Small time heat kernel asymptotics at the cut locus on surfaces of revolution
Barilari, Davide ; Jendrej, Jacek
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014), p. 281-295 / Harvested from Numdam
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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},
     volume = {31},
     year = {2014},
     pages = {281-295},
     doi = {10.1016/j.anihpc.2013.03.003},
     mrnumber = {3181670},
     zbl = {1301.53035},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 31 (2014) pp. 281-295. doi : 10.1016/j.anihpc.2013.03.003. http://gdmltest.u-ga.fr/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