An upper gradient approach to weakly differentiable cochains

Abstract The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub)additive functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio–Kirchheimʼs theory of metric currents. The notion of weak differentiability we introduce is in analogy with Heinonen–Koskelaʼs concept of upper gradients of functions. In one of the main results of our paper, we prove continuity estimates for cochains with p-integrable upper gradient in n-dimensional Lie groups endowed with a left-invariant Finsler metric. Our result general…

Bonnesenʼs inequality for John domains in Rn

Abstract We prove sharp quantitative isoperimetric inequalities for John domains in R n . We show that the Bonnesen-style inequalities hold true in R n under the John domain assumption which rules out cusps. Our main tool is a proof of the isoperimetric inequality for symmetric domains which gives an explicit estimate for the isoperimetric deficit. We use the sharp quantitative inequalities proved in Fusco et al. (2008) [7] and Fuglede (1989) [4] to reduce our problem to symmetric domains.

Mappings of finite distortion: Removable singularities

We show that certain small sets are removable for bounded mappings of finite distortion for which the distortion function satisfies a suitable subexponential integrability condition. We also give an example demonstrating the sharpness of this condition.

Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces

Abstract We use the heat equation to establish the Lipschitz continuity of Cheeger-harmonic functions in certain metric spaces. The metric spaces under consideration are those that are endowed with a doubling measure supporting a (1,2)-Poincare inequality and in addition supporting a corresponding Sobolev–Poincare-type inequality for the modification of the measure obtained via the heat kernel. Examples are given to illustrate the necessity of our assumptions on these spaces. We also provide an example to show that in the general setting the best possible regularity for the Cheeger-harmonic functions is Lipschitz continuity.

Quasisymmetric Koebe uniformization with weak metric doubling measures

We give a characterization of metric spaces quasisymmetrically equivalent to a finitely connected circle domain. This result generalizes the uniformization of Ahlfors 2-regular spaces by Merenkov and Wildrick. peerReviewed

A lower bound for the Bloch radius of 𝐾-quasiregular mappings

We give a quantitative proof to Eremenko’s theorem (2000), which extends Bloch’s classical theorem to the class of n n -dimensional K K -quasiregular mappings.

Quasiregular ellipticity of open and generalized manifolds

We study the existence of geometrically controlled branched covering maps from \(\mathbb R^3\) to open \(3\)-manifolds or to decomposition spaces \(\mathbb {S}^3/G\), and from \(\mathbb {S}^3/G\) to \(\mathbb {S}^3\).

Wolfe's theorem for weakly differentiable cochains

Abstract A fundamental theorem of Wolfe isometrically identifies the space of flat differential forms of dimension m in R n with the space of flat m -cochains, that is, the dual space of flat chains of dimension m in R n . The main purpose of the present paper is to generalize Wolfe's theorem to the setting of Sobolev differential forms and Sobolev cochains in R n . A suitable theory of Sobolev cochains has recently been initiated by the second and third author. It is based on the concept of upper norm and upper gradient of a cochain, introduced in analogy with Heinonen–Koskela's concept of upper gradient of a function.

Mappings of finite distortion and asymmetry of domains

We establish an anisotropic Bonnesen inequality for images of balls under homeomorphisms with exponentially integrable distortion. Mathematics Subject Classification (2000): 30C65, 46E35.

Mappings of finite distortion : removable singularities

Mappings of finite distortion: The Rickman-Picard theorem for mappings of finite lower order

We show that an entire mappingf of finite distortion with finite lower order can omit at most finitely many points when the distortion function off is suitably controlled. The proof uses the recently established modulus inequalities for mappings of finite distortion [15] and comparison inequalities for the averages of the counting function. A similar technique also gives growth estimates for mappings having asymptotic values.

Surface families and boundary behavior of quasiregular mappings

We study the boundary behavior of bounded quasiregular mappings f : Bn(0, 1) → Rn, n ≥ 3. We show that there exists a large family of cusps, with vertices on the boundary sphere S n−1 (0, 1), so that the images of these cusps under f have finite (n − 1)-measure. peerReviewed

Reciprocal lower bound on modulus of curve families in metric surfaces

We prove that any metric space $X$ homeomorphic to $\mathbb{R}^2$ with locally finite Hausdorff 2-measure satisfies a reciprocal lower bound on modulus of curve families associated to a quadrilateral. More precisely, let $Q \subset X$ be a topological quadrilateral with boundary edges (in cyclic order) denoted by $\zeta_1, \zeta_2, \zeta_3, \zeta_4$ and let $\Gamma(\zeta_i, \zeta_j; Q)$ denote the family of curves in $Q$ connecting $\zeta_i$ and $\zeta_j$; then $\text{mod} \Gamma(\zeta_1, \zeta_3; Q) \text{mod} \Gamma(\zeta_2, \zeta_4; Q) \geq 1/\kappa$ for $\kappa = 2000^2\cdot (4/\pi)^2$. This answers a question concerning minimal hypotheses under which a metric space admits a quasiconfor…

Uniformization with infinitesimally metric measures

We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces $X$ homeomorphic to $\mathbb R^2$. Given a measure $\mu$ on such a space, we introduce $\mu$-quasiconformal maps $f:X \to \mathbb R^2$, whose definition involves deforming lengths of curves by $\mu$. We show that if $\mu$ is an infinitesimally metric measure, i.e., it satisfies an infinitesimal version of the metric doubling measure condition of David and Semmes, then such a $\mu$-quasiconformal map exists. We apply this result to give a characterization of the metric spaces admitting an infinitesimally quasisymmetric parametrization.

Mappings of finite distortion: Removable singularities for locally homeomorphic mappings

Let f be a locally homeomorphic mapping of finite distortion in dimension larger than two. We show that when the distortion of f satisfies a certain subexponential integrability condition, small sets are removable. The smallness is measured by a weighted modulus.

Invertibility of Sobolev mappings under minimal hypotheses

Abstract We prove a version of the Inverse Function Theorem for continuous weakly differentiable mappings. Namely, a nonconstant W 1 , n mapping is a local homeomorphism if it has integrable inner distortion function and satisfies a certain differential inclusion. The integrability assumption is shown to be optimal.

Quasispheres and metric doubling measures

Applying the Bonk-Kleiner characterization of Ahlfors 2-regular quasispheres, we show that a metric two-sphere $X$ is a quasisphere if and only if $X$ is linearly locally connected and carries a weak metric doubling measure, i.e., a measure that deforms the metric on $X$ without much shrinking.

Uniformization of two-dimensional metric surfaces

We establish uniformization results for metric spaces that are homeomorphic to the Euclidean plane or sphere and have locally finite Hausdorff 2-measure. Applying the geometric definition of quasiconformality, we give a necessary and sufficient condition for such spaces to be QC equivalent to the Euclidean plane, disk, or sphere. Moreover, we show that if such a QC parametrization exists, then the dilatation can be bounded by 2. As an application, we show that the Euclidean upper bound for measures of balls is a sufficient condition for the existence of a 2-QC parametrization. This result gives a new approach to the Bonk-Kleiner theorem on parametrizations of Ahlfors 2-regular spheres by qu…