Abstract
We present an explicit form of the subelliptic heat kernel of the intrinsic sublaplacian \(\Delta _{\textrm{sub}}^5\) induced by a rank 5 trivializable subriemannian structure on the Euclidean seven dimensional sphere \(\mathbb {S}^7\). This completes the heat kernel analysis of trivializable subriemannian structures on \(\mathbb {S}^7\) induced by a Clifford module action on \(\mathbb {R}^8\). As an application we derive the spectrum of \(\Delta _{\textrm{sub}}^5\) and the Green function of the conformal sublaplacian in an explicit form.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Availability of data and materials
Not applicable.
Code availability
Not applicable.
Notes
Chebyshev polynomials of the first kind fulfill the recursive relations: \(T_0(z)=1\), \(T_1(z)=z\) and
$$\begin{aligned} T_{\ell +1}(z) =2zT_{\ell }(z)-T_{\ell -1}(z). \end{aligned}$$Using induction one easily finds the estimate \(|T_{\ell }(z)| \le 4^{\ell } (1+R)^{\ell }\) if \(|z| \le R\) for \(R>0\). Therefore the above series converge uniformly on compact sets in \(\mathbb {C}\).
References
Adams, J.F.: Vector fields on spheres. Ann. Math. 75, 603–632 (1962)
Agrachev, A., Barilari, D., Boscain, U.: A comprehensive introduction to sub-Riemannian geometry. Cambridge University Press, Cambridge (2019)
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., Rizzi, R.: A formula for Popp’s volume in sub-Riemannian geometry. Anal. Geom. Metric Spaces. 1, 42–57 (2013)
Bariliari, D., Rizzi, R.: Sharp measure contraction property for generalized H-type Carnot groups. Commun. Contemp. Math. 20(6), 1750081 (2018)
Baudoin, F., Bonnefont, M.: The subelliptic heat kernel on \(\rm SU (2)\): representations, asymptotics and gradient bounds. Math. Z. 263, 647–672 (2009)
Baudoin, F., Wang, J.: The subelliptic heat kernels of the quaternionic Hopf fibration. Potential Anal. 41, 959–982 (2014)
Baudoin, F., Wang, J.: The subelliptic heat kernel on the CR sphere. Math. Z. 275, 135–150 (2013)
Bauer, W.: Subriemannian geometries on \(\mathbb{S}^{7}\) and spectral analysis. In: RIMS Kôkyûroku
Bauer, W., Furutani, K., Iwasaki, C.: Trivializable sub-Riemannian structures on spheres. Bull. Sci. Math. 137, 361–385 (2013)
Bauer, W., Laaroussi, A.: Trivializable and quaternionic subriemannian structure on \(\mathbb{S}^{7}\) and subelliptic heat kernel. J. Geom. Anal. 32(8), 219 (2022)
Bauer, W., Tarama, D.: Subriemannian geodesic flow on \(\mathbb{S}^{7}\). RIMS Kôkyûroku. 2137, 25–41 (2018)
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)
Beals, R., Gaveau, B., Greiner, P.C.: The Green function of model step two hypoelliptic operators and the analysis of certain tangential Cauchy Riemann complexes. Adv. Math. 121, 288–345 (1996)
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. Birkhäuser/Springer, New York (2011)
Calin, O., Chang, D.-C., Markina, I.: Subriemannian geometry on the sphere \(S^3\). Canad. J. Math. 61(4), 721–739 (2009)
Chow, W.L.: Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math. Ann. 117(1), 98–105 (1939)
Cowling, M., Sikora, A.: A spectral multiplier theorem for a sublaplacian on \(SU(2)\). Math. Z. 238, 1–36 (2001)
Erdélyi, T., Magnus, A.P., Navai, P.: Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials. SIAM J. Math. Anal. 25, 602–614 (1994)
Gradshteyn, I.S., Ryzhik, I. M.: Table of Integrals, Series, and Products, 7th edn. In: Jeffrey, A., Zwillinger, D. (eds.). Academic Press, New York (2007)
Greiner, P.: A Hamiltonian approach to the heat kernel of a sub-Laplacian on \(S^{2n+1}\). Anal. Appl. (Singap.). 11(6), 1350035 (2013). arXiv:1303.0457
Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)
