Related papers: Measure contraction properties of Carnot groups

A composite quantum system comprising a finite number k of subsystems which are described with position and momentum variables in Z_{n_{i}}, i=1,...,k, is considered. Its Hilbert space is given by a k-fold tensor product of Hilbert spaces…

Let $M=G/H$ be a compact connected isotropy irreducible Riemannian homogeneous manifold, where $G$ is a compact Lie group (may be, disconnected) acting on $M$ by isometries. This class includes all compact irreducible Riemannian symmetric…

We prove that an arbitrary convex body $C \subseteq \mathbf{R}^{n+1} $, whose $ k $-th anisotropic curvature measure (for $ k =0, \ldots , n-1 $) is a multiple constant of the anisotropic perimeter of C, must be a rescaled and translated…

Let L\subset V=\bR^{k,l} be a maximally isotropic subspace. It is shown that any simply connected Lie group with a bi-invariant flat pseudo-Riemannian metric of signature (k,l) is 2-step nilpotent and is defined by an element \eta \in…

The Whitney extension theorem is a classical result in analysis giving a necessary and sufficient condition for a function defined on a closed set to be extendable to the whole space with a given class of regularity. It has been adapted to…

We provide two new proofs of a theorem of Cooper, Long and Reid which asserts that, apart from an explicit finite list of exceptional manifolds, any compact orientable irreducible 3-manifold with non-empty boundary has large fundamental…

We study semi-Riemannian submanifolds of arbitrary codimension in a Lie group $G$ equipped with a bi-invariant metric. In particular, we show that, if the normal bundle of $M \subset G$ is closed under the Lie bracket, then any normal…

In the setting of Carnot groups, we exhibit examples of intrinisc Lipschitz curves of positive $\mathcal{H}^1$-measure that intersect every connected intrinsic Lipschitz curve in a $\mathcal{H}^1$-negligible set. As a consequence such…

Measure contraction property is a synthetic Ricci curvature lower bound for metric measure spaces. We consider Sasakian manifolds with non-negative Tanaka-Webster Ricci curvature equipped with the metric measure space structure defined by…

We show that in an $m$-step Carnot group, a probability measure with finite $m^{th}$ moment has a well-defined Buser-Karcher center-of-mass, which is a polynomial in the moments of the measure, with respect to exponential coordinates. Using…

This paper is devoted to the study of tangential properties of measures with density in the Heisenberg groups $\mathbb{H}^n$. Among other results we prove that measures with $(2n+1)$-density have only flat tangents and conclude the…

Let $G$ be a compact connected Lie group and $H$ a closed subgroup of $G$. Suppose the homogeneous space $G/H$ is effective and has dimension 3 or higher. Consider a $G$-invariant, symmetric, positive-semidefinite, nonzero (0,2)-tensor…

For a closed subset $K$ of a compact metric space $A$ possessing an $\alpha$-regular measure $\mu$ with $\mu(K)>0$, we prove that whenever $s>\alpha$, any sequence of weighted minimal Riesz $s$-energy configurations…

For a closed cocompact subgroup $\Gamma$ of a locally compact group $G$, given a compact abelian subgroup $K$ of $G$ and a homomorphism $\rho:\hat{K}\to G$ satisfying certain conditions, Landstad and Raeburn constructed equivariant…

We prove that various spaces of constrained positive scalar curvature metrics on compact 3-manifolds with boundary, when not empty, are contractible. The constraints we mostly focus on are given in terms of local conditions on the mean…

We prove that all (generalized) Higman groups on at least $5$ generators are superrigid for measure equivalence. More precisely, let $k\ge 5$, and let $H$ be a group with generators $a_1,\dots,a_k$, and Baumslag-Solitar relations given by…

We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are…

We classify all closed, aspherical Riemannian manifolds M whose universal cover has indiscrete isometry group. One sample application is the theorem that any such M with word-hyperbolic fundamental group must be isometric to a negatively…

We study positive kernels on $X\times X$, where $X$ is a set equipped with an action of a group, and taking values in the set of $\mathcal A$-sesquilinear forms on a (not necessarily Hilbert) module over a $C^*$-algebra $\mathcal A$. These…

Let $\mathcal{K}(n, V)$ be the set of $n$-dimensional compact Kahler-Einstein manifolds $(X, g)$ satisfying $Ric(g)= - g$ with volume bounded above by $V$. We prove that after passing to a subsequence, any sequence $\{ (X_j,…