Advertisement

Geometriae Dedicata

, Volume 178, Issue 1, pp 21–47 | Cite as

Stability properties and topology at infinity of \(f\)-minimal hypersurfaces

Original Paper

Abstract

We study stability properties of \(f\)-minimal hypersurfaces isometrically immersed in weighted manifolds with non-negative Bakry–Émery Ricci curvature under volume growth conditions. Moreover, exploiting a weighted version of a finiteness result and the adaptation to this setting of Li–Tam theory, we investigate the topology at infinity of \(f\)-minimal hypersurfaces. On the way, we prove a new comparison result in weighted geometry and we provide a general weighted \(L^1\)-Sobolev inequality for hypersurfaces in Cartan–Hadamard weighted manifolds, satisfying suitable restrictions on the weight function.

Keywords

\(f\)-minimal hypersurfaces Weighted manifolds Stability Finite index Topology at infinity 

Mathematics Subject Classification

53C42 53C21 

Notes

Acknowledgments

Part of this work was done while we were visiting the Institut Henri Poincaré, Paris. We would like to thank the institute for the warm hospitality. Moreover we are deeply grateful to Stefano Pigola for useful conversations during the preparation of the manuscript. We would also like to thank Marcio Batista and Heudson Mirandola and the anonymous referee for useful comments.