Hörmander, L.: The analysis of partial differential operators I, 2nd edn. Springer, Berlin-Tokyo (1990)
James, I. M.: The topology of Stiefel manifolds. London Math. Soc. Lecture Note Ser., vol. 24, Cambridge Univ. Press, Cambridge (1977)
Krasikov, I.: An upper bound on Jacobi polynomials. J. Approx. Theory. 149, 116–130 (2007)
Laaroussi, A.: Heat kernel asymptotic for quaternionic contact manifolds. arXiv:2103.00892v1 (2021)
Markina, I.: Geodesics in geometry with constraints and applications. In: Quantization, PDEs, and geometry, 153 - 314, Oper. Theory Adv. Appl., 251, Adv. Partial Differ. Equ. (Basel), Birkhäuser/Springer, Cham (2016)
Markina, I., Godoy Molina, M.: Sub-Riemannian geodesics and heat operator on odd dimensional spheres. Anal. Math. Phys. 2, 123–147 (2012)
Markina, I., Godoy Molina, M.: Sub-Riemannian geometry of parallelizable spheres. Rev. Mat. Iberoam. 27(3), 997–1022 (2011)
Montgomery, R.: A tour of subriemannian geometries, their geodesics and applications. Mathematical Surveys and Monographs, 91. AMS, Providence (2002)
Rashevskii, P.K.: About connecting two points of complete nonholonomic space by admissible curve. Uch. Zapiski ped. inst. Libknexta. 2, 83 - 94 (in Russian) (1938)
Strichartz, R.: Sub-Riemannian Geometry. J. Diff. Geom. 24(2), 221–263 (1986)
Strichartz, R.: Corrections to “Sub-Riemannian Geometry”. J. Diff. Geom. 30(2), 595–596 (1989)
Taylor, M.E.: Partial differential equations, qualitative studies of linear equations. Applied Mathematical Sciences, Springer, Heidelberg (1996)
Verdiére, Y., Hillairet, L., Trélat, E.: Small-time asymptotics of hypoelliptic heat kernels near the diagonal, nilpotentization and related results. Ann. H. Lebesgue. 4, 897–971 (2021)
Whittaker, E.T., Watson, G.N.: A course of modern analysis, 4th edn. Cambridge University Press, Cambridge (2005)
Funding
Open Access funding enabled and organized by Projekt DEAL. The first and second author have been supported by the priority programm SPP 2026 geometry at infinity of Deutsche Forschungsgemeinschaft (project number BA 3793/6-1). The third author has been supported by JSPS KAKENHI, grant number JP19K14540, JP22H01138, JP23H04481. This work was partially supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing interests
The authors have no relevant financial or non-financial interests to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
In this appendix we provide some estimates on the heat kernel of the Laplacian on \(\mathbb {S}^7\) which justify the convergence of the integral expressions in Theorem 4.10.
Recall the following closed form of the heat kernel of \(\Delta _{\mathbb {S}^7}\), (c.f. [34, (4.26), p. 97]). With the notation in (4.12) and \(t>0\):
and \(\vartheta (s,t)\) denotes the theta function:
With \(s\in \mathbb {C}\) and \(r\in \mathbb {N}_0\) consider the function
If we define \(F_{-1}(t,s) \equiv 0\), then one easily verifies the recursive relation:
Formula (6.3) shows that \(k^{\mathbb {S}^7}_t(s)\) is expressed with respect to \(F_r\) as follows:
Since \(F_0(t,\cdot )\) is even it follows that for all \(m \in \mathbb {N}\) the functions
are even as well and holomorphically extend to \(s=0\). Therefore, for fixed \(t>0\) the right hand side of (6.6) is holomorphic in s on the strip \(S_{\pi }:=\{ s \, : \, -\pi<\textrm{Re }s<\pi \}\). We keep denoting this extension to \(\mathbb {R}_+ \times S_{\pi }\) by
From the expression of \(k_t^{\mathbb {S}^7}(s)\) we find:
Lemma 1.24 For all \(t>0\) the function \(k_t^{\mathbb {S}^7}(\cdot )\) is even. In particular, \(k_t^{\mathbb {S}^7}(is) \in \mathbb {R}\) for all \(t >0\) and \(s \in \mathbb {R}\).
