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Distortion theorems in families of harmonic mappings

January 2010

Authors:

Sergey Graf

Tver State University

Differential inequalities in linear- and affine-invariant families of harmonic mappings

Article

Oct 2010

In linear- and affine-invariant families of harmonic mappings of the unit disk we prove some differential inequalities such as the sharp two-sided estimate of the Jacobian and an estimate of the curvature of the image of the circle. Key words and phrasesharmonic mappings-linear- and affine-invariant families of functions-order of a family

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Bi-Lipschicity of quasiconformal harmonic mappings in the plane

February 2009 · Filomat

Vesna Todorcevic

We show that quasiconformal harmonic mappings on the proper domains in R 2 are bi-Lipschitz with respect to the quasihyperbolic metric.

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One system of quasilinear hyperbolic equations: Uniqueness and global solvability of the Cauchy prob...

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Andrzej Łada

In a coordinate patch, the harmonic maps from the two-dimensional Minkowski space into a complete Riemannian manifold satisfy a nonlinear system of hyperbolic partial differential equations. The Cauchy problem for such systems was studied by Gu Chao-Hao [Comm. Pure Appl. Math.33 (1980), 727–737] and O. A. Ladyzhenskaya and V. I. Shubov [Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov. ... [Show full abstract] (LOMI)110 (1981), 81–94 [in Russian]]. In this paper we investigate uniqueness and global existence of C2 solutions of the Cauchy problem for one generalization of the previous hyperbolic system, when the map takes its values in Rm endowed with the Riemannian metric.

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December 1988 · Tokyo Journal of Mathematics

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September 1982 · Siberian Mathematical Journal

O. V. Ivanov

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Holomorphic maps: Geometric aspects

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We study holomorphic, in particular biholomorphic, maps between domains in c and, in one case, in cn , n > 1. These maps are for n = 1 conformal (angle and orientation preserving); so we shall use the terms biholomorphic and conformal interchangeably in this case. For n > 1 we consistently use biholomorphic. Automorphisms of domains, i.e. biholomorphic self-maps, are determined for disks resp. ... [Show full abstract] half-planes, the entire plane, and the sphere: they form groups consisting of Möbius transformations (VII.1). The proof of this fact relies on an important growth property of bounded holomorphic functions: the Schwarz lemma 1.3. Because the automorphism group of the unit disk (or upper half plane) acts transitively, it gives rise – according to F. Klein's Erlangen programme – to a geometry, which turns out to be the hyperbolic (non-euclidean) geometry (VII.2 and 3). The unit disk is conformally equivalent to almost all simply connected plane domains: Riemann's mapping theorem, proved in VII.4. For n > 1 even the immediate generalisations of the disk – the polydisk and the unit ball – are not biholomorphically equivalent (VII.4). Riemann's mapping theorem can be generalised to the general uniformization theorem (VII.4): a special but exceedingly useful case of this is the modular map λ which we introduce in VII.7. Its construction uses tools that are also expedient for other purposes: harmonic functions (with a solution of the Dirichlet problem for disks) and Schwarz's reflection principle (VII.5 and 6). The existence of λ finally yields two important classical results: Montel's and Picard's “big theorems”.

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The Gauss map of a harmonic surface

March 2011 · Indagationes Mathematicae

David Kalaj

We prove that the distortion function of the Gauss map of a surface parametrized by harmonic coordinates coincides with the distortion function of the parametrization. Consequently, the Gauss map of a harmonic surface is K quasiconformal if and only if its harmonic parametrization is K quasiconformal, provided that the Gauss map is regular or what is shown to be the same, provided that the ... [Show full abstract] surface is non-planar. This generalizes the classical result that the Gauss map of a minimal surface is a conformal mapping.

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Product of Harmonic Maps is Harmonic: A Stochastic Approach

July 2007

Key words: Harmonic mappings, Lie groups, Semimartingales

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Leslie Samuel Smith

As part of an investigation of how sound processing may be driven by the characteristics of the sound, we consider the processing of voiced sounds. We consider the (physical) parameters characterising such sounds, how these map into the perceptual space, and how early auditory processing may perform this mapping. The way in which the auditory system is sensitive to phase relationships between ... [Show full abstract] nearby harmonics is investigated using using computer generated stimuli in which phase differences between the partíais alter only the amplitude modulation envelope. The results fit with a model in which cochlear nucleus neurons detect this amplitude modulation.

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Differential as a harmonic morphism with respect to Cheeger–Gromoll-type metrics

April 2010 · Annals of Global Analysis and Geometry

We investigate horizontal conformality of a differential of a map between Riemannian manifolds, where the tangent bundles are equipped with Cheeger–Gromoll-type metrics. As a corollary, we characterize the differential of a map as a harmonic morphism. KeywordsHorizontally conformal mapping-Harmonic morphism-Tangent bundle-Cheeger–Gromoll-type metric Mathematics Subject Classification ... [Show full abstract] (2000)53C07-53C43-53A30

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Singular points of harmonic maps from 4-dimensional domains into 3-spheres

April 2006 · Duke Mathematical Journal

Tôru Nakajima

In this article we show that if a stable-stationary harmonic map u from a domain in

\mathbb{R}^4

into

\mathbb{S}^3

has an isolated singular point

\xi

, then the mapping degree

\degree (u , \xi)

of u at

\xi

is +1 or

-1

. Furthermore, the complete characterization of stable-stationary tangent maps from

\mathbb{B}^4

to

\mathbb{S}^3

is given

Distortion theorems in families of harmonic mappings

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