Curvature Dimension Inequalities and Subelliptic Heat Kernel Gradient Bounds on Contact Manifolds

Baudoin, Fabrice; Wang, Jing

doi:10.1007/s11118-013-9345-x

Cited by 27 publications

(25 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A compactness result for contact structures is also obtained [13] by applying the classical Bonnet-Myers theorem to a suitable Riemannian extension of the metric.…”

Section: Relation Between the Two Approachesmentioning

confidence: 99%

Sub-Riemannian Curvature in Contact Geometry

2016

View full text Add to dashboard Cite

We compare different notions of curvature on contact sub-Riemannian manifolds. In particular, we introduce canonical curvatures as the coefficients of the sub-Riemannian Jacobi equation. The main result is that all these coefficients are encoded in the asymptotic expansion of the horizontal derivatives of the subRiemannian distance. We explicitly compute their expressions in terms of the standard tensors of contact geometry. As an application of these results, we obtain a sub-Riemannian version of the Bonnet-Myers theorem that applies to any contact manifold.

show abstract

“…A compactness result for contact structures is also obtained [13] by applying the classical Bonnet-Myers theorem to a suitable Riemannian extension of the metric.…”

Section: Relation Between the Two Approachesmentioning

confidence: 99%

Sub-Riemannian Curvature in Contact Geometry

2016

View full text Add to dashboard Cite

show abstract

“…Proof. The identities (13) follow directly from the definitions while (14) is precisely (12) written in terms of A, C and v. For (15), working in the Fermi frame along γ, we havė…”

Section: Lemma 16 the Non-zero Brackets Between ∂mentioning

confidence: 99%

“…The identities (16) follow directly from (13) and (15). For the first one in (17), we take the derivative of (15) applying (13) and (14) to get thaẗ…”

Section: Lemma 16 the Non-zero Brackets Between ∂mentioning

confidence: 99%

A Bonnet–Myers type theorem for quaternionic contact structures

Barilari

Ivanov

2019

Calc. Var.

View full text Add to dashboard Cite

We prove a Bonnet-Myers type theorem for quaternionic contact manifolds of dimension bigger than 7. If the manifold is complete with respect to the natural sub-Riemannian distance and satisfies a natural Ricci-type bound expressed in terms of derivatives up to the third order of the fundamental tensors, then the manifold is compact and we give a sharp bound on its sub-Riemannian diameter.for some (and then any) set X 1 , . . . , X k ∈ Γ(D) of local generators for D.Given a sub-Riemannian structure on M , the sub-Riemannian distance is defined by:d SR (x, y) = inf{ℓ(γ) | γ(0) = x, γ(T ) = y, γ horizontal}.

show abstract

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

Section: Future Prospectsmentioning

confidence: 99%

Geometry of subelliptic diffusions

Thalmaier

2016

EMS Series of Lectures in Mathematics

View full text Add to dashboard Cite

Abstract. These lectures focus on some probabilistic aspects related to subRiemannian geometry. The main intention is to give an introduction to hypoelliptic and subelliptic diffusions. The notes are written from a geometric point of view trying to minimize the weight of "probabilistic baggage" necessary to follow the arguments. We discuss in particular the following topics: stochastic flows to second order differential operators; smoothness of transition probabilities under Hörmander's brackets condition; control theory and Stroock-Varadhan's support theorems; Malliavin calculus; Hörmander's theorem. The notes start from well-known facts in Geometric Stochastic Analysis and guide to recent on-going research topics, like hypoelliptic heat kernel estimates; gradient estimates and Harnack type inequalities for subelliptic diffusion semigroups; notions of curvature related to sub-Riemannian diffusions.

show abstract

Curvature Dimension Inequalities and Subelliptic Heat Kernel Gradient Bounds on Contact Manifolds

Cited by 27 publications

References 26 publications

Sub-Riemannian Curvature in Contact Geometry

Sub-Riemannian Curvature in Contact Geometry

A Bonnet–Myers type theorem for quaternionic contact structures

Geometry of subelliptic diffusions

Contact Info

Product

Resources

About