Related papers: Lipschitz Carnot-Carath\'eodory structures and the…

We investigate local and metric geometry of weighted Carnot-Carath\'eodory spaces which are a wide generalization of sub-Riemannian manifolds and arise in nonlinear control theory, subelliptic equations etc. For such spaces the intrinsic…

We consider the multi-marginal optimal transport of aligning several compactly supported marginals on the Heisenberg group to minimize the total cost, which we take to be the sum of the squared Carnot-Carath\'eodory distances from the…

We introduce a new distance dist_oq between compact quantum metric spaces. We show that dist_oq is Lipschitz equivalent to Rieffel's distance dist_q, and give criteria for when a parameterized family of compact quantum metric spaces is…

We determine when a quasi-isometry between discrete spaces is at bounded distance from a bilipschitz map. From this we prove a geometric version of the Von Neumann conjecture on amenability. We also get some examples in geometric groups…

We construct a Banach space satisfying that the nearest point map (also called proximity mapping or metric projection) onto any compact and convex subset is continuous but not uniformly continuous. The space we construct is locally…

We introduce a natural definition of $L^p$-convergence of maps, $p \ge 1$, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a…

The distances between flats of a Poisson $k$-flat process in the $d$-dimensional Euclidean space with $k<d/2$ are discussed. Continuing an approach originally due to Rolf Schneider, the number of pairs of flats having distance less than a…

We prove that if a family of metrics, $g_i$, on a compact Riemannian manifold, $M^n$, have a uniform lower Ricci curvature bound and converge to $g_\infty$ smoothly away from a singular set, $S$, with Hausdorff measure, $H^{n-1}(S) = 0$,…

We study transversality for Lipschitz-Fredholm maps in the context of bounded Fr\'{e}chet manifolds. We show that the set of all Lipschitz-Fredholm maps of a fixed index between Fr\'{e}chet spaces has the transverse stability property. We…

We prove Cheeger-Gromov convergence for a subsequence of a given sequence of manifolds-with-boundary of bounded geometry. The method of the proof is to reduce, via height functions, the problem to the setting of Hamilton's compactnes…

We extend the recent result of G. Godefroy which concerns the existence of non-norm attaining Lipschitz maps in order to characterize the norm attainment toward vectors for Lipschitz maps in the general setting of underlying space. The main…

In this paper we prove convergence and compactness results for Ricci flows with bounded scalar curvature and entropy. More specifically, we show that Ricci flows with bounded scalar curvature converge smoothly away from a singular set of…

Consider a Hausdorff space (X,T) and a set C of converging nets in X. By virtue of the limit uniqueness, the relation Lim which assigns each member x of X to every net N lying in C that converges to x is a map. Of course, structuring C with…

This book explores geometries defined by left-invariant distance functions on Lie groups, with a particular focus on nilpotent groups and Carnot groups equipped with geodesic distances. Geodesic left-invariant metrics are either…

Let $X$ be a subset of a Hilbert space. We prove that if $v\colon X\to \mathbb{R}^m$ is such that \begin{equation*} \Big\lVert v(x)-\sum_{i=1}^m t_iv(x_i)\Big\rVert\leq \Big\lVert x-\sum_{i=1}^m t_ix_i\Big\rVert \end{equation*} for all…

In this paper we show that if $Y=N \times \mathbb{Q}_m$ is a metric space where $N$ is a Carnot group endowed with the Carnot-Caratheodory metric then any quasisymmetric map of $Y$ is actually bilipschitz. The key observation is that $Y$ is…

Given a compact Riemannian manifold with boundary, we prove that the limit of a sequence of embedded, almost properly embedded free boundary minimal hypersurfaces, with uniform area and Morse index upper bound, always inherits a non-trivial…

A compact quantum metric space is a unital $C^*$-algebra equipped with a Lip-norm. Let $\{(A_n, L_n)\}$ be a sequence of compact quantum metric spaces, and let $\phi_n:A_n\to A_{n+1}$ be a unital $^*$-homomorphism preserving Lipschitz…

In this paper we study the main geometric properties of the Carnot-Carath\'eodory (abbreviated CC) distance $\dc$ in the setting of $k$-step sub-Riemannian Carnot groups from many different points of view. An extensive study of the…

A compactness theorem is proved for a family of K\"{a}hler surfaces with constant scalar curvature and volume bounded from below, diameter bounded from above, Ricci curvature bounded and the signature bounded from below. Furthermore, a…