Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds

A classical aspect of Riemannian geometry is the study of estimates that hold uniformly over some class of metrics. The best known examples are eigenvalue bounds under curvature assumptions. In this paper, we study the family of all left-invariant geometries on SU(2). We show that left-invariant geometries on SU(2) are uniformly doubling and give a detailed estimate of the volume of balls that is valid for any of these geometries and any radius. We discuss a number of consequences concerning the spectrum of the associated Laplacians and the corresponding heat kernels.1991 Mathematics Subject Classification. Primary 53C21; Secondary 35K08, 53C17, 58J35, 58J60, 22C05, 22E30.

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“…In this article, we prove that Conjecture 1.1 is valid for K = SU (2). Our main result is as follows.…”

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confidence: 83%

“…A conjecture and the main result. This work is devoted to the uniform analysis of the family of all left-invariant Riemannian metrics on the Lie group SU (2). This is the simplest case of a natural problem we now describe.…”

mentioning

confidence: 99%

Left-invariant geometries on SU(2) are uniformly doubling

Eldredge¹,

Gordina²,

Saloff‐Coste³

2018

Geom. Funct. Anal.

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“…In [31, §4], Lee showed a Riccati type equation with respect to the Laplacian ∆ H in (4.3) (see also [4], [5] and [32] for the sub-Riemannian case). We repeat his argument for completeness, and derive a generalization of the Bochner-Weitzenböck formula along the line of [46] in the next subsection.…”

Section: Riccati Equationmentioning

confidence: 99%

“…A geometric counterpart to the Laplacian comparison is the measure contraction property (see [38], [57, §5] for the precise de nition on metric measure spaces, and [4], [32] for related works on sub-Riemannian manifolds). To state it, we x c > , z ∈ M, α ∈ T * z M and η : [ , Tα] −→ M as in Theorem 6.2, and take a local coordinate (…”

Section: Measure Contraction Propertymentioning

confidence: 99%

“…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.…”

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confidence: 99%

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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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Curvature-Dimension Condition

Ohta

2021

Springer Monographs in Mathematics

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Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds

Abstract: Abstract. Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In this paper, we discover sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property.

Cited by 63 publications

References 43 publications

Left-invariant geometries on SU(2) are uniformly doubling

Left-invariant geometries on SU(2) are uniformly doubling

On the Curvature and Heat Flow on Hamiltonian Systems

Curvature-Dimension Condition

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