https://doi.org/10.1007/s00208-017-1549-8

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