Advertisement

Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

The Horizontal Heat Kernel on the Quaternionic Anti-De Sitter Spaces and Related Twistor Spaces

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 to check access.

References

  1. 1.

    Baditoiu, G.: Classification of Pseudo-Riemannian submersions with totally geodesic fibres from pseudo-hyperbolic spaces. Proc. Lond. Math. Soc. 105(6), 1315-1338 (2010)

  2. 2.

    Baditoiu, G., Ianus, S.: Semi-Riemannian submersions from real and complex hyperbolic spaces. Differential Geometry and its Applications 16(1), 79–94 (2002)

  3. 3.

    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)

  4. 4.

    Baudoin, F., Bonnefont, M.: The subelliptic heat kernel on SU(2): representations, asymptotics and gradient bounds, Math. Z. 263, 647–672 (2009)

  5. 5.

    Baudoin, F., Demni, N.: Integral representation of the sub-elliptic heat kernel on the complex anti-de Sitter fibration. To appear in ArKiv. Math

  6. 6.

    Baudoin, F., Grong, E., Kuwada, K., Thalmaier, A: Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations. Arxiv preprint (2017)

  7. 7.

    Baudoin, F., Wang, J.: The Subelliptic Heat Kernels of the Quaternionic Hopf Fibration, Potential Analysis. J. Potential Anal. 41(3), 959–982 (2014)

  8. 8.

    Bérard-Bergery, L., Bourguignon, J. P.: Laplacians and Riemannian submersions with totally geodesic fibres. Illinois. J. Math. 26(2), 181–200 (1982)

  9. 9.

    Biquard, O.: Quaternionic contact structures. In: Quaternionic structures in mathematics and physics (Rome, 1999), 23-30 (electronic). Univ. Studi Roma “La Sapienza”, Rome (1999)

  10. 10.

    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)

  11. 11.

    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)

  12. 12.

    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)

  13. 13.

    Faraut, J.: Analysis on Lie groups, an introduction. Cambridge University Press, Cambridge (2008)

  14. 14.

    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

  15. 15.

    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)

  16. 16.

    Jelonek, W.: Positive and negative 3-K contact structures, Proceedings of the AMS, vol. 129, number 1, pp. 247–256

  17. 17.

    Wang, J.: The subelliptic heat kernel on the anti-de Sitter spaces. J. Potential Anal. 45(4), 635–653 (2016)

  18. 18.

    Serre, J.-P.: Complex semisimple lie algebras, Springer Science & Business Media (2012)

Download references

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

Correspondence to Fabrice Baudoin.

Additional information

Author supported in part by the NSF Grant DMS 1660031

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

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

Download citation

Keywords

  • Heat kernels
  • Subelliptic operators
  • Anti-de Sitter spaces
  • Symmetric spaces

Mathematics Subject Classification (2010)

  • 53C17
  • 53C26
  • 53C28
  • 58J65
  • 60J60