Let \(\sigma \in \mathbb {R}\) and assume that \(\lambda _0\in (-\pi ,\pi )\) is fixed. Then we estimate the growth of the function \(F_r(t,\lambda _0+i\sigma )\) along the variable \(\sigma \) and for fixed \(r\in \mathbb {N}_0\):
The expression in the first bracket on the right hand side is a polynomial in the variable \(\sigma \) of degree \(\le 2r\). Hence, there is \(C_t>0\) independent of \(\sigma \) such that:
The constant \(C_t\) has at most polynomial growth as a function of \(t^{-1} \in \mathbb {R}_+\) and as \(t \downarrow 0\). Recall the Poisson summation formula:
which allows to express \(F_0(t,s)\) in the form:
Applying the recursive formula (6.5) we find:
We can rewrite \((\sin s)\cdot F_1\) and \((\sin s) \cdot F_3\) as functions of \(\cos (\ell s)\) by applying the relation \(2\sin (s)\sin (\ell s)=\cos ((\ell -1)s)-\cos ((\ell +1)s)\) for all \(\ell \in \mathbb {N}\):
Let \(\ell \in \mathbb {N}_0\) and by \(T_{\ell }(x)\) denote the \(\ell \)-th Chebyshev polynomial of the first kind. Then \(T_{\ell }(\cos s)=\cos (\ell s)\) and we can express \(F_0\), \(F_2\), \(G_1\), \(G_3\) in the form:
where \(j=0,2\) and \(r=1,3\). Here the functions \(\widetilde{F}_j\) and \(\widetilde{G}_r\) are given in the form:
Note that for each \(t>0\) these functions are entire on \(\mathbb {C}\).Footnote 1 With the notation in (6.8) we rewrite the heat kernel (6.6) in the form \(k_t^{\mathbb {S}^7}(s)= q_t \circ \cos s\), where
Remark 1.25 Note that for fixed \(t>0\) the function \(q_t\) is defined in \(z=1\) since the following limit exists:
Therefore, for each \(t>0\) the function \(q_t\) is holomorphic on \(\mathbb {C} \setminus \{1\}\).
Next, we estimate the growth of \(\widetilde{F}_0(t,x)\), \(\widetilde{F}_2(t,x)\), \(\widetilde{G}_1(t,x)\) and \(\widetilde{G}_3(t,x)\) for \(x\in \mathbb {R}\).
Lemma 1.26 Let \(j \in \mathbb {N}_0\) and \(t>0\), then there is a constant \(C_t>0\) only depending on t but independent of R such that
Proof
Taking the j-th derivative, \(j=1,2,3\) with respect to z of the Poisson summation formula (6.9) gives the relations:
where \(p_j(\ell , z)\) are polynomials of degree j with t-dependent coefficients. Let \(R>0\) and choose \(z= \frac{i}{2\pi } \log R\). If we replace t by \(\frac{\pi ^2}{t}\), then it follows:
Hence there is a constant \(C_t>0\) independent of \(R>0\) such that
Note that the expression in the first brackets on the right is a polynomial in \(\log R\) of degree 2j. From this the estimate follows. \(\square \)
We can estimate the growth of \(\widetilde{F}_0(t,z)\), \(\widetilde{F}_2(t,z)\), \(\widetilde{G}_1(t,z)\) and \(\widetilde{G}_3(t,z)\) with respect to the variable z. Recall the inequality:
Then Lemma 6.4 shows that there are constants \(C_{j,t}>0\) for \(j=0,1,2,3\) independent of z such that:
Note: there is a constant \(\beta >0\) such that for \(s>0\):
In conclusion, we can find constants \(\widetilde{C}_{j,t}\) independent of \(s \in \mathbb {R}\) such that:
Lemma 1.27 Let \(r \in (0, \frac{\pi }{2})\) and \(s \in \mathbb {R}\). Then there is a constant \(D_{r,t}\) independent of s such that
Proof
Since \(\cosh \) is even, we can assume without loss of generality that \(s>1\) and \(\cos r \cosh s>s\). Then the statement follows from the representation (6.11) of \(q_t\) together with the estimates in (6.12). \(\square \)
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bauer, W., Laaroussi, A. & Tarama, D. Rank 5 Trivializable Subriemannian Structure on \(\mathbb {S}^7\) and Subelliptic Heat Kernel. Potential Anal (2023). https://doi.org/10.1007/s11118-023-10110-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11118-023-10110-8