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A new characterization of normalized potentials in dimension two. (English) Zbl 0946.58017

The paper extends certain special dressing transformations on the normalized potentials for harmonic maps, which are already well studied and extensively used in [J. Dorfmeister and 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.

MSC:

58E20 Harmonic maps, etc.
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C43 Differential geometric aspects of harmonic maps
22E67 Loop groups and related constructions, group-theoretic treatment

Citations:

Zbl 0898.53008
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References:

[1] J. Dorfmeister & G. Haak: Meromorphic potentials and smooth constant mean curvature surfaces. Math. Z. 224 (1997), 603–640. · Zbl 0898.53008 · doi:10.1007/PL00004295
[2] J. Dorfmeister, F. Pedit & H. Wu: Weierstrass-type representation of harmonic maps into symmetric spaces. Commun. Anal. Geom., to appear. · Zbl 0932.58018
[3] H. Wu: A simple way for determining the normalized potentials for harmonic maps. Ann. Global Anal. Geom., to appear. · Zbl 0954.58017
[4] H. Wu: On the dressing action of loop groups on constant mean curvature surfaces. Tôhoku Math. J. 49 (1997), 599–621. · Zbl 0912.53010 · doi:10.2748/tmj/1178225065
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