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Journal of Mathematical Sciences

, Volume 195, Issue 3, pp 391–411 | Cite as

Trace heat kernel asymptotics in 3D contact sub-Riemannian geometry

  • D. Barilari
Article

Abstract

In this paper, we study the small time asymptotics for the heat kernel on a sub-Riemannian manifold, using a perturbative approach. We explicitly compute, in the case of a 3D contact structure, the first two coefficients of the small time asymptotics expansion of the heat kernel on the diagonal, expressing them in terms of the two basic functional invariants χand κ defined on a 3D contact structure.

Keywords

Heat Kernel Heisenberg Group Orthonormal Frame Reeb Vector Local Orthonormal Frame 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A. A. Agrachev, R. V. Gamkrelidze, and A. V. Sarychev, “Local invariants of smooth control systems,” Acta Appl. Math., 14, No. 3, 191–237 (1989).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    A. A. Agrachev, “Exponential mappings for contact sub-Riemannian structures,” J. Dynam. Control Syst., 2, No. 3, 321–358 (1996).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    A. A. Agrachev, El-A. El-H. Chakir, and J.-P. Gauthier, “Sub-Riemannian metrics on R 3,” in: CMS Conf. Proc., 25, Amer. Math. Soc., Providence (1998), pp. 29–78.Google Scholar
  4. 4.
    A. A. Agrachev and J.-P. A. Gauthier, “On the Dido problem and plane isoperimetric problems,” Acta Appl. Math., 57, No. 3, 287–338 (1999).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    A. A. Agrachev and Yu. L. Sachkov, “Control theory from the geometric viewpoint,” in: Encyclopaedia of Mathematical Sciences, 87, Springer-Verlag, Berlin (2004).Google Scholar
  6. 6.
    A. Agrachev, U. Boscain, J.-P. Gauthier, and F. Rossi, “The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups,” J. Funct. Anal., 256, No. 8, 2621–2655 (2009).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    A. A. Agrachev, D. Barilari, and U. Boscain, Introduction to Riemannian and sub-Riemannian geometry, SISSA, Trieste (2011).Google Scholar
  8. 8.
    A. A. Agrachev and D. Barilari, “Sub-Riemannian structures on 3D Lie groups,” J. Dynam. Control Syst., 18, No. 1, 21–44 (2012).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    A. A. Agrachev, D. Barilari, and U. Boscain, “On the Hausdorff volume in sub-Riemannian geometry,” Calc. Var. Partial Differ. Equ., 43, Nos. 3–4, 355–388 (2012).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    F. Baudoin, An Introduction to the Geometry of Stochastic Flows, Imperial College Press, London (2004).CrossRefMATHGoogle Scholar
  11. 11.
    F. Baudoin and M. Bonnefont, “The subelliptic heat kernel on SU(2): representations, asymptotics and gradient bounds,” Math. Z., 263, No. 3, 647–672 (2009).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    F. Baudoin and N. Garofalo, Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries, arXiv:1101.3590 (2012).Google Scholar
  13. 13.
    R. Beals, B. Gaveau, and P. Greiner, “The Green function of model step two hypoelliptic operators and the analysis of certain tangential Cauchy Riemann complexes,” Adv. Math., 121, No. 2, 288–345 (1996).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    A. Bellaïche, “The tangent space in sub-Riemannian geometry,” in: Prog. Math., 144, Birkhäuser, Basel (1996), pp. 1–78.Google Scholar
  15. 15.
    G. Ben Arous, “Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus,” Ann. Sci. École Norm. Sup. (4), 21, No. 3, 307–331 (1988).MathSciNetMATHGoogle Scholar
  16. 16.
    G. Ben Arous, “Développement asymptotique du noyau de la chaleur hypoelliptique sur la diagonale,” Ann. Inst. Fourier (Grenoble), 39, No. 1, 73–99 (1989).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    G. Ben Arous and R. Léandre, “Décroissance exponentielle du noyau de la chaleur sur la diagonale. II,” Probab. Theory Relat. Fields, 90, No. 3, 377–402 (1991).CrossRefMATHGoogle Scholar
  18. 18.
    M. Berger, P. Gauduchon, and E.Mazet, Le Spectre d’une Variété Riemannienne, Springer-Verlag, Berlin (1971).MATHGoogle Scholar
  19. 19.
    J.-M. Bismut, Large Deviations and the Malliavin Calculus, Birkhäuser, Boston (1984).MATHGoogle Scholar
  20. 20.
    R. W. Brockett and A.Mansouri, “Short-time asymptotics of heat kernels for a class of hypoelliptic operators,” Am. J. Math., 131, No. 6, 1795–1814 (2009).MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Springer-Verlag, Berlin (1987).MATHGoogle Scholar
  22. 22.
    G. B. Folland, “Subelliptic estimates and function spaces on nilpotent Lie groups,” Ark. Mat., 13, No. 2, 161–207 (1975).MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    N. Garofalo and E. Lanconelli, “Asymptotic behavior of fundamental solutions and potential theory of parabolic operators with variable coefficients,” Math. Ann., 283, No. 2, 211–239 (1989).MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    B. Gaveau, “Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents,” Acta Math., 139, Nos. 1–2, 95–153 (1977).MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    L. Hörmander, “Hypoelliptic second-order differential equations,” Acta Math., 119, 147–171 (1967).MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    S. Kusuoka and D. Stroock, “Applications of the Malliavin calculus. II,” J. Fac. Sci. Univ. Tokyo Sect. IA Math., 32, No. 1, 1–76 (1985).MathSciNetMATHGoogle Scholar
  27. 27.
    R. Léandre, “Développement asymptotique de la densité d’une diffusion dégénérée,” Forum Math., 4, No. 1, 45–75 (1992).MathSciNetMATHGoogle Scholar
  28. 28.
    J. Mitchell, “On Carnot—Carathéodory metrics,” J. Differ. Geom., 21, No. 1, 35–45 (1985).MATHGoogle Scholar
  29. 29.
    R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, American Mathematical Society, Providence (2002).Google Scholar
  30. 30.
    R. Neel and D. Stroock, “Analysis of the cut locus via the heat kernel,” in: Surv. Differ. Geom., IX, Int. Press, Somerville (2004), pp. 337–349.Google Scholar
  31. 31.
    S. Rosenberg, The Laplacian on a Riemannian Manifold, Cambridge Univ. Press, Cambridge (1997).CrossRefMATHGoogle Scholar
  32. 32.
    R. S. Strichartz, “Sub-Riemannian geometry,” J. Differ. Geom., 24, No. 2, 221–263 (1986).MathSciNetMATHGoogle Scholar
  33. 33.
    T. Taylor, “A parametrix for step-two hypoelliptic diffusion equations,” Trans. Am. Math. Soc., 296, No. 1, 191–215 (1986).CrossRefMATHGoogle Scholar
  34. 34.
    T. J. S. Taylor, “Off diagonal asymptotics of hypoelliptic diffusion equations and singular Riemannian geometry,” Pacific J. Math., 136, No. 2, 379–399 (1989).MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Y. Yu, The Index Theorem and the Heat Equation Method, World Scientific Publishing, River Edge (2001).MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.SISSA, Via Bonomea 265TriesteItaly

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