Geometric Inequalities and Generalized Ricci Bounds in the Heisenberg Group

Juillet, Nicolas

doi:10.1093/imrn/rnp019

Cited by 81 publications

(128 citation statements)

References 24 publications

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“…Note also that, under the same assumptions as in Theorem 1.1, it was shown in [10] that (M, d CC , vol P ) satisfies MCP (0, 2n + 3), where d CC is the Carnot-Caratheordory distance and vol P is the Popp measure (see also [8,1] for the earlier results).…”

Section: Introductionmentioning

confidence: 86%

On Measure Contraction Property without Ricci Curvature Lower Bound

Lee

2015

Potential Anal

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Abstract. Measure contraction properties M CP (K, N ) are synthetic Ricci curvature lower bounds for metric measure spaces which do not necessarily have smooth structures. It is known that if a Riemannian manifold has dimension N , then M CP (K, N ) is equivalent to Ricci curvature bounded below by K. On the other hand, it was observed in [20] that there is a family of left invariant metrics on the three dimensional Heisenberg group for which the Ricci curvature is not bounded below. Though this family of metric spaces equipped with the Harr measure satisfy M CP (0, 5).In this paper, we give sufficient conditions for a 2n + 1 dimensional weakly Sasakian manifold to satisfy M CP (0, 2n + 3). This extends the above mentioned result on the Heisenberg group in [20].

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Section: Introductionmentioning

confidence: 86%

On Measure Contraction Property without Ricci Curvature Lower Bound

Lee

2015

Potential Anal

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“…The point is that u c z (η(t)) is not proportional to t since (u c z • η) (t) = c+L(η(t)) and L(η) is not necessarily constant. We also remark that even the stronger inequality (6.2) does not imply the curvature bound Ric [57,Remark 5.6] for a simple example, and [27] for a related work on the gap between the measure contraction property and the curvature-dimension condition in Heisenberg groups).…”

Section: Laplacian Comparisonmentioning

confidence: 94%

On the Curvature and Heat Flow on Hamiltonian Systems

Ohta

2014

Analysis and Geometry in Metric Spaces

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Abstract:We develop the di erential geometric and geometric analytic studies of Hamiltonian systems. Key ingredients are the curvature operator, the weighted Laplacian, and the associated Riccati equation. We prove appropriate generalizations of the Bochner-Weitzenböck formula and Laplacian comparison theorem, and study the heat ow.Keywords: Hamiltonian, curvature, comparison theorem, heat ow MSC: Primary: 53C21; Secondary: 58J35, 49Q20 DOI 10.2478/agms-2014-0003 Received August 20, 2013 accepted February 14, 2014 IntroductionThe aim of this article is to apply the recently developed technique in Finsler geometry to the study of Hamiltonian systems. A Finsler manifold carries a (Minkowski) norm on each tangent space. Although Finsler manifolds form a much wider class than Riemannian manifolds, the notion of curvature makes sense and we can consider various comparison theorems similarly to the Riemannian case (see, e.g., [54]). Especially, the weighted Ricci curvature introduced by the author [41] has fruitful applications including the curvature-dimension condition ([41]), Laplacian comparison theorem for the natural nonlinear Laplacian ([44]), Bochner-Weitzenböck formula and gradient estimates ([46]), and generalizations of the CheegerGromoll splitting theorem ([43]). To be precise, the weighted Ricci curvature is de ned for a pair consisting of a Finsler manifold and a measure on it, and our Laplacian depends on the choice of the measure (see Subsections 2.3, 4.1 for details).Then, it is natural to expect that the theory of curvature can be applied beyond Finsler manifolds, and a class of manifolds M endowed with Lagragians L or, equivalently, Hamiltonians H is a natural choice. In fact, on the one hand, we know that Agrachev and Gamkrelidze [3] (see also [2]) have developed the theory of curvature operator for Hamiltonian systems in connection with optimal control theory (see [1] for a dynamical application). On the other hand, optimal transport theory (which is related to the curvature-dimension condition) for Lagrangian cost functions has been well investigated ([13], [19], [58]). Furthermore, Lee [31] recently showed a Riccati equation (see also [4], [5], [32] for the sub-Riemannian case) as well as convexity estimates for entropy functionals along smooth optimal transports for general (time-dependent) Hamiltonian systems, by means of the curvature operator. His uni ed approach recovers both the curvature-dimension condition (CD(K, ∞) and CD( , N) to be precise) for Riemannian or Finsler manifolds and various monotonicity formulas along ows in Riemannian metrics related to the Ricci ow.Our Hamiltonian will always be time-independent and non-negative. Compared with the Finsler situation, the lack of the homogeneity causes many di culties, for example, we need to take care of the di erence between the Lagrangian and the Hamiltonian (they coincide as functions via the Legendre transform in the Finsler case). Nevertheless, by combining the Riccati equation and the technique in the Finsler case, we prove ...

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“…This approach has also been conducted on the Heisenberg group [Jui09]: no Ricci curvature lower bound CD(K, N ) with K > −∞ holds in this space. On the other hand, in [Jui09] it is shown that a related but weaker property holds, the measure contraction property [Stu06,Oht07].…”

Section: Nmentioning

confidence: 99%

“…On the other hand, in [Jui09] it is shown that a related but weaker property holds, the measure contraction property [Stu06,Oht07].…”

Section: Nmentioning

confidence: 99%

A visual introduction to Riemannian curvatures and some discrete generalizations

Ollivier¹

2013

CRM Proceedings and Lecture Notes

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Abstract. We try to provide a visual introduction to some objects used in Riemannian geometry: parallel transport, sectional curvature, Ricci curvature, Bianchi identities... We then explain some of the strategies used to define analogues of curvature in non-smooth or discrete spaces, beginning with Alexandrov curvature and δ-hyperbolic spaces, and insisting on various notions of generalized Ricci curvature, which we briefly compare.The first part of this text covers in a hopefully intuitive and visual way some of the usual objects of Riemannian geometry: parallel transport, sectional curvature, Ricci curvature, the Riemann tensor, Bianchi identities... For each of those we try to provide one or several pictures that convey the meaning (or at least one possible interpretation) of the formal definition.Next, we consider the problem of defining analogues of curvature for nonsmooth or discrete objets. This is in the spirit of "synthetic" or "coarse" geometry, in which large-scale properties of metric spaces are investigated instead of fine smallscale properties; one of the aims being to be impervious to small perturbations of the underlying space. Motivation for these non-smooth or discrete extensions often comes from various other fields of mathematics, such as group theory or optimal transport. From this viewpoint is emerging a theory of metric measure spaces, see for instance [Gro99].Thus we cover the definitions and mention some applications of δ-hyperbolic spaces and Alexandrov curvature (generalizing sectional curvature), and of two notions of generalized Ricci curvature: displacement convexity of entropy introduced by Sturm and Lott-Villani (following Renesse-Sturm and others), and coarse Ricci curvature as used by the author. We mention new theorems obtained for the original Riemannian case thanks to this more general viewpoint. We also briefly discuss the differences between these two approaches to Ricci curvature.Note that scalar curvature is not covered, for lack of the author's competence about recent developments and applications in this field.

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Geometric Inequalities and Generalized Ricci Bounds in the Heisenberg Group

Cited by 81 publications

References 24 publications

On Measure Contraction Property without Ricci Curvature Lower Bound

On Measure Contraction Property without Ricci Curvature Lower Bound

On the Curvature and Heat Flow on Hamiltonian Systems

A visual introduction to Riemannian curvatures and some discrete generalizations

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