A new characterization of normalized potentials in dimension two
type: Item
instance of: scholarly article
title: A new characterization of normalized potentials in dimension two
zbMATH Open document ID: 0946.58017
DOI: 10.1007/BF03322110
author: Hongyou Wu
published in: Results in Mathematics
publication date: 2000-10-26T00:00:00Z
review text: The paper extends certain special dressing transformations on the normalized potentials for harmonic maps, which are already well studied and extensively used in [\textit{J. Dorfmeister} and \textit{G. Haak}, Math. Z. 224, No. 4, 603-640 (1997; Zbl 0898.53008)]. The analytic characterization at a pole \(z_0\) of a normalized potential \(P\) for the harmonic maps \(\varphi: M\to S^2\), where \(M\) is a Riemann surface other than the Riemann sphere, amounts to the zero residue condition on \(p\), \(E/p\), and their first few integrals, where \(E\) is the Hopf differential of \(\varphi\) and \(p\) is the holomorphic part of the conformal factor in the ``metric'' on the universal cover \(\widetilde M\) of \(M\), induced by \(\varphi\). Each of the considered residue conditions is in fact an explicit polynomial equation on the coefficients of negative powers of \(z-z_0\) in the Laurent expansion of \(p\) or \(E/p\) about \(z_0\) and the first few coefficients in the Taylor expansion of \(E\) about \(z_0\). An application of this explicit characterization is also provided.
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C43
Mathematics Subject Classification ID: 22E67
zbMATH DE Number: 1346922
zbMATH Keywords: dressing transformations
zbMATH Keywords: normalized potentials
zbMATH Keywords: harmonic maps
reviewed by: Vladimir Balan
MaRDI profile type: Publication
cites work: Meromorphic potentials and smooth surfaces of constant mean curvature
cites work: Weierstrass type representation of harmonic maps into symmetric spaces
cites work: A simple way for determining the normalized potentials for harmonic maps
cites work: On the dressing action of loop groups on constant mean curvature surfaces
full work available at URL: https://doi.org/10.1007/bf03322110
OpenAlex ID: W2004804868
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description: scientific article; zbMATH DE number 1346922