Skip to main content
SpringerLink
Log in

The Subelliptic Heat Kernel of the Octonionic Hopf Fibration

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

We study the sub-Laplacian of the 15-dimensional unit sphere which is obtained by lifting with respect to the Hopf fibration the Laplacian of the octonionic projective space. We obtain in particular explicit formulas for its heat kernel and deduce an expression for the Green function of a related sub-Laplacian. As a byproduct we also obtain the spectrum of the sub-Laplacian, the small-time asymptotics of the heat kernel and explicitly compute the sub-Riemannian distance.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baudoin, F.: Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations. Geometry, analysis and dynamics on sub-Riemannian manifolds 1, 259–321 (2016). MR3587668

    MathSciNet  MATH  Google Scholar 

  2. Baudoin, F., Bonnefont, M.: The subelliptic heat kernel on SU(2): representations, asymptotics and gradient bounds. Math. Z. 263(3), 647–672 (2009). https://doi.org/10.1007/s00209-008-0436-0. MR2545862

    Article  MathSciNet  MATH  Google Scholar 

  3. Baudoin, F., Grong, E., Molino, G., Rizzi, L.: H-type foliations (2018)

  4. Baudoin, F., Wang, J.: The subelliptic heat kernel on the CR sphere. Math. Z. 275(1-2), 135–150 (2013). https://doi.org/10.1007/s00209-012-1127-4. MR3101801

    Article  MathSciNet  MATH  Google Scholar 

  5. Baudoin, F., Wang, J.: The subelliptic heat kernels of the quaternionic Hopf fibration. Potential Anal. 41(3), 959–982 (2014). https://doi.org/10.1007/s11118-014-9403-z. MR3264830

    Article  MathSciNet  MATH  Google Scholar 

  6. Baudoin, F., Wang, J.: Stochastic areas, winding numbers and Hopf fibrations. Probab. Theory Related Fields 169(3-4), 977–1005 (2017). https://doi.org/10.1007/s00440-016-0745-x. MR3719061

    Article  MathSciNet  MATH  Google Scholar 

  7. Beals, R., Gaveau, B., Greiner, P.C.: Hamilton-Jacobi theory and the heat kernel on Heisenberg groups. J. Math. Pures Appl. (9) 79(7), 633–689 (2000). MR1776501

    Article  MathSciNet  Google Scholar 

  8. Bérard-Bergery, L., Bourguignon, J.-P.: Laplacians and Riemannian submersions with totally geodesic fibres. Illinois J. Math. 26(2), 181–200 (1982). MR650387

    MathSciNet  MATH  Google Scholar 

  9. Besse, A.L.: Manifolds all of whose geodesics are closed. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93. Springer-Verlag, Berlin-New York (1978). With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan. MR496885

  10. Besse, A.L.: Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10. Springer-Verlag, Berlin (1987). MR867684

  11. Demni, N., Zani, M.: Large deviations for statistics of the Jacobi process. Stochastic Process. Appl. 119(2), 518–533 (2009). https://doi.org/10.1016/j.spa.2008.02.015. MR2494002

    Article  MathSciNet  MATH  Google Scholar 

  12. Escobales, R., Jr.: Riemannian submersions with totally geodesic fibers. J. Differential Geometry 10, 253–276 (1975). https://doi.org/http://projecteuclid.org/euclid.jdg/1214432793. MR370423

    Article  MathSciNet  Google Scholar 

  13. Escobales, R.H., Jr.: Riemannian foliations of the rank one symmetric spaces. Proc. Amer. Math. Soc. 95(3), 495–498 (1985). MR806095

    Article  MathSciNet  Google Scholar 

  14. Faraut, J.: Analysis on Lie groups. Cambridge Studies in Advanced Mathematics, vol. 110. Cambridge University Press, Cambridge (2008). An introduction. MR2426516

  15. Geller, D.: The Laplacian and the Kohn Laplacian for the sphere. J. Differential Geometry 15(3), 417–435 (1981) (1980). MR620896

    Article  MathSciNet  Google Scholar 

  16. Greiner, P.C.: A Hamiltonian approach to the heat kernel of a sublaplacian on S2n+ 1. Anal. Appl. (Singap.) 11(6), 1350035 (2013). https://doi.org/10.1142/S0219530513500358. 62 MR3130140

    Article  MathSciNet  MATH  Google Scholar 

  17. Helgason, S.: The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds. Acta Math. 113, 153–180 (1965). https://doi.org/10.1007/BF02391776. MR172311

    Article  MathSciNet  MATH  Google Scholar 

  18. Helgason, S.: Groups and geometric analysis. Pure and Applied Mathematics, vol. 113. Academic Press, Inc., Orlando, FL (1984). Integral geometry, invariant differential operators, and spherical functions. MR754767

  19. Léandre, R.: Majoration en temps petit de la densité d’une diffusion dégénérée. Probab. Theory Related Fields 74(2), 289–294 (1987). MR871256

    Article  MathSciNet  Google Scholar 

  20. Léandre, R.: Minoration en temps petit de la densité d’une diffusion dégénérée. J. Funct. Anal. 74(2), 399–414 (1987). MR904825

    Article  MathSciNet  Google Scholar 

  21. Ornea, L., Parton, M., Piccinni, P., Vuletescu, V.: Spin(9) geometry of the octonionic Hopf fibration. Transform. Groups 18(3), 845–864 (2013). https://doi.org/10.1007/s00031-013-9233-x. MR3084336

    Article  MathSciNet  MATH  Google Scholar 

  22. Ranjan, A.: Riemannian submersions of spheres with totally geodesic fibres. Osaka J. Math. 22(2), 243–260 (1985). http://projecteuclid.org/euclid.ojm/1200778256 MR800969

    MathSciNet  MATH  Google Scholar 

  23. Ucci, J.: On the nonexistence of Riemannian submersions from C P(7) and QP(3). Proc. Amer. Math. Soc. 88(4), 698–700 (1983). https://doi.org/10.2307/2045465. MR702302

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang, J.: Conformal transforms and Doob’s h-processes on Heisenberg groups, Stochastic analysis and related topics, pp. 165–177. MR3737629 (2017)

Download references

Acknowledgments

The authors thank the anonymous referee for a careful reading and remarks that greatly helped improve the presentation of the paper. We point out that due to their complexity, some of the symbolic computations were computer assisted using the software Mathematica.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, 196 Auditorium Road, Storrs, CT, 06269-3009, USA

    Fabrice Baudoin & Gunhee Cho

Authors
  1. Fabrice Baudoin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gunhee Cho
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gunhee Cho.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

F.B is partially funded by the Simons Foundation and NSF grant DMS-1901315.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baudoin, F., Cho, G. The Subelliptic Heat Kernel of the Octonionic Hopf Fibration. Potential Anal 55, 211–228 (2021). https://doi.org/10.1007/s11118-020-09854-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-020-09854-4

Keywords

Search

Navigation