Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries

By Fabrice Baudoin, Nicola Garofalo. 2011

215 citations recorded by the source corpus.

Let $\M$ be a smooth connected manifold endowed with a smooth measure $\mu$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $\mu$. Associated with $L$ one has \textit{le carr\'e du champ} $\Gamma$ and a canonical distance $d$, with respect to which we suppose that $(M,d)$ be complete. We assume that $\M$ is also equipped with another first-order differential bilinear form $\Gamma^Z$ and we assume that $\Gamma$ and $\Gamma^Z$ satisfy the Hypothesis below. With these forms we introduce in \eqref{cdi} below a generalization of the curvature-dimension inequality from Riemannian geometry, see Definition \ref{D:cdi}. In our main results we prove that, using solely \eqref{cdi}, one can develop a theory which parallels the celebrated works of Yau, and Li-Yau on complete manifolds with Ricci bounded from below. We also obtain an analogue of the Bonnet-Myers theorem. In Section \ref{S:appendix} we construct large classes of sub-Riemannian manifolds with transverse symmetries which satisfy the generalized curvature-dimension inequality \eqref{cdi}. Such classes include all Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is bounded from below, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is bounded from below.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/bdb02ed9e4078ae7780dfdf97cf768236dfd6461

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates (2018, background): “ove weak Bakry-Emery condition, see for example [´ 65]. It should also be noted that, in the setting of complete sub-Riemannian manifolds with generalized non-negative Ricci curvature in the sense of [17], while the weak Bakry-Emery curvature condition´ is known to be satisﬁed (see [14,19]), the 1-P”
Sub-Riemannian Curvature in Contact Geometry (2015, methodology): “For example, with heat semigroup approaches: for Yang–Mills type structures with transverse symmetries [11] and Riemannian foliations with totally geodesic leaves [12].”
Gradient estimates for heat kernels and harmonic functions (2017, background): “Below we will mainly concentrate on three different settings, and we refer the readers to [4, 5, 7, 12, 17, 46, 47, 79, 110] for more examples.”
Heat semigroup and singular PDEs (2015, background): “Assume that (M,d, μ) is locally Ahlfors regular, with homogeneous dimension d = ν ∈ [2, 4).”