Martin Chuaqui

Curvature Properties of Planar Harmonic Mappings

A geometric interpretation of the Schwarzian of a harmonic mapping is given in terms of geodesic curvature on the associated minimal surface, generalizing a classical formula for analytic functions. A formula for curvature of image arcs under harmonic mappings is applied to derive a known result on concavity of the boundary. It is also used to characterize fully convex mappings, which are related to fully starlike mappings through a harmonic analogue of Alexander’s theorem.

Univalence criteria for lifts of harmonic mappings to minimal surfaces.

A general criterion in terms of the Schwarzian derivative is given for global univalence of the Weierstrass-Enneper lift of a planar harmonic mapping. Results on distortion and boundary regularity are also deduced. Examples are given to show that the criterion is sharp. The analysis depends on a generalized Schwarzian defined for conformai metrics and on a Schwarzian introduced by Ahlfors for curves. Convexity plays a central role.

Schwarzian derivative criteria for valence of analytic and harmonic mappings

For analytic functions in the unit disk, general bounds on the Schwarzian derivative in terms of Nehari functions are shown to imply uniform local univalence and in some cases finite and bounded valence. Similar results are obtained for the Weierstrass–Enneper lifts of planar harmonic mappings to their associated minimal surfaces. Finally, certain classes of harmonic mappings are shown to have finite Schwarzian norm.

OSCILLATION OF SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS

In this note we study the zeros of solutions of differential equations of the formu !! +pu=0. A criterion for oscillation is found, and some sharper forms of the Sturm comparison theorem are given.

AN EXTENSION OF A THEOREM OF GEHRING AND POMMERENKE

Gehring and Pommerenke have shown that if the Schwarzian derivativeSf of an analytic functionf in the unit diskD satisfies |Sf(z)|≤, 2(1 - |z|2)–2 thenf(D) is a Jordan domain except whenf(D) is the image under a Möbius transformation of an infinite parallel strip. The condition |Sf(z)|≤ 2(1 - |z|2)–2 is the classical sufficient condition for univalence of Nehari. In this paper we show that the same type of phenomenon established by Gehring and Pommerenke holds for a wider class of univalence criteria of the form|Sf(z)|≤p(|z|) also introduced by Nehari. These include|Sf((z)|≤π2/2 and|Sf((z)|≤4(1-|z|2)–1. We also obtain results on Hölder continuity and quasiconformal extensions.

Schwarzian Derivatives and Uniform Local Univalence

Quantitative estimates are obtained for the (finite) valence of functions analytic in the unit disk with Schwarzian derivative that is bounded or of slow growth. A harmonic mapping is shown to be uniformly locally univalent with respect to the hyperbolic metric if and only if it has finite Schwarzian norm, thus generalizing a result of B. Schwarz for analytic functions. A numerical bound is obtained for the Schwarzian norms of univalent harmonic mappings.

Ellipses, near ellipses, and harmonic Möbius transformations

It is shown that an analytic function taking circles to ellipses must be a Mobius transformation. It then follows that a harmonic mapping taking circles to ellipses is a harmonic Mobius transformation.

Constant principal strain mappings on 2-manifolds.

We study mappings between Riemannian 2-manifolds which have constant principal stretching factors (cps-mappings). Such mappings f can be described in terms of the relationship between the geodesic curvature of the curves of principal strain at p and that of their images at f(p). In the context of local coordinates this relationship takes the form of a nonlinear hyperbolic system, the blow-up properties of which depend on the Gaussian curvatures of the two manifolds. We use the theory of such systems to study global existence when both manifolds are the hyperbolic plane

Concave conformal mappings and pre-vertices of Schwarz-Christoffel mappings

\Bbb{H}^2

Functions with prescribed quasisymmetry quotients

and obtain a simple description of all cps-mappings of