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Variations of the complex structure of Riemann surfaces by vector fields on a contour and objects of the KP theory. The Krichever-Novikov problem of the action on the Baker-Akhieser functions. (English. Russian original) Zbl 0719