A new characterization of normalized potentials in dimension two
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.
scientific article; zbMATH DE number 1346922