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The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 1–32 | Cite as

Rigidity Phenomena in Manifolds with Boundary Under a Lower Weighted Ricci Curvature Bound

  • Yohei SakuraiEmail author
Article
  • 77 Downloads

Abstract

We study Riemannian manifolds with boundary under a lower N-weighted Ricci curvature bound for N at most 1, and under a lower weighted mean curvature bound for the boundary. We examine rigidity phenomena in such manifolds with boundary. We conclude a volume growth rigidity theorem for the metric neighborhoods of the boundaries, and various splitting theorems. We also obtain rigidity theorems for the smallest Dirichlet eigenvalues for the weighted p-Laplacians.

Keywords

Manifold with boundary Weighted Ricci curvature Weighted p-Laplacian 

Mathematics Subject Classification

Primary 53C20 

Notes

Acknowledgements

The author would like to express his gratitude to Professor Koichi Nagano for his constant advice and suggestions. The author would also like to thank Professor Shin-ichi Ohta for his valuable comments. The author would like to thank Professor William Wylie for his valuable advice concerning Proposition 2.9.

References

  1. 1.
    Allegretto, W., Huang, Y.X.: A Picone’s identity for the \(p\)-Laplacian and applications. Nonlinear Anal. 32(7), 819–830 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bakry, D., Émery, M.: Diffusions hypercontractives, Séminaire de Probabilités, XIX, 1983/84, 177–206. Lecture Notes in Mathematics, vol. 1123. Springer, Berlin (1985)Google Scholar
  3. 3.
    Bayle, V.: Propriétés de concavité du profil isopérimétrique et applications, PhD Thesis, Université Joseph-Fourier-Grenoble I (2003)Google Scholar
  4. 4.
    Besse, A.L.: Einstein Manifolds. Springer, Berlin (1987)CrossRefzbMATHGoogle Scholar
  5. 5.
    Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence, RI (2001)Google Scholar
  6. 6.
    Calabi, E.: An extension of E. Hopf’s maximum principle with an application to Riemannian geometry. Duke Math. J. 25, 45–56 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cheeger, J.: Degeneration of Riemannian metrics under Ricci curvature bounds. Accademia Nazionale dei Lincei. Scuola Normale Superiore, Lezione Fermiane (2001)Google Scholar
  8. 8.
    Cheeger, J., Gromoll, D.: The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differ. Geom. 6, 119–128 (1971/1972)Google Scholar
  9. 9.
    Croke, C., Kleiner, B.: A warped product splitting theorem. Duke Math. J. 67(3), 571–574 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fang, F., Li, X., Zhang, Z.: Two generalizations of Cheeger–Gromoll splitting theorem via Bakry–Emery Ricci curvature. Ann. Inst. Fourier (Grenoble) 59(2), 563–573 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Heintze, E., Karcher, H.: A general comparison theorem with applications to volume estimates for submanifolds. Ann. Sci. École Norm. Sup. (4) 11(4), 451–470 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ichida, R.: Riemannian manifolds with compact boundary. Yokohama Math. J. 29(2), 169–177 (1981)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kasue, A.: Ricci curvature, geodesics and some geometric properties of Riemannian manifolds with boundary. J. Math. Soc. Japan 35(1), 117–131 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kasue, A.: On a lower bound for the first eigenvalue of the Laplace operator on a Riemannian manifold. Ann. Sci. École Norm. Sup. (4) 17(1), 31–44 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kasue, A.: Applications of Laplacian and Hessian Comparison Theorems, Geometry of geodesics and related topics (Tokyo, 1982), 333–386, Advanced Studies in Pure Mathematics, vol. 3, North-Holland, Amsterdam (1984)Google Scholar
  16. 16.
    Klartag, B.: Needle decompositions in Riemannian geometry. Mem. Am. Math. Soc. 249(1180), v + 77 pp (2017)Google Scholar
  17. 17.
    Kolesnikov, A.V., Milman, E.: Isoperimetric inequalities on weighted manifolds with boundary. Dokl. Akad. Nauk 464(2), 136–140 (2015)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Li, P., Yau, S.T.: Estimates of eigenvalues of a compact Riemannian manifold. Proc. Symp. Pure Math. 36, 205–239 (1980)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lichnerowicz, A.: Variétés riemanniennes à tenseur C non négatif. C. R. Acad. Sci. Paris Sér. A-B 271, A650–A653 (1970)zbMATHGoogle Scholar
  20. 20.
    Lott, J.: Some geometric properties of the Bakry–Émery–Ricci tensor. Comment. Math. Helv. 78(4), 865–883 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lott, J., Villani, C.: Weak curvature conditions and functional inequalities. J. Funct. Anal. 245(1), 311–333 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. (2) 169(3), 903–991 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Milman, E.: Beyond traditional curvature-dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension. Trans. Am. Math. Soc. 369(5), 3605–3637 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Morgan, F.: Manifolds with density. Not. Am. Math. Soc. 52(8), 853–858 (2005)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Ohta, S.: \((K, N)\)-convexity and the curvature-dimension condition for negative \(N\). J. Geom. Anal. 26(3), 2067–2096 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ohta, S., Takatsu, A.: Displacement convexity of generalized relative entropies. Adv. Math. 228(3), 1742–1787 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ohta, S., Takatsu, A.: Displacement convexity of generalized relative entropies. II. Commun. Anal. Geom. 21(4), 687–785 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Perales, R.: Volumes and limits of manifolds with Ricci curvature and mean curvature bound. Differ. Geom. Appl. 48, 23–37 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Petersen, P.: Riemannian Geometry: Second Edition. Graduate Texts in Mathematics, vol. 171. Springer, New York (2006)Google Scholar
  30. 30.
    Qian, Z.: Estimates for weighted volumes and applications. Q. J. Math. Oxf. Ser. (2) 48(190), 235–242 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Sakai, T.: Riemannian Geometry. Translations of Mathematical Monographs, vol. 149. American Mathematical Society, Providence, RI (1996)Google Scholar
  32. 32.
    Sakurai, Y.: Rigidity of manifolds with boundary under a lower Ricci curvature bound. Osaka J. Math. 54(1), 85–119 (2017)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Sakurai, Y.: Rigidity of manifolds with boundary under a lower Bakry–Émery Ricci curvature bound. Tohoku Math. J. arXiv preprint arXiv:1506.03223v4 (2016)
  34. 34.
    Sturm, K.-T.: On the geometry of metric measure spaces. I. Acta Math. 196(1), 65–131 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Sturm, K.-T.: On the geometry of metric measure spaces. II. Acta Math. 196(1), 133–177 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51(1), 126–150 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Villani, C.: Optimal Transport: Old and New. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar
  38. 38.
    Wei, G., Wylie, W.: Comparison geometry for the Bakry–Emery Ricci tensor. J. Differ. Geom. 83(2), 377–405 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Wylie, W.: A warped product version of the Cheeger-Gromoll splitting theorem. Trans. Am. Math. Soc. 369(9), 6661–6681 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Wylie, W., Yeroshkin, D.: On the geometry of Riemannian manifolds with density. arXiv preprint arXiv:1602.08000 (2016)

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Authors and Affiliations

  1. 1.Advanced Institute for Materials ResearchTohoku UniversitySendaiJapan

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