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.
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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).
A. A. Agrachev, “Exponential mappings for contact sub-Riemannian structures,” J. Dynam. Control Syst., 2, No. 3, 321–358 (1996).
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.
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).
A. A. Agrachev and Yu. L. Sachkov, “Control theory from the geometric viewpoint,” in: Encyclopaedia of Mathematical Sciences, 87, Springer-Verlag, Berlin (2004).
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).
A. A. Agrachev, D. Barilari, and U. Boscain, Introduction to Riemannian and sub-Riemannian geometry, SISSA, Trieste (2011).
A. A. Agrachev and D. Barilari, “Sub-Riemannian structures on 3D Lie groups,” J. Dynam. Control Syst., 18, No. 1, 21–44 (2012).
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).
F. Baudoin, An Introduction to the Geometry of Stochastic Flows, Imperial College Press, London (2004).
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).
F. Baudoin and N. Garofalo, Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries, arXiv:1101.3590 (2012).
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).
A. Bellaïche, “The tangent space in sub-Riemannian geometry,” in: Prog. Math., 144, Birkhäuser, Basel (1996), pp. 1–78.
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).
G. Ben Arous, “Développement asymptotique du noyau de la chaleur hypoelliptique sur la diagonale,” Ann. Inst. Fourier (Grenoble), 39, No. 1, 73–99 (1989).
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).
M. Berger, P. Gauduchon, and E.Mazet, Le Spectre d’une Variété Riemannienne, Springer-Verlag, Berlin (1971).
J.-M. Bismut, Large Deviations and the Malliavin Calculus, Birkhäuser, Boston (1984).
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).
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).
G. B. Folland, “Subelliptic estimates and function spaces on nilpotent Lie groups,” Ark. Mat., 13, No. 2, 161–207 (1975).
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).
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).
L. Hörmander, “Hypoelliptic second-order differential equations,” Acta Math., 119, 147–171 (1967).
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).
R. Léandre, “Développement asymptotique de la densité d’une diffusion dégénérée,” Forum Math., 4, No. 1, 45–75 (1992).
J. Mitchell, “On Carnot—Carathéodory metrics,” J. Differ. Geom., 21, No. 1, 35–45 (1985).
R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, American Mathematical Society, Providence (2002).
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.
S. Rosenberg, The Laplacian on a Riemannian Manifold, Cambridge Univ. Press, Cambridge (1997).
R. S. Strichartz, “Sub-Riemannian geometry,” J. Differ. Geom., 24, No. 2, 221–263 (1986).
T. Taylor, “A parametrix for step-two hypoelliptic diffusion equations,” Trans. Am. Math. Soc., 296, No. 1, 191–215 (1986).
T. J. S. Taylor, “Off diagonal asymptotics of hypoelliptic diffusion equations and singular Riemannian geometry,” Pacific J. Math., 136, No. 2, 379–399 (1989).
Y. Yu, The Index Theorem and the Heat Equation Method, World Scientific Publishing, River Edge (2001).
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 82, Nonlinear Control and Singularities, 2012.
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Barilari, D. Trace heat kernel asymptotics in 3D contact sub-Riemannian geometry. J Math Sci 195, 391–411 (2013). https://doi.org/10.1007/s10958-013-1585-1
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DOI: https://doi.org/10.1007/s10958-013-1585-1