Abstract
The geometry of the quaternionic anti-de Sitter fibration is studied in details. As a consequence, we obtain formulas for the horizontal Laplacian and subelliptic heat kernel of the fibration. The heat kernel formula is explicit enough to derive small time asymptotics. Related twistor spaces and corresponding heat kernels are also discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baditoiu, G.: Classification of Pseudo-Riemannian submersions with totally geodesic fibres from pseudo-hyperbolic spaces. Proc. Lond. Math. Soc. 105(6), 1315-1338 (2010)
Baditoiu, G., Ianus, S.: Semi-Riemannian submersions from real and complex hyperbolic spaces. Differential Geometry and its Applications 16(1), 79–94 (2002)
Baudoin, F.: Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations, Geometry, analysis and dynamics on sub-Riemannian manifolds, vol. 1, pp. 259–321. EMS Ser. Lect. Math., Eur. Math. Soc., Zürich (2016)
Baudoin, F., Bonnefont, M.: The subelliptic heat kernel on SU(2): representations, asymptotics and gradient bounds, Math. Z. 263, 647–672 (2009)
Baudoin, F., Demni, N.: Integral representation of the sub-elliptic heat kernel on the complex anti-de Sitter fibration. To appear in ArKiv. Math
Baudoin, F., Grong, E., Kuwada, K., Thalmaier, A: Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations. Arxiv preprint (2017)
Baudoin, F., Wang, J.: The Subelliptic Heat Kernels of the Quaternionic Hopf Fibration, Potential Analysis. J. Potential Anal. 41(3), 959–982 (2014)
Bérard-Bergery, L., Bourguignon, J. P.: Laplacians and Riemannian submersions with totally geodesic fibres. Illinois. J. Math. 26(2), 181–200 (1982)
Biquard, O.: Quaternionic contact structures. In: Quaternionic structures in mathematics and physics (Rome, 1999), 23-30 (electronic). Univ. Studi Roma “La Sapienza”, Rome (1999)
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, C.P., Galicki, K.: 3-Sasakian manifolds, Surveys in differential geometry: essays on Einstein manifolds, 123-184. Surv. Differ Geom., VI, Int. Press, Boston (1999)
Davies, E.B., Mandouvalos, N.: Heat Kernel Bounds on Hyperbolic Space and Kleinian Groups, Proceedings of the London Mathematical Society, vol. 3-57, Issue 1, pp. 182–208 (1988)
Faraut, J.: Analysis on Lie groups, an introduction. Cambridge University Press, Cambridge (2008)
Gibbons, G.W.: Anti-de-Sitter spacetime and its uses. Mathematical and Quantum Aspects of Relativity and Cosmology 537, 102–142 (2000). Lecture Notes in Physics
Intissar, A., Ould Moustapha, M. V.: Explicit formulae for the wave kernels for the laplacians Δβ in the Bergman Ball Bn,n1. Ann. Glo. Anal. Geom. 15, 221–234 (1997)
Jelonek, W.: Positive and negative 3-K contact structures, Proceedings of the AMS, vol. 129, number 1, pp. 247–256
Wang, J.: The subelliptic heat kernel on the anti-de Sitter spaces. J. Potential Anal. 45(4), 635–653 (2016)
Serre, J.-P.: Complex semisimple lie algebras, Springer Science & Business Media (2012)
Acknowledgments
The authors would like to thank Brian Hall for general discussions about complexification of symmetric spaces and the notion of Cartan dual.
Author information
Authors and Affiliations
Corresponding author
Additional information
Author supported in part by the NSF Grant DMS 1660031
Rights and permissions
About this article
Cite this article
Baudoin, F., Demni, N. & Wang, J. The Horizontal Heat Kernel on the Quaternionic Anti-De Sitter Spaces and Related Twistor Spaces. Potential Anal 52, 281–300 (2020). https://doi.org/10.1007/s11118-018-9746-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-018-9746-y