Related papers: Remarks on conformal modulus in metric spaces

Let $(Z,d,\mu)$ be a compact, connected, Ahlfors $Q$-regular metric space with $Q>1$. Using a hyperbolic filling of $Z$, we define the notions of the $p$-capacity between certain subsets of $Z$ and of the weak covering $p$-capacity of path…

In this paper we survey a number of recent results concerning the existence and moduli spaces of solutions of various geometric problems on noncompact manifolds. The three problems which we discuss in detail are: I. Complete properly…

Quasisymmetry is a well-studied property of homeomorphisms between metric spaces, and Ahlfors regular conformal dimension is a quasisymmetric invariant. In the present paper, we consider the Ahlfors regular conformal dimension of metrics on…

Given a homeomorphism $f\colon X\to Y$ between $Q$-dimensional spaces $X,Y$, we show that $f$ satisfying the metric definition of quasiconformality outside suitable exceptional sets implies that $f$ belongs to the Sobolev class…

Directional notions in topology and analysis naturally lead to nonsymmetric structures such as quasi-metrics, quasi-uniformities, and modular spaces. In these settings, classical notions of connectedness and completion based on symmetric…

We study a quasisymmetric version of the classical Koebe uniformization theorem in the context of Ahlfors regular metric surfaces. In particular, we prove that an Ahlfors 2-regular metric surface X homeomorphic to a finitely connected…

We study the existence of a metric with zero scalar curvature maximizing the isoperimetric ratio among all zero scalar curvature metrics in a fixed conformal class of metrics on a compact manifold with boundary. The question may be reduced…

We study conformal invariants that arise from functions in the nullspace of conformally covariant differential operators. The invariants include nodal sets and the topology of nodal domains of eigenfunctions in the kernel of GJMS operators.…

Apart from global topological problems an affine homogeneous space is locally described by its curvature, its torsion and a slightly less tangible object called its connection in a given base point. Using this description of the local…

We study constant Q-curvature metrics conformal to the round metric on the sphere with finitely many point singularities. We show that the moduli space of solutions with finitely many punctures in fixed positions, equipped with the…

We discuss the analogy between collapsing Conformal Field Theories and measured Gromov-Hausdorff limit of Riemannian manifolds with non-negative Ricci curvature. Motivated by this analogy we propose the notion of non-commutative…

We generalize a result of J. C. Kelly to the setting of Ahlfors $Q$-regular metric measure spaces supporting a $1$-Poincar\'e inequality. It is shown that if $X$ and $Y$ are two Ahlfors $Q$-regular spaces supporting a $1$-Poincar\'e…

A detailed study is made of the noncommutative geometry of $R^3_q$, the quantum space covariant under the quantum group $SO_q(3)$. For each of its two $SO_q(3)$-covariant differential calculi we find its metric, the corresponding frame and…

We show that for an orientable non-spin manifold with fundamental group $\mathbb{Z}_2$ and universal cover $S^2\times S^3,$ the moduli space of metrics of nonnegative sectional curvature has infinitely many path components. The…

In an attempt to develop higher-dimensional quasiconformal mappings on metric measure spaces with curvature conditions, i.e. from Ahlfors to Alexsandrov, we show that a non-collapsed $\mathrm{RCD}(0,n)$ space ($n\geq2$) with Euclidean…

We show that a homeomorphism of Euclidean space is quasiconformal if and only if at each point there exists a sequence of uncentered open sets with bounded eccentricity shrinking to that point whose images also have bounded eccentricity.…

The modulus metric between two points in a subdomain of $\mathbb{R}^n, n\ge 2,$ is defined in terms of moduli of curve families joining the boundary of the domain with a continuum connecting the two points. This metric is one of the…

Let k be an algebraically closed field of characteristic 0, and let $A = k[x,y]/(f)$ be a quasi-homogeneous plane curve. We show that for any graded torsion free A-module M, there exists a natural graded integrable connection, i.e. a graded…

We give an example of a totally disconnected set E in R^3 which is not removable for quasiconformal homeomorphisms, i.e., there is a homeomorphism f of R^3 to itself which is quasiconformal off E, but not quasiconformal on all of R^3. The…

Let $M$ be a Milnor sphere or, more generally, the total space of a linear $S^3$-bundle over $S^4$ with $H^4(M;\mathbb{Q})=0$. We show that the moduli space of metrics of nonnegative sectional curvature on $M$ has infinitely many path…