Advertisement

Heat conservation for generalized Dirac Laplacians on manifolds with boundary

  • Levi Lopes de LimaEmail author
Article
  • 22 Downloads

Abstract

We consider a notion of conservation for the heat semigroup associated with a generalized Dirac Laplacian acting on sections of a vector bundle over a noncompact manifold with a (possibly noncompact) boundary under mixed boundary conditions. Assuming that the geometry of the underlying manifold is controlled in a suitable way and imposing uniform lower bounds on the zero-order piece (Weitzenböck potential) of the Dirac Laplacian, and on the endomorphism defining the mixed boundary condition, we show that the corresponding conservation principle holds. A key ingredient in the proof is a domination property for the heat semigroup which follows from an extension to this setting of a Feynman–Kac formula recently proved by the author de Lima (Pac J Math 292(1):177–201, 2018) in the context of differential forms. When applied to the Hodge Laplacian acting on differential forms satisfying absolute boundary conditions, this extends previous results by Vesentini (Ann Math Pura Appl 182:1–19, 2003) and Masamune (Atti Accad Naz Lincei Rend Lincei Mat Appl 18(4):351–358, 2007) in the boundaryless case. Along the way, we also prove a vanishing result for \(L^2\) harmonic sections in the broader context of generalized (not necessarily Dirac) Laplacians. These results are further illustrated with applications to the Dirac Laplacian acting on spinors and to the Jacobi operator acting on sections of the normal bundle of a free boundary minimal immersion.

Keywords

Heat conservation principle Generalized Dirac Laplacians Mixed boundary conditions Feynman–Kac formula Reflected Brownian motion 

Mathematics Subject Classification

58J35 58J32 58J65 

Notes

Acknowledgements

The author would like to heartily thank an anonymous referee for his/her careful reading of the manuscript, which substantially contributed to improve its presentation.

Funding

Funding was provided by Fundação Cearense de Apoio ao Desenvolvimento Científico e Tecnológico (Grant Number 00068.01.00/15) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (Grant Number 311258/2014-0).

