Advertisement

The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3906–3927 | Cite as

Comparison Results, Exit Time Moments, and Eigenvalues on Riemannian Manifolds with a Lower Ricci Curvature Bound

  • Don Colladay
  • Jeffrey J. LangfordEmail author
  • Patrick McDonald
Article
  • 95 Downloads

Abstract

We study the relationship between the geometry of smoothly bounded domains in complete Riemannian manifolds and the associated sequence of \(L^1\)-norms of exit time moments for Brownian motion. We establish bounds for Dirichlet eigenvalues and, for closed manifolds, we establish a comparison result for elements of the moment sequence using lower bounds on Ricci curvature.

Keywords

Torsional rigidity Heat content Dirichlet Problem Brownian motion 

Mathematics Subject Classification

58J65 58J50 35P15 

References

  1. 1.
    Aizenman, M., Simon, B.: Brownian motion and Harnack inequality for Schrödinger operators. Commun. Pure Appl. Math. 35(2), 209–273 (1982)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bérard, P., Meyer, D.: Inégalités isopérimétriques et applications. Ann. Sci. École Norm. Sup. (4) 15(3), 513–541 (1982)Google Scholar
  3. 3.
    Bérard, P., Besson, G., Gallot, S.: Sur une inégalité isopérimétrique qui généralise celle de Paul Lévy-Gromov. Invent. Math. 80(2), 295–308 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bessa, G., Gimeno, V., Jorge, L.: Dirichlet spectrum and Green function. arXiv:1605.04355
  5. 5.
    Besson, G.: From isoperimetric inequalities to heat kernels via symmetrisation. Surv. Differ. Geom., 9, vol. IX, 27–51. Int. Press, Somerville, MA (2004)Google Scholar
  6. 6.
    Burchard, A., Schmuckenschläger, M.: Comparison theorems for exit time moments. Geom. Funct. Anal. 11(4), 651–692 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cadeddu, L., Gallot, S., Loi, A.: Maximizing mean exit-time of the Brownian motion on Riemannian manifolds. Monatsh. Math. 176(4), 551–570 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cheng, S.Y.: Eigenvalue comparison theorems and its geometric applications. Math. Z. 143(3), 289–297 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Debiard, A., Gaveau, B., Mazet, E.: Thèorèmes de comparaison en géométrie riemannienne. Publ. Res. Inst. Math. Sci. 12(2), 391–425 (1976/77)Google Scholar
  10. 10.
    Dryden, E., Langford, J., McDonald, P.: Exit time moments and eigenvalue estimates. Bull. Lond. Math. Soc. 49(3), 480–490 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hassannezhad, A., Kokarev, G., Polterovich, I.: Eigenvalue inequalities on Riemannian manifolds with a lower Ricci curvature bound. J. Spectr. Theory 6(4), 807–835 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hurtado, A., Markorvsen, S., Palmer, V.: Torsional rigidity of submanifolds with controlled geometry. Math. Ann. 344(3), 511–542 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hurtado, A., Markorvsen, S., Palmer, V.: Comparison of exit moment spectra for extrinsic metric balls. Potential Anal. 36(1), 137–153 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hurtado, A., Markorvsen, S., Palmer, V.: Estimates of the first Dirichlet eigenvalue from exit time moment spectra. Math. Ann. 365(3–4), 1603–1632 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kesavan, S.: Some remarks on a result of Talenti. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15(3), 453–465 (1988)Google Scholar
  16. 16.
    Kinateder, K.J., McDonald, P., Miller, D.: Exit time moments, boundary value problems and the geometry of domains in Euclidean space. Prob. Theory Relat. 111(4), 469–487 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    McDonald, P.: Isoperimetric conditions, Poisson problems, and diffusions in Riemannian manifolds. Potential Anal. 16, 115–138 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    McDonald, P.: Exit times, moment problems and comparison theorems. Potential Anal. 38(4), 1365–1372 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    McDonald, P., Meyers, R.: Dirichlet spectrum and heat content. J. Funct. Anal. 200(1), 150–159 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Myers, S.B.: Riemannian manifolds with positive mean curvature. Duke Math. J. 8, 401–404 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Pólya, G.: Torsional rigidity, principal frequency, electrostatic capacity and symmetrization. Q. Appl. Math. 6, 267–277 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Talenti, G.: Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3(4), 697–718 (1976)Google Scholar
  23. 23.
    van den Berg, M., Buttazzo, G., Velichkov, B.: Optimization problems involving the first Dirichlet eigenvalue and the torsional rigidity. New trends in shape optimization, Internat. Ser. Numer. Math., vol. 166, pp. 19–41. Birkhäuser/Springer, Cham (2015)Google Scholar
  24. 24.
    van den Berg, M., Gilkey, P.: Heat content asymptotics for a Riemannian manifold with boundary. J. Funct. Anal. 120(1), 48–71 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    van den Berg, M., Ferone, V., Nitsch, C., Trombetti, C.: On Pólya’s inequality for torsional rigidity and first Dirichlet eigenvalue. Integral Eq. Oper. Theory 86(4), 579–600 (2016)CrossRefzbMATHGoogle Scholar
  26. 26.
    Wei, G.: Manifolds with a lower Ricci curvature bound, Surveys in differential geometry. Surv. Differ. Geom., 11, vol. XI, 203–227. Int. Press, Somerville, MA (2007)Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  • Don Colladay
    • 2
  • Jeffrey J. Langford
    • 1
    Email author
  • Patrick McDonald
    • 2
  1. 1.Department of MathematicsBucknell UniversityLewisburgUSA
  2. 2.Division of Natural ScienceNew College of FloridaSarasotaUSA

Personalised recommendations