The Subelliptic Heat Kernels of the Quaternionic Hopf Fibration
West Lafayette, IN, USA
Abstract
The main goal of this work is to study the subLaplacian 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 subLaplacian and smalltime asymptotics. As a byproduct of our study we also obtain several results related to the subLaplacian of a projected Hopf fibration.
1 Introduction
The first objective of this work is to study and give meaningful formulas for the heat kernel of the subLaplacian which is associated to the quaternionic Hopf fibration on :
This fibration originates from the natural action of on , which identifies as a bundle over the projective quaternionic space . The subLaplacian we are interested in appears then as the lift on of the LaplaceBeltrami operator of . As we will see, it is intrinsically associated to the quaternionic contact geometry of .
Let us briefly describe our main results. One of the first observations is that, due to the symmetries of the above fibration, the heat kernel of the subLaplacian only depends on two variables: the variable which is the Riemannian distance on and the variable which is the Riemannian distance on the fiber . We prove that in these coordinates, the cylindric part of the subLaplacian writes
As a consequence of this expression for the subLaplacian, we are able to derive two expressions for the heat kernel:

A MinakshisundaramPleijel spectral expansion:
where and is a Jacobi polynomial. This formula is useful to study the longtime behavior of the heat kernel but seems difficult to use in the study of smalltime asymptotics or for the purpose of proving upper and lower bounds. In order to derive smalltime asymptotics of the kernel, we give another analytic expression for which is much more geometrically meaningful.

An integral representation:
where is the heat kernel of the Riemannian structure of . We obtain this formula by employing a similar idea that was developed in the usual Hopf fibrations (see [3], [5], [10], [20]). The key point is the commutation between the subLaplacian and the transversal directions in . From this formula we are able to deduce the fundamental solution of the conformal subLaplacian . Furthermore, we also derive three different behaviors of the smalltime asymptotics of the heat kernel: on the diagonal, on the vertical cutlocus, and outside of cutlocus. An interesting byproduct of this smalltime asymptotics we obtain, is a previously unknown explicit formula for the subRiemannian distance on the quaternionic unit sphere.
The second main objective of this work is the study of another subLaplacian that we now define. The natural action of on induces the classical Hopf fibration
whose subLaplacian and corresponding heat kernel were studied in details in our previous work [5]. Identifying with a subgroup of , defines a fibration
that makes the following diagram commutative
{diagram}
& & S^1 & &
& \ldTo& \dTo& &
SU(2) & \rTo&S^4n+3 & \rTo& HP^n
\dTo& &\dTo& \ruTo&
CP^1 & \rTo& CP^2n+1 & &
We are interested in the subLaplacian of the complex projective space which is obtained by lifting the LaplaceBeltrami operator of . Again, our main goal will be to provide meaningful formulas for the heat kernel of this subLaplacian. For the very same reasons as above, the heat kernel of this subLaplacian only depends on two variables: which is again the Riemannian distance on and which is the Riemannian distance on . The expression of the cylindric part of the subLaplacian is then given by:
The MinakshisundaramPleijel expansion of can then be deduced in the same fashion as on , and by comparing it with the previous spectral decomposition of , we prove the following intertwining between the two kernels:
As a consequence we obtain in particular the smalltime asymptotics of .
To put our work in perspective, we mention that the study of subelliptic heat kernels on model spaces has generated quite a lot of interest in the past and is still nowadays an active domain of research which lies at the intersection of harmonic analysis, partial differential equations, control theory, differential geometry and probability theory (see the book [12] for an overview). One of the first studies goes back to Gaveau [15] who provided an expression for the subelliptic heat kernel on the simplest subRiemannian model space, the 3dimensional Heisenberg group. The subelliptic heat kernel of the 3dimensional Hopf fibration on was first studied by Bauer [6] and then, in more details by Baudoin and Bonnefont [3]. The subelliptic heat kernel of the 3dimensional Hopf fibration on was then studied by Bonnefont [10]. A general study of heat kernels on any 3dimensional contact manifold is then presented in [4]. The dimensional generalization of the work by Gaveau [15] was made by Beals, Gaveau and Greiner in [7]. The dimensional generalization of the work by BaudoinBonnefont [3] was made by the two present authors in [5]. We also mention the work [19] and the work by Greiner [16] who recovers results of [5] by using the Hamiltonian method. The dimensional generalization of the work by Bonnefont [10] was made by the second author of the present paper in [20]. We also point out the work by AgrachevBoscain and Gauthier [1] that studies a general class of subelliptic heat kernels on Lie groups.
