Quasiconformal extension for harmonic mappings on finitely connected domains

Quasiconformal Extension of Harmonic Mappings in the Plane

Let f be a sense-preserving harmonic mapping in the unit disk. We give a sufficient condition in terms of the pre-Schwarzian derivative of f to ensure that it can be extended to a quasiconformal map in the complex plane. Introduction A well-known criterion due to Becker [5] states that if a locally univalent analytic function φ in the unit disk D satisfies (1) sup z∈D ∣∣∣∣φ′′(z) φ′(z) ∣∣∣∣ (1− ...

Harmonic Univalent Mappings and Linearly Connected Domains

We investigate the relationship between the univalence of f and of h in the decomposition f = h+g of a sense-preserving harmonic mapping defined in the unit disk D ⊂ C. Among other results, we determine the holomorphic univalent maps h for which there exists c > 0 such that every harmonic mapping of the form f = h + g with |g| < c|h| is univalent. The notion of a linearly connected domain appea...

Harmonic Coordinates on Finitely Connected Fractafolds

We define finitely connected fractafolds, which are generalizations of p.c.f. self-similar sets introduced by Kigami and of fractafolds introduced by Strichartz. Any self-similarity is not assumed, and countably infinite ramification is allowed. We prove that if a fractafold has a resistance form in the sense of Kigami that satisfies certain assumptions, then there exists a weak Riemannian metr...

Harmonic Quasiconformal Mappings of Riemannian Manifolds

Locally Uniform Domains and Quasiconformal Mappings

We document various properties of the classes of locally uniform and weakly linearly locally connected domains. We describe the boundary behavior for quasiconformal ho-meomorphisms of these domains and exhibit certain metric conditions satissed by such maps. We characterize the quasiconformal homeomorphisms from locally uniform domains onto uniform domains. We furnish conditions which ensure th...