David A. Herron

UNIFORM, SOBOLEV EXTENSION AND QUASICONFORMAL CIRCLE DOMAINS

This paper contributes to the theory of uniform domains and Sobolev extension domains. We present new features of these domains and exhibit numerous relations among them. We examine two types of Sobolev extension domains, demonstrate their equivalence for bounded domains and generalize known sufficient geometric conditions for them. We observe that in the plane essentially all of these domains possess the trait that there is a quasiconformal self-homeomorphism of the extended plane which maps a given domain conformally onto a circle domain. We establish a geometric condition enjoyed by these plane domains which characterizes them among all quasicircle domains having no large and no small boundary components.

Conformal deformations of uniform Loewner spaces

We study conformal deformations of uniform Loewner spaces, verify the Gehring-Hayman Inequality, demonstrate uniform continuity of the identity mapping, and provide results which describe the boundary of the deformed space. In addition, we establish two connections between Hausdorff content and modulus.

Quasiextremal distance domains and conformal mappings onto circle domains

This paper contributes to the theory of quasiextremal distance domains. We present some new properties for these domains and point out results concerning the extension of quasiconformal homeomorphisms.For example, we establish a continuity property for mod(E, F; D) and use this to demonstrate that mod(E, F; D) = cap(E, F; D) whenever D is a QED domain and E, F are disjoint compacta in D. Our final result is that if each boundary component of a plane domain is either a point or a Jordan curve and if the domain satisfies a boundary quasiextremal distance property, then there exists a quasiconformal self-homeomorphism of the entire plane which maps the given domain conformally onto a circle domain.

Möbius invariant metrics bilipschitz equivalent to the hyperbolic metric

We study three Möbius invariant metrics, and three affine invariant analogs, all of which are bilipschitz equivalent to the Poincaré hyperbolic metric. We exhibit numerous illustrative examples.

Quasiconformal Deformations and Volume Growth

We determine when a metric measure space admits a uniformizing density which has Ahlfors regular volume growth. A characterization of uniformizing conformal densities in terms of doubling and a lifting procedure are the key ingredients in our presentation. We also furnish an application characterizing metric doubling measures.

Bilipschitz homogeneous Jordan curves

We characterize bilipschitz homogeneous Jordan curves by utilizing quasihomogeneous parameterizations. We verify that rectifiable bilipschitz homogeneous Jordan curves satisfy a chordarc condition. We exhibit numerous examples including a bilipschitz homogeneous quasicircle which has lower Hausdorff density zero. We examine homeomorphisms between Jordan curves.

FRACTAL INNER CHORDARC DISKS

We introduce a class of fractal inner chordarc domains and characterize them as inner Ahlfors regular John disks. We analyze ‘global’ parametrizations for such regions, describe the associated Riemann maps and exhibit several illustrative examples.

Uniform spaces and weak slice spaces

We characterize uniform spaces in terms of a slice condition. We also establish the Gehring–Osgood–Vaisala theorem for uniformity in the metric space context.

Estimates for Conformal Metric Ratios

We present uniform and pointwise estimates for various ratios of the hyperbolic, quasihyperbolic and Möbius metrics. We determine when these ratios are constant. We exhibit numerous illustrative examples.

COMPARING INVARIANT DISTANCES AND CONFORMAL METRICS ON RIEMANN SURFACES

We exhibit sharp inequalities comparing the hyperbolic distance (and hyperbolic metric) to distances (and associated metrics) defined via positve harmonic functions as well as bounded harmonic functions. In the simply connected case, all four inequalities are identities. For the non-simply connected case, we determine precisely when equality can hold: for a pair of points in a distance inequality, or a single point in a metric inequality.