Small time heat kernel asymptotics at the cut locus on surfaces of revolution
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 2, p. 281-295
The full text of recent articles is available to journal subscribers only. See the article on the journal's website
In this paper we investigate the small time heat kernel asymptotics on the cut locus on a class of surfaces of revolution, which are the simplest two-dimensional Riemannian manifolds different from the sphere with non-trivial cut-conjugate locus. We determine the degeneracy of the exponential map near a cut-conjugate point and present the consequences of this result to the small time heat kernel asymptotics at this point. These results give a first example where the minimal degeneration of the asymptotic expansion at the cut locus is attained.
@article{AIHPC_2014__31_2_281_0,
author = {Barilari, Davide and Jendrej, Jacek},
title = {Small time heat kernel asymptotics at the cut locus on surfaces of revolution},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {31},
number = {2},
year = {2014},
pages = {281-295},
doi = {10.1016/j.anihpc.2013.03.003},
zbl = {1301.53035},
mrnumber = {3181670},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2014__31_2_281_0}
}

Barilari, Davide; Jendrej, Jacek. Small time heat kernel asymptotics at the cut locus on surfaces of revolution. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 2, pp. 281-295. doi : 10.1016/j.anihpc.2013.03.003. http://www.numdam.org/item/AIHPC_2014__31_2_281_0/

[1] A. Agrachev, U. Boscain, J.-P. Gauthier, F. Rossi, The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups, J. Funct. Anal. 256 (2009), 2621-2655 | MR 2502528 | Zbl 1165.58012

[2] D. Barilari, U. Boscain, R.W. Neel, Small time heat kernel asymptotics at the sub-Riemannian cut locus, J. Differential Geom. 92 no. 3 (2012), 373-416 | MR 3005058 | Zbl 1270.53066

[3] G. Ben Arous, Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus, Ann. Sci. École Norm. Sup. (4) 21 (1988), 307-331 | MR 974408

[4] G. Ben Arous, Développement asymptotique du noyau de la chaleur hypoelliptique sur la diagonale, Ann. Inst. Fourier (Grenoble) 39 (1989), 73-99 | MR 1011978

[5] M. Berger, A Panoramic View of Riemannian Geometry, Springer-Verlag, Berlin (2003) | MR 2002701 | Zbl 1038.53002

[6] M. Berger, P. Gauduchon, E. Mazet, Le spectre d'une variété riemannienne, Lecture Notes in Math. vol. 194, Springer-Verlag, Berlin (1971) | MR 282313 | Zbl 0223.53034

[7] B. Bonnard, J.-B. Caillau, R. Sinclair, M. Tanaka, Conjugate and cut loci of a two-sphere of revolution with application to optimal control, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 1081-1098 | Numdam | MR 2542715 | Zbl 1184.53036

[8] M.P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall (1976) | MR 394451 | Zbl 0326.53001

[9] H.R. Fischer, J.J. Jungster, F.L. Williams, The heat kernel on the two-sphere, Adv. Math. 54 (1984), 226-232 | MR 762513 | Zbl 0549.58032

[10] S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Universitext, Springer (2004) | MR 2088027 | Zbl 0636.53001

[11] A. Grigor'Yan, Heat Kernel and Analysis on Manifolds, AMS/IP Stud. Adv. Math. vol. 47, Amer. Math. Soc., Providence, RI (2009) | MR 2569498 | Zbl 1206.58008

[12] D. Gromoll, W. Meyer, On differentiable functions with isolated critical points, Topology (1969), 361-369 | MR 246329 | Zbl 0212.28903

[13] E.P. Hsu, Stochastic Analysis on Manifolds, Grad. Stud. Math. vol. 38, Amer. Math. Soc., Providence, RI (2002) | MR 1882015

[14] J. Itoh, K. Kiyohara, The cut loci and the conjugate loci on ellipsoids, Manuscripta Math. (2004), 247-264 | MR 2067796 | Zbl 1076.53042

[15] J.-I. Itoh, K. Kiyohara, The cut loci on ellipsoids and certain Liouville manifolds, Asian J. Math. 14 (2010), 257-289 | MR 2746124 | Zbl 1210.53044

[16] R. Léandre, Développement asymptotique de la densité d'une diffusion dégénérée, Forum Math. 4 (1992), 45-75 | MR 1142473

[17] S.A. Molčanov, Diffusion processes, and Riemannian geometry, Uspekhi Mat. Nauk 30 (1975), 3-59 | MR 413289

[18] R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, Math. Surveys Monogr. vol. 91, Amer. Math. Soc., Providence, RI (2002) | MR 1867362 | Zbl 1044.53022

[19] R. Neel, The small-time asymptotics of the heat kernel at the cut locus, Comm. Anal. Geom. 15 (2007), 845-890 | MR 2395259 | Zbl 1154.58020

[20] S. Rosenberg, The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds, London Math. Soc. Stud. Texts vol. 31, Cambridge University Press, Cambridge (1997) | MR 1462892 | Zbl 0868.58074

[21] R. Sinclair, M. Tanaka, The cut locus of a two-sphere of revolution and Topogonov's comparison theorem, Tohoku Math. J. (2) 59 no. 3 (2007), 379-399 | MR 2365347 | Zbl 1158.53033

[22] S.R.S. Varadhan, On the behavior of the fundamental solution of the heat equation with variable coefficients, Comm. Pure Appl. Math. 20 (1967), 431-455 | MR 208191 | Zbl 0155.16503