To conclude, we can observe that, up to an exotic example, due to the work of Escobales [13] the submersions and are the only examples of Riemannian submersions of the sphere with totally geodesic fibers. As a consequence our work is a perfect complement of [5] and completes hence the study of all the natural subelliptic heat kernels of the unit sphere that come from a Riemannian submersion.
2 The subelliptic heat kernel on
The subRiemannian geometry of we are interested in, may be defined in at least two ways. The first one is to consider the quaternionic Hopf fibration
Lifting the Laplace Beltrami operator of with respect to this submersion gives the subLaplacian of we want to study. This point of view gives a quick way to prove Proposition 2.2 below.
The second way to study the subRiemannian geometry of is to see as a quaternionic contact manifold (more precisely a 3Sasakian manifold). The subLaplacian on is then simply defined as the trace of the horizontal Hessian for the Biquard connection. For the sake of completeness we give a proof of Proposition 2.2 by using the quaternionic contact point of view, because it is natural to really see as the model space of a positively curved quaternionic contact manifold.
The reader more interested in the analysis of the subLaplacian than in the geometry associated to it may skip Section 2.1. and admit Proposition 2.2. Once Proposition 2.2 is admitted the remainder of the paper may be read without further references to quaternionic contact geometry.
2.1 The quaternionic contact structures and the unit spheres
Quaternionic spheres appear as the model spaces of quaternionic contact manifolds and 3Sasakian manifolds (see [11, 17, 18]) . We introduce them as follows: the quaternionic unit sphere is given by
where we denote the quaternionic field by
where are the Pauli matrices:
The quaternionic norm is
The subRiemannian structure of the quaternionic spheres comes from the quaternionic Hopf fibration:
that we now describe. There is natural and isometric group action of the Lie group on which is given by,
For any , the infinitesimal generator of the left translation by is given by
that is,
where we use the real coordinates in , . Similarly, since
and
we have that
and
In quaternionic coordinates since
and
They form a basis of the fibers of a  bundle structure on .
We denote the one form . Simple computations show that for all , ,
and
Therefore, we have that for all ,
We choose the contact form . It has only imaginary part and we denote by where are real contact forms.
Let and clearly . If we assume that , then , then
We can easily compute that
and
Therefore, the real contact forms given by
satisfy that
The horizontal distribution of is given by the kernel of and induces the quaternionic Hermitian structure on : for all ,
It is compatible with the Hermitian structure in the sense that
Notice that , and we can obtain the semiRiemannian metric on by extending as follows:
for all .
2.2 The subLaplacian on
On a general contact quaternionic manifold whose vertical space is generated by three Reeb vector fields, there exists a canonical connection which preserves the metric and the almost complex structure (see [9]). It is called the Biquard connection. We now introduce the canonical subLaplacian on as follows: for any ,
(2.1) 
where is the pseudoHermitian Hessian of with respect to the Biquard connection, and is the restriction of to for any bilinear form on .
To compute in local coordinates, we project the vector fields onto and obtain the horizontal vector fields: since
Therefore
Similarly, we can obtain , , by projecting , , and as follows: since
Similarly, plug in
and
Therefore we obtain
The subLaplacian is then given by
that is
It can be written in quaternionic coordinates as follows:
(2.2) 
It is hence seen that is essentially selfadjoint on with respect to the volume measure and related to the LaplaceBeltrami operator of the standard Riemannian structure on by the formula:
where
Moreover, we can observe, a fact which will be important for us, that and commute, that is, on smooth functions .
To study we now introduce a new set of coordinates that reflect the symmetries of the quaternionic contact structure. Keeping in mind the submersion , we let be local coordinates for , where are the local inhomogeneous coordinates for given by , and are local coordinates on the fiber. More explicitly, these coordinates are given by
(2.3) 
where , , and . Therefore by considering the diffeomorphism
and restrict to
we have that on , for all :
Moreover, obviously we have that
and when restricted to , we have . Thus
It is obvious that the subelliptic heat kernel is cylindric symmetric, i.e. it only depends on two coordinates where and are the Riemannian distance from the north pole on and respectively. Therefore we just need to write the cylindrical part of , denote as . We define it rigorously as follows:
Definition 2.1
Let us denote by the map from to such that
where and is the Riemannian distance from the identity on . We denote by the space of smooth and compactly supported functions on . Then the cylindrical part of is defined by such that for every , we have
We now compute the subLaplacian in cylindric coordinates.
Proposition 2.2
The cylindric part of the subLaplacian on is given in the coordinates by
(2.4) 
Proof. Noticing that
and
Moreover,
and we have that
By the same reason we have that and vanish as well. This yields,
Hence we obtain after simplifications