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We investigate isoperimetric inequalities for Lipschitz 2-spheres in CAT(0) spaces, proving bounds on the volume of efficient null-homotopies. In one dimension lower, it is known that a quadratic inequality with a constant smaller than…

We consider an area-minimizing integral current $T$ of codimension higher than $1$ in a smooth Riemannian manifold $\Sigma$. In a previous paper we have subdivided the set of interior singular points with at least one flat tangent cone…

We show that there exists a strong uniform embedding from any proper metric space into any Banach space without cotype. Then we prove a result concerning the Lipschitz embedding of locally finite subsets of $\mathcal{L}_{p}$-spaces. We use…

For mappings in metric spaces satisfying one inequality with respect to modulus of families of curves, there is proved a lightness of the uniform limit of these mappings. It is proved that, the uniform limit of these mappings is light…

The Lipschitz geometry of segments of the infinite Hamming cube is studied. Tight estimates on the distortion necessary to embed the segments into spaces of continuous functions on countable compact metric spaces are given. As an…

In this paper, we establish compactness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined a priori on the class of compact, embedded $m$-dimensional Lipschitz…

On a compact stratified space (X, g) there exists a metric of constant scalar curvature in the conformal class of g, if the scalar curvature satisfies an integrability condition and if the Yamabe constant of X is strictly smaller than the…

In this paper we study a class of quasi--variational--hemi\-va\-ria\-tio\-nal inequalities in reflexive Banach spaces. The inequalities contain a convex potential, a locally Lipschitz superpotential, and a solution-dependent set of…

We consider the Dirichlet problem for elliptic systems with periodically distributed inclusions whose conduction parameter exhibits a significant contrast compared to the background media. We develop a unified method to quantify the…

Riemannian manifolds of quasi-constant sectional curvatures (QC-manifolds) are divided into two basic classes: with positive or negative horizontal sectional curvatures. We prove that the Riemannian QC-manifolds with positive horizontal…

The optimal Orlicz target space is exhibited for embeddings of fractional-order Orlicz-Sobolev spaces in $\mathbb R^n$. An improved embedding with an Orlicz-Lorentz target space, which is optimal in the broader class of all…

For a k-flat F inside a locally compact CAT(0)-space X, we identify various conditions that ensure that F bounds a (k+1)-dimensional half flat in X. Our conditions are formulated in terms of the ultralimit of X. As applications, we obtain…

Let $T$ be a compact, metrisable and strongly countable-dimensional topological space. Let $\mathcal{M}^T$ be the set of all metrics $d$ on $T$ compatible with its topology, and equip $\mathcal{M}^T$ with the topology of uniform…

We study metric measure spaces that admit "thick" families of rectifiable curves or curve fragments, in the form of Alberti representations or curve families of positive modulus. We show that such spaces cannot be bi-Lipschitz embedded into…

We prove the local Lipschitz continuity of sub-elliptic harmonic maps between certain singular spaces, more specifically from the $n$-dimensional Heisenberg group into $CAT(0)$ spaces. Our main theorem establishes that these maps have the…

The phase space of a compact, irreducible, simply connected, Riemannian symmetric space admits a natural family of K\"ahler polarizations parametrized by the upper half plane $S$. Using this family, geometric quantization, including the…

The main result states that a connected conic singular sub-manifold of a Riemannian manifold, compact when the ambient manifold is non-Euclidean, is Lipschitz Normally Embedded: the outer and inner metric space structures are metrically…

This paper presents an existence result and maximal regularity estimates for distributional solutions to degenerate/singular elliptic systems of $p$-Laplacian type with absorption and (prescribed) locally integrable forcing posed in…

We prove that if a quasiconvex subset $X$ of a metric space $Y$ has finite Nagata dimension and is Lipschitz $k$-connected or admits Euclidean isoperimetric inequalities up to dimension $k$ for some $k$ then $X$ is isoperimetrically…

We investigate CAT(0) metric spaces whose associated Tits boundary is compact. Prominent examples of such spaces are of course the euclidean ones. However there exist non trivial geodesically complete CAT(0) spaces with compact Tits…