Abstract
The main goal of this work is to study the sub-Laplacian of the unit sphere which is obtained by lifting with respect to the Hopf fibration the Laplacian of the quaternionic projective space. We obtain in particular explicit formulas for its heat kernel and deduce an expression for the Green function of the conformal sub-Laplacian and small-time asymptotics. As a byproduct of our study we also obtain several results related to the sub-Laplacian of a projected Hopf fibration.
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Agrachev, A., Boscain, U., Gauthier, J.P., Rossi, F.: The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups. J. Funct. Anal. 256(8), 2621–2655 (2009)
Barilari, D., Boscain, U., Neel, R.: Small time heat kernel asymptotics at the sub -Riemannian cut locus. JDG 92(3), 373–416 (2012)
Baudoin, F., Bonnefont, M.: The subelliptic heat kernel on S U(2): representations, asymptotics and gradient bounds. Math. Z 263, 647–672 (2009)
Baudoin, F., Cecil, M.: The subelliptic heat kernel on the three-dimensional solvable groups. To appear in Forum Mathematicum (2013)
Baudoin. F., Wang, J: The subelliptic heat kernel on the CR sphere. Math. Z 275(1-2), 135–150 (2013)
Bauer, R.O.: Analysis of the horizontal Laplacian for the Hopf fibration. Forum Math. 17(6), 903–920 (2005)
Beals, R., Gaveau, B., Greiner, P.C.: Hamilton-Jacobi theory and the heat kernel on Heisenberg groups. J. Math. Pures Appl. 79(7), 633–689 (2000)
Ben Arous, G.: Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus. Ann. Sci. École Norm. Sup. (4) 21, 307–331 (1988)
Biquard, O.: Quaternionic contact structures. Univ. Studi Roma “La Sapienza”, Rome (1999). In: Quaternionic Structures in Mathematics and Physics. (Rome, 1999), 23–30 (electronic)
Bonnefont, M.: The subelliptic heat kernel on SL(2,R) and on its universal covering: integral representations and some functional inequalities. Potential Anal. 36(2), 275–300 (2012)
Boyer, CP, Galicki, K: 3-Sasakian manifolds, Surveys in differential geometry: essays on Einstein manifolds, pp. 123–184. Surv. Differ. Geom., VI, Int. Press, Boston (1999)
Calin, O, Chang, D C, Furutani, K., Iwasaki, C.: Heat kernels for elliptic and sub-elliptic operators. Methods and techniques. In: Applied and Numerical Harmonic Analysis. Birkhuser/Springer, New York (2011)
Escobales, R.H.: Riemannian submersions with totally geodesic fibers. J. Differ. Geom. 10, 253–276 (1975)
Faraut, J: Analysis on Lie Groups, an Introduction. Cambridge University Press (2008)
Gaveau B.: Principe de moindre action, propagation de la chaleur et estiméees sous elliptiques sur certains groupes nilpotents. Acta Math. 139(1), 95–153 (1977)
Greiner, P.: A Hamiltonian Approach to the Heat Kernel of a Sub-Laplacian on S(2n+1). arXiv:1303.0457
Ivanov, S., Minchev, I., Vassilev, D: Quaternionic Contact Einstein Structures and the Quaternionic Contact Yamabe Problem. (to appear in Mem. of AMS) (2013)
Ivanov, S., Petkov, A., Vassilev, D: The Obata sphere theorems on a quaternionic contact manifold of dimension bigger than seven. arXiv:1303.0409 (2013)
Molina, M, Markina, I.: Sub-Riemannian geodesics and heat operator on odd dimensional spheres. Anal. Math. Phys. 2(2), 123–147 (2012)
Wang, J.: The subelliptic heat kernel on the CR hyperbolic spaces. arXiv:1204.3642 (2012)
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Baudoin, F., Wang, J. The Subelliptic Heat Kernels of the Quaternionic Hopf Fibration. Potential Anal 41, 959–982 (2014). https://doi.org/10.1007/s11118-014-9403-z
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DOI: https://doi.org/10.1007/s11118-014-9403-z