Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations

Baudoin, Fabrice; Grong, Erlend; Kuwada, Kazumasa; Thalmaier, Anton

doi:10.1007/s00526-019-1570-8

Cited by 26 publications

(48 citation statements)

References 46 publications

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“…Other works have obtained geometric inequalities, including volume doubling and the stronger measure contraction property M CP (k, n) introduced by [41], that hold uniformly over a one-parameter family of Riemannian geometries approximating a sub-Riemannian geometry [2,5,29,32,33,45]. However, these works use very different techniques, and all known results appear to rely on assumptions of horizontal curvature bounds or additional symmetry, such as Sasakian structure.…”

mentioning

confidence: 99%

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

Eldredge¹,

Gordina²,

Saloff‐Coste³

2018

Geom. Funct. Anal.

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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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confidence: 99%

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

Eldredge¹,

Gordina²,

Saloff‐Coste³

2018

Geom. Funct. Anal.

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“…An analysis based on the canonical variation of the index form has been successfully implemented on Sasakian foliations in [BGKT19]. The same idea is applied in [BGMR] to H-type foliations with parallel Clifford structure, introduced in [BGMR18], which extends to higher corank all the nice features of Sasakian foliations.…”

Section: Maximal Length Boundsmentioning

confidence: 99%

Bakry–Émery curvature and model spaces in sub-Riemannian geometry

Barilari

Rizzi

2020

Math. Ann.

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We prove comparison theorems for the sub-Riemannian distortion coefficients appearing in interpolation inequalities. These results, which are equivalent to a sub-Laplacian comparison theorem for the sub-Riemannian distance, are obtained by introducing a suitable notion of sub-Riemannian Bakry-Émery curvature. The model spaces for comparison are variational problems coming from optimal control theory. As an application we establish the sharp measure contraction property for 3-Sasakian manifolds satisfying a suitable curvature bound.

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“…Example 1.1 illustrates that such constants can be found if the Riemannian foliation is given by the Reeb foliation of a Sasakian structure with curvature bounded below. This observation is based on the Laplacian comparison theorem recently proved by Baudoin, Grong, Kuwada and Thalmaier in [2]. The Itô formula is taken from their next article [1], currently in preparation.…”

Section: Introductionmentioning

confidence: 99%

Exponential integrability and exit times of diffusions on sub-Riemannian and metric measure spaces

Thalmaier

Thompson

2020

Bernoulli

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In this article we derive moment estimates, exponential integrability, concentration inequalities and exit times estimates for canonical diffusions in two settings each beyond the scope of Riemannian geometry. Firstly, we consider sub-Riemannian limits of Riemannian foliations. Secondly, we consider the non-smooth setting of RCD * (K, N ) spaces. In each case the necessary ingredients are an Itô formula and a comparison theorem for the Laplacian, for which we refer to the recent literature. As an application, we derive pointwise Carmona-type estimates on eigenfunctions of Schrödinger operators.

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Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations

Cited by 26 publications

References 46 publications

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

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

Bakry–Émery curvature and model spaces in sub-Riemannian geometry

Exponential integrability and exit times of diffusions on sub-Riemannian and metric measure spaces

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