arXiv subjects

David A. Herron

Publications and source records attributed to David A. Herron.

4 recordsLinked to original sources

arXiv2020

Subharmonic Functions, Conformal Metrics, and CAT(0)

We present an analytical proof that certain natural metric planar universal covers are Hadamard metric spaces. In particular if $ρ=φ\circ u$ where $u$ is locally Lipschitz and subharmonic in $Ω$, $φ$ is positive and increasing on an interval containing $u(Ω)$ with $\logφ$ convex, and if the metric space $(Ω,ρ(z)|dz|)$ is complete, then it has universal cover $(\tildeΩ,\tilde{d})$ which is a Hadamard space for which geodesics have Lipschitz continuous first derivatives.

math.CV↗

arXiv2020

Quasihyperbolic Metric Universal Covers

We present a simple analytical proof that the natural metric universal cover of a quasihyperbolic planar domain is a complete Hadamard metric space.

math.CV↗

arXiv2017

Quasiconvexity in the Heisenberg group

We show that if $A$ is a closed subset of the Heisenberg group whose vertical projections are nowhere dense, then the complement of $A$ is quasiconvex. In particular, closed sets which are null sets for the cc-Hausdorff $3$-measure have quasiconvex complements. Conversely, we exhibit a compact totally disconnected set of Hausdorff dimension three whose complement is not quasiconvex.

math.MG↗

arXiv2010

Quasicircles and Bounded Turning Circles Modulo bi-Lipschitz Maps

We construct a catalog, of snowflake type metric circles, that describes all metric quasicircles up to \bl\ equivalence. This is a metric space analog of a result due to Rohde. Our construction also works for all bounded turning metric circles; these need not be doubling. As a byproduct, we show that a metric quasicircle with Assouad dimension strictly less than two is bi-Lipschitz equivalent to a planar quasicircle.

math.CV↗