References

  1. 1.
    Arnaudon, M., Li, X.-M.: Reflected Brownian motion: selection, approximation and linearization. Electron. J. Probab. 22(31), 55 (2017)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Arnaudon, M., Thalmaier, A.: Li-Yau type gradient estimates and Harnack inequalities by stochastic analysis. In: Probabilistic Approach to Geometry, Adv. Stud. Pure Math., 57, pp. 29–48. Mathematical Society of Japan, Tokyo (2010)Google Scholar
  3. 3.
    Avramidi, I.G., Esposito, G.: Gauge theories on manifolds with boundary. Commun. Math. Phys. 200(3), 495–543 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bakry, D., Gentil, I., Ledoux, M.: Analysis and geometry of Markov diffusion operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 348. Springer (2014)Google Scholar
  5. 5.
    de Lima, L.L.: A Feynman-Kac formula for differential forms on manifolds with boundary and applications. Pac. J. Math. 292–1, 177–201 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    de Lima, L.L.: A probabilistic proof of the Gauss-Bonnet formula for manifolds with boundary. arXiv:1709.03772
  7. 7.
    de Lima, L.L.: The conservative principle for differential forms on manifolds with boundary, Lectures Notes on Geometric Analysis, celebrating Barnabé P. Lima’s \(60^{\rm th}\) birthday, edited by Leandro F. PessoaGoogle Scholar
  8. 8.
    Dollard, J.D., Friedman, C.N.: Product integration with applications to differential equations. With a foreword by Felix E. Browder. With an appendix by P. R. Masani. Encyclopedia of Mathematics and its Applications, vol. 10. Addison-Wesley Publishing Co., Reading, MA (1979)Google Scholar
  9. 9.
    Elworthy, K.D.: Geometric aspects of diffusions on manifolds. École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–87, 277-425, Lecture Notes in Math., vol. 1362, Springer, Berlin (1988)Google Scholar
  10. 10.
    Elworthy, K.D., Rosenberg, S.: Generalized Bochner theorems and the spectrum of complete manifolds. Acta Appl. Math. 12(1), 1–33 (1988)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Friedrich, T.: Dirac operators in Riemannian geometry. Graduate Studies in Mathematics, vol. 25. American Mathematical Society, Providence, RI (2000)Google Scholar
  12. 12.
    Gilkey, P.: Asymptotic formulae in spectral geometry. Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton, FL (2004)Google Scholar
  13. 13.
    Grigor’yan, 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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Grubb, G.: Spectral boundary conditions for generalizations of Laplace and Dirac operators. Commun. Math. Phys. 240(1–2), 243–280 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Güneysu, B.: On the semimartingale property of Brownian bridges on complete manifolds arXiv:1604.08351
  16. 16.
    Güneysu, B.: On generalized Schrödinger semigroups. J. Funct. Anal. 262(11), 4639–4674 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hijazi, O., Montiel, S., Zhang, X.: Dirac operator on embedded hypersurfaces. Math. Res. Lett. 8(1–2), 195–208 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hsu, E.P.: Stochastic analysis on manifolds. Graduate Studies in Mathematics, vol. 38. American Mathematical Society, Providence, RI (2002)Google Scholar
  19. 19.
    Hsu, E.P.: Multiplicative functional for the heat equation on manifolds with boundary. Mich. Math. J. 50(2), 351–367 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo (1989)Google Scholar
  21. 21.
    Lawson, Jr., H.B., Michelsohn, M.-L.: Spin geometry. Princeton Mathematical Series, vol. 38. Princeton University Press, Princeton, NJ (1989)Google Scholar
  22. 22.
    Lörinczi, J., Hiroshima, F., Betz, V.: Feynman-Kac-type theorems and Gibbs measures on path space. With applications to rigorous quantum field theory. De Gruyter Studies in Mathematics, vol. 34. Walter de Gruyter & Co., Berlin (2011)Google Scholar
  23. 23.
    Masamune, J.: Conservative principle for differential forms. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 18(4), 351–358 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Nakad, R., Roth, J.: The \({\rm Spin}^c\) Dirac operator on hypersurfaces and applications. Differ. Geom. Appl. 31(1), 93–103 (2013)CrossRefzbMATHGoogle Scholar
  25. 25.
    Nicolaescu, L.I.: Lectures on the Geometry of Manifolds, 2nd edn. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2007)CrossRefzbMATHGoogle Scholar
  26. 26.
    Oh, Y.-G.: Second variation and stabilities of minimal Lagrangian submanifolds in Kähler manifolds. Invent. Math. 101(2), 501–519 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Rosenberg, S.: Semigroup domination and vanishing theorems. Geometry of random motion (Ithaca, N.Y., 1987), Contemp. Math., vol. 73, pp. 287–302. American Mathematical Society, Providence, RI (1988)Google Scholar
  28. 28.
    Rosenberg, S.: The Laplacian on a Riemannian manifold. An introduction to analysis on manifolds. London Mathematical Society Student Texts, vol. 31. Cambridge University Press, Cambridge (1997)Google Scholar
  29. 29.
    Schick, T.: Analysis and Geometry of Boundary-Manifolds of Bounded Geometry. arXiv:math/9810107
  30. 30.
    Schoen, R.: Minimal submanifolds in higher codimension. Mat. Contemp. 30, 169–199 (2006)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Vassilevich, D.V.: Heat kernel expansion: user’s manual. Phys. Rep. 388(5–6), 279–360 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Vesentini, E.: Heat conservation on Riemannian manifolds. Ann. Math. Pura Appl. 182, 1–19 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Wang, F.-Y.: Analysis for diffusion processes on Riemannian manifolds. Advanced Series on Statistical Science & Applied Probability, vol. 18. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2014)Google Scholar
  34. 34.
    Yau, S.-T.: On the heat kernel of a complete Riemannian manifold. J. Math. Pures Appl., Ser. 9(57), 191–201 (1978)MathSciNetzbMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal do CearáFortalezaBrazil

Personalised recommendations