Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations: part II

Abstract

Abstract. Using the curvature-dimension inequality proved in Part I, we look at consequences of this inequality in terms of the interaction between the subRiemannian geometry and the heat semigroup Pt corresponding to the subLaplacian. We give bounds for the gradient, entropy, a Poincaré inequality and a Li-Yau type inequality. These results require that the gradient of Ptf remains uniformly bounded whenever the gradient of f is bounded and we give several sufficient conditions for this to hold.

Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations

“…Going forward, we will always assume in the sequel of the paper that the horizontal distribution H satisfies the Yang-Mills condition, meaning that δ H T = 0 (see [12,24,25] for the geometric significance of this condition).…”

“…In this section, we show that Theorems 2.7 and 2.15 may also be obtained as a consequence of Bochner type identities. The methods developed in the previous sections are more powerful to understand second derivatives of the distance functions (see Section 3), but using Bochner type identities and the resulting curvature dimension estimates has the advantage to be applicable in more general situations (see [16,24,25] for the general framework on curvature dimension inequalities).…”

Integration by parts and quasi-invariance for the horizontal Wiener measure on foliated compact manifolds

“…The main reason why we use these connections is that we can not make use of the Bott connection since the adjoint connection to the Bott connection is not metric. We refer to [7,22,30,31] and especially the books [23,24] for a discussion on Weitzenböck-type identities and adjoint connections. Instead we make use of the family of connections first introduced in [2] and only keep the Bott connection as a reference connection.…”

Geometry of subelliptic diffusions

“…Defining curvature in sub-Riemannian geometry however is an intriguing problem [1]. Up to now, for instance, no direct probabilistic proof for non-explosion in finite time of sub-Riemannian diffusion by controlling the radial process (8.3) under sub-Riemannian curvature bounds is known [21]. During the last years, several results have appeared, linking sub-Riemannian geometric invariants to properties of diffusions of corresponding second order operators and their heat semi-group, see [6,7,22,23].…”

“…Conversely, functional inequalities of the type as (8.4) can be used to deduce nonexplosion of the underlying diffusion [9,21].…”