Sub-Riemannian balls in CR Sasakian manifolds

We prove global estimates for the sub-Riemannian distance of CR Sasakian manifolds with non negative horizontal Webster-Tanaka Ricci curvature. In particular, in this setting, large sub-Riemannian balls are comparable to Riemannian balls.


Introduction
Let M be a complete strictly pseudo convex CR Sasakian manifold with real dimension 2n+1. Let θ be a pseudo-hermitian form on M with respect to which the Levi form L θ is positive definite. The kernel of θ determines an horizontal bundle H. Denote now by T the Reeb vector field on M, i.e., the characteristic direction of θ. We denote by ∇ the Tanaka-Webster connection of M.
We recall that the CR manifold (M, θ) is called Sasakian if the pseudo-hermitian torsion of ∇ vanishes, in the sense that T(T, X) = 0, for every X ∈ H. For instance the standard CR structures on the Heisenberg group H 2n+1 and the sphere S 2n+1 are Sasakian. In every Sasakian manifold the Reeb vector field T is a sub-Riemannian Killing vector field (see Theorem 1.5 on p. 42 and Lemma 1.5 on p. 43 in [3]).
We consider the family of scaled Riemannian metrics g τ , τ > 0, such that for X, Y ∈ H: where J is the complex structure on M. We denote by d τ the distance corresponding to the Riemannian structure g τ and by d the sub-Riemannian distance on M. It is well known that d τ (x, y) → d(x, y), when τ → 0. Our goal is to prove the following theorem: Theorem 1.1. Let R be the Ricci curvature of the Webster-Tanaka connection ∇. If for every X ∈ H, R(X, X) ≥ 0, then for every x, y ∈ M, where A n and B n are two positive universal constants depending only on n.
First author supported in part by NSF Grant DMS 0907326.
To put things in perspective, estimates between the sub-Riemannian distance and Riemannian ones have been extensively studied in the litterature (see for instance [4], [5], [6], [7], [8]). But in these cited works, such estimates are local in nature. To the knowledge of the authors Theorem 1.1 is the first result that gives global and uniform estimates for a large class of sub-Riemannian metrics. It is consistent with the well known Nagel-Stein-Wainger estimate [7], that implies at small scales d(x, y) ≤ Cd τ (x, y) 1 2 and shows that due to curvature effects at big scales we have d(x, y) ≃ d τ (x, y).

2.
Li-Yau and Harnack estimates for the heat kernel on Sasakian manifolds 2.1. Curvature dimension inequalities and heat kernel bounds. We first recall some results that will be needed in the sequel and that can be found in [1] and [2]. We denote by ∆ the sub-Laplacian on M and by ∇ H the horizontal gradient. For smooth functions f : M → R, set The following result was obtained in [2] by means of a Bochner's type formula.
Then for every f ∈ C ∞ (M) and any ν > 0, We denote by p(t, x, y) the heat kernel of M, that is the fundamental solution of the heat equation ∂f ∂t = ∆f . The following global lower and upper bounds were proved in [1]. Theorem 2.2. Assume that for every X ∈ H, For any 0 < ε ≤ 1 there exists a constant C(ε) = C(n, ε) > 0, which tends to ∞ as ε → 0 + , such that for every x, y ∈ M and t > 0 one has In the above Theorem, d is the sub-Riemannian distance, B(x, √ t) is the sub-Riemannian ball with center x and radius √ t and µ is the volume corresponding to the volume form θ ∧ (dθ) n .

2.2.
Harnack type estimates. From now on and in all the sequel we assume that for every X ∈ H, R(X, X) ≥ 0. We first have the following Li-Yau type estimate for the heat kernel.
Proof. The result is essentially proved in [2], but due to the simplicity of the argument we reproduce, without the details, the proof by sake of completeness. Fix T > 0 and consider the functional where P t is the heat semigroup associated with ∆. Since T is a Killing vector field, for any smooth function f we have Differentiating Φ and using the above yields Therefore we obtain Now, for every γ(t), we have Therefore we get 3 Setting γ(t) = − n+3 T −t , leads then to By integrating the last inequality from 0 to T , we obtain −Φ(0) ≥ − 3(n + 3) n 2 T 2 ∆p T − 3 n 2 (n + 3) 2 T, which is the required inequality.
We can deduce from the previous Li-Yau type inequality the following Harnack inequality. where d τ denotes the Riemannian metric introduced in (1.1).
Choosing λ = a(u) 2 and using then (2.4) yields By integrating this inequality from s to t we get as a result.
We now minimize the quantity t s a(u) γ ′ (u) 2 du over the set of absolutely continuous paths such that γ(s) = y, γ(t) = z. By using reparametrization of paths, it is seen that