Skip to main content
SpringerLink
Log in

On Witten Laplacians and Brascamp–Lieb’s Inequality on Manifolds with Boundary

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

In this paper, we derive from the supersymmetry of the Witten Laplacian Brascamp–Lieb’s type inequalities for general differential forms on compact Riemannian manifolds with boundary. In addition to the supersymmetry, our results essentially follow from suitable decompositions of the quadratic forms associated with the Neumann and Dirichlet self-adjoint realizations of the Witten Laplacian. They moreover imply the usual Brascamp–Lieb’s inequality and its generalization to compact Riemannian manifolds without boundary.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bach, V., Jecko, T., Sjöstrand, J.: Correlation asymptotics of classical lattice spin systems with nonconvex Hamilton function at low temperature. Ann. Henri Poincaré 1(1), 59–100 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bach, V., Møller, J.S.: Correlation at low temperature: I. Exponential decay. J. Funct. Anal. 203(1), 93–148 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bach, V., Møller, J.S.: Correlation at low temperature: II. Asymptotics. J. Stat. Phys. 116(1–4), 591–628 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bakry, D., Émery, M.: Diffusions Hypercontractives. Sem. Probab. XIX, Lecture Notes in Math. 1123, pp. 177–206. Springer, Berlin (1985)

  5. Bakry, D., Gentil, I., Ledoux, M.: Analysis and Geometry of Markov Diffusion Operators. Grund. der Math. Wiss. 348. Springer, Berlin (2014)

  6. Bochner, S.: Curvature and Betti numbers. Ann. Math. 49(2), 379–390 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brascamp, H.J., Lieb, E.H.: On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22(4), 366–389 (1976)

    Article  MATH  Google Scholar 

  8. Di Gesù, G., Le Peutrec, D.: Small noise spectral gap asymptotics for a large system of nonlinear diffusions. To appear in J. Spectr. Theory, preprint on http://arxiv.org/abs/1506.04434, 47 pages (2015)

  9. Gallot, S., Meyer, D.: Opérateur de courbure et laplacien des formes différentielles d’une variété riemannienne. J. Math. Pures Appl. 54(9), no. 3, 259–284 (1975)

  10. Helffer, B.: Remarks on the decay of correlations and Witten Laplacians—the Brascamp–Lieb inequality and semiclassical limit. J. Funct. Anal. 155, 571–586 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Helffer, B., Nier, F.: Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary. Mémoire 105, Société Mathématique de France (2006)

  12. Helffer, B., Sjöstrand, J.: On the correlations for Kac like models in the convex case. J. Stat. Phys. 74, 349–409 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jammes, P.: Sur la multiplicité des valeurs propres du laplacien de Witten. Trans. Am. Math. Soc. 364(6), 2825–2845 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Johnsen, J.: On the spectral properties of Witten Laplacians, their range projections and Brascamp-Lieb’s inequality. Integr. Equ. Oper. Theory 36(3), 288–324 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kolesnikov, A.V., Milman, E.: Brascamp–Lieb-type inequalities on weighted Riemannian manifolds with boundary. J. Geom. Anal. (2016). doi:10.1007/s12220-016-9736-5

  16. Kolesnikov, A.V., Milman, E.: Riemannian metrics on convex sets with applications to Poincaré and log-Sobolev inequalities. Calc. Var. 55, 36 (2016). doi:10.1007/s00526-016-1018-3

    Article  MATH  Google Scholar 

  17. Le Peutrec, D.: Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian. Ann. de la Faculté des Sciences de Toulouse 19(3–4), 735–809 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lichnerowicz, A.: Variétés riemanniennes à tenseur C non négatif. C. R. Acad. Sci. Paris Sér. A-B 271, A650–A653 (1970)

  19. Lott, J.: Some geometric properties of the Bakry–Émery–Ricci tensor. Comment. Math. Helv. 78(4), 865–883 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. (2) 169(3), 903–991 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Naddaf, A., Spencer, T.: On homogenization and scaling limit of some gradient perturbations of a massless free field. Commun. Math. Phys. 183(1), 55–84 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sjöstrand, J.: Correlation asymptotics and Witten Laplacians. Algebra i Analiz 8(1), 160–191 (1996)

    MathSciNet  MATH  Google Scholar 

  23. Schwarz, G.: Hodge decomposition: a method for solving boundary value problems. Lecture Notes in Mathematics 1607, Springer, Berlin (1995)

  24. Witten, E.: Supersymmetry and Morse inequalities. J. Differ. Geom. 17, 661–692 (1982)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405, Orsay, France

    Dorian Le Peutrec

Authors
  1. Dorian Le Peutrec
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dorian Le Peutrec.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Le Peutrec, D. On Witten Laplacians and Brascamp–Lieb’s Inequality on Manifolds with Boundary. Integr. Equ. Oper. Theory 87, 411–434 (2017). https://doi.org/10.1007/s00020-017-2354-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00020-017-2354-1

Mathematics Subject Classification

Keywords

Search

Navigation