References

  1. 1.
    Bakry, D., Émery, M.: Diffusions Hypercontractives, Séminaire de Probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, pp. 177–206Google Scholar
  2. 2.
    Bayle, V.: Propriétés de concavité du profil isopérimétrique et applications, Thèse de Doctorat (2003)Google Scholar
  3. 3.
    Bianchini, B., Mari, L., Rigoli, M.: Spectral radius, index estimates for Schrödinger operators and geometric applications. J. Funct. Anal. 256(6), 1769–1820 (2009)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Brooks, R.: A relation between growth and the spectrum of the Laplacian. Math. Z. 178(4), 501–508 (1981)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Buckley, S.M., Koskela, P.: Ends of metric measure spaces and Sobolev inequalities. Math. Z. 252(2), 275–285 (2006)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Carron, G.: \(L^2\)-cohomologie et inégalités de Sobolev. Math. Ann. 314(4), 613–639 (1999)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Cheng, X., Mejia, T., Zhou, D.: Eigenvalue estimate and compactness for closed \(f\)-minimal surfaces, arXiv:1210.8448. To appear on Pacific J. Math.
  8. 8.
    Cheng, X., Mejia, T., Zhou, D.: Stability and compactness for complete \(f\)-minimal surfaces, arXiv:1210.8076. To appear on Trans. Am. Math. Soc.
  9. 9.
    Cheng, X., Zhou, D.: Volume estimates about shrinkers. Proc. Am. Math. Soc. 141(2), 687–696 (2013)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Colding, T.H., Minicozzi, W.P.: Generic mean curvature flow I; generic singularities. Ann. Math. 2(175), 755–833 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Devyver, B.: On the finiteness of the Morse index for Schrödinger operators. Manuscr. Math. 139(1–2), 249–271 (2012)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    do Carmo, M.P., Zhou, D.: Eigenvalue estimate on complete noncompact Riemannian manifolds and applications. Trans. Am. Math. Soc. 351(4), 1391–1401 (1999)Google Scholar
  13. 13.
    Espinar, J.M.: Manifolds with density, applications and gradient Schrödinger operators, arXiv:1209.6162v6
  14. 14.
    Fan, E.M.: Topology of three-manifolds with positive \(P\)-scalar curvature. Proc. Am. Math. Soc. 136(9), 3255–3261 (2008)MATHCrossRefGoogle Scholar
  15. 15.
    Fischer-Colbrie, D., Schoen, R.: The structure of complete stable minimal surfaces in \(3\)-manifolds of nonnegative scalar curvature. Commun. Pure Appl. Math. 33(2), 199–211 (1980)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Grigor’yan, A.A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. (N.S.) 36(2), 135–249 (1999)MATHCrossRefGoogle Scholar
  17. 17.
    Gromov, M.: Isoperimetry of waists and concentration of maps. Geom. Funct. Anal. 13(1), 178–215 (2003)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Gursky, M.J., Chang, S.-Y.A., Yang, P.: Conformal invariants associated to a measure. Proc. Natl. Acad. Sci. USA 103(8), 2535–2540 (2006)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Ho, P.T.: The structure of \(\phi \)-stable minimal hypersurfaces in manifolds of nonnegative \(p\)-scalar curvature. Math. Ann. 348(2), 319–332 (2010)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Hoffman, D., Spruck, J.: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Commun. Pure Appl. Math. 27, 715–727 (1974)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Li, P., Yau, S.T.: On the Schrödinger equation and the eigenvalue problem. Commun. Math. Phys. 88(3), 309–318 (1983)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Li, P., Tam, L.-F.: Harmonic functions and the structure of complete manifolds. J. Differ. Geom. 35(2), 359–383 (1992)MATHMathSciNetGoogle Scholar
  23. 23.
    Li, P., Wang, J.: Minimal hypersurfaces with finite index. Math. Res. Lett. 9(1), 95–103 (2002)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Li, P., Wang, J.: Stable minimal hypersurfaces in a nonnegatively curved manifold. J. Reine Angew. Math. 566, 215–230 (2004)MATHMathSciNetGoogle Scholar
  25. 25.
    Lichnerowicz, A.: Variétés riemanniennes à tenseur C non négatif. C. R. Acad. Sci. Paris Sér. A-B 271, A650–A653 (1970)MathSciNetGoogle Scholar
  26. 26.
    Liu, G.: Stable weighted minimal surfaces in manifolds with nonnegative Bakry-Emery Ricci tensor. Comm. Anal. Geom. 21(5), 1061–1079 (2013)Google Scholar
  27. 27.
    Michael, J.H., Simon, L.M.: Sobolev and mean-value inequalities on generalized submanifolds of \(R^{n}\). Commun. Pure Appl. Math. 26, 361–379 (1973)MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Milman, E.: Sharp isoperimetric inequalities and model spaces for curvature-dimension- diameter condition, arXiv: 1108.4609. To appear on J. Eur. Math. Soc.
  29. 29.
    Morgan, F.: Manifolds with density. Notices Am. Math. Soc. 52(8), 853–858 (2005)MATHGoogle Scholar
  30. 30.
    Morgan, F., Pratelli, A.: Existence of isoperimetric regions in \(\mathbb{R}^n\) with density. Ann. Global Anal. Geom. 43(4), 331–365 (2013)MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Moss, W.F., Piepenbrink, J.: Positive solutions of elliptic equations. Pac. J. Math. 75(1), 219–226 (1978)MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Munteanu, O., Wang, J.: Smooth metric measure spaces with nonnegative curvature. Commun. Anal. Geom. 19(3), 451–486 (2011)MATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    O’Neill, B.: Semi-Riemannian geometry. In: Pure and Applied Mathematics, vol. 103, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983, With applications to relativityGoogle Scholar
  34. 34.
    Pigola, S., Rigoli, M., Setti, A.G.: Vanishing and finiteness results in geometric analysis. In: Progress in Mathematics, vol. 266, Birkhäuser Verlag, Basel. A Generalization of the Bochner Technique (2008)Google Scholar
  35. 35.
    Pigola, S., Rigoli, M., Setti, A.G.: A finiteness theorem for the space of \(L^p\) harmonic sections. Rev. Math. Iberoamericana 24(1), 91–116 (2008)MATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Pigola, S., Rigoli, M., Rimoldi, M., Setti, A.G.: Ricci almost solitons. Ann. Sci. Norm. Super. Pisa Cl. Sci. (5) 10(4), 757–799 (2011)MATHMathSciNetGoogle Scholar
  37. 37.
    Pigola, S., Rimoldi, M.: Complete self-shrinkers confined into some regions of the space. Ann. Global Anal. Geom. 45(1), 47–65 (2014)MATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Rigoli, M., Setti, A.G.: Liouville type theorems for \(\varphi \)-subharmonic functions. Rev. Math. Iberoamericana 17(3), 471–520 (2001)MATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Rimoldi, M.: On a classification theorem for self-shrinkers. Proc. Am. Math. Soc. 142(10), 3605–3613 (2014)Google Scholar
  40. 40.
    Rimoldi, M., Veronelli, G.: Topology of steady and expanding gradient Ricci solitons via \(f\)-harmonic maps. Differ. Geom. Appl. 31(5), 623–638 (2013)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Rosales, C., Cañete, A., Bayle, V., Morgan, F.: On the isoperimetric problem in Euclidean space with density. Calc. Var. Partial Differ. Equ. 31(1), 27–46 (2008)MATHCrossRefGoogle Scholar
  42. 42.
    Shen, Y., Cao, H.-D., Zhu, S.: The structure of stable minimal hypersurfaces in \({ R}^{n+1}\). Math. Res. Lett. 4(5), 637–644 (1997)MATHMathSciNetCrossRefGoogle Scholar
  43. 43.
    Sheng, W., Yu, H.: \(f\)-stability of \(f\)-minimal hypersurfaces, to appear on Proc. Am. Math. Soc.Google Scholar
  44. 44.
    Sinaei, Z.: Harmonic maps on smooth metric measure spaces and their convergence, arXiv:1209.5893
  45. 45.
    Volpi, S.: Proprietá spettrali di operatori di Schrödinger, Degree ThesisGoogle Scholar
  46. 46.
    Wei, G., Wylie, W.: Comparison geometry for the Bakry-Emery Ricci tensor. J. Differ. Geom. 83(2), 377–405 (2009)MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Dipartimento di Matematica e ApplicazioniUniversità degli Studi di Milano BicoccaMilanItaly

Personalised recommendations