Brad Osgood

Extremals of determinants of Laplacians

On etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

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.

A generalization of Nehari's univalence criterion.

This note is a sequel to our paper [OS] where we generalized the Schwarzian derivative to conformal mappings of Riemannian manifolds. There we found that many of the phenomena familiar from the classical theory have counterparts in the more general setting. Here we advance this another step by giving a generalization of the well known univalence criterion of Nehari [N]. Despite its relatively advanced age, this result continues to generate interest, see [L]. The argument used here in the general case, if specialized to the situation considered by Nehari, gives a somewhat different and a more geometric proof of his theorem than is often presented. We want to keep this note short, since the proof of the Theorem is really quite simple, and also fairly self-contained. We shall need a number of facts from our earlier paper and we collect them here with very little additional discussion. We refer the reader to that paper for more details.

Univalence criteria in multiply-connected domains

Theorems due to Nehari and Ahlfors give sufficient conditions for the univalence of an analytic function in relation to the growth of its Schwarzian derivative. Neharis theorem is for the unit disc and was generalized by Ahifors to any simply-connected domain bounded by a quasiconformal circle. In both cases the growth is measured in terms of the hyperbolic metric of the domain. In this paper we prove a corresponding theorem for finitely-connected domains bounded by points and quasiconformal circles. Metrics other than the hyperbolic metric are also considered and similar results are obtained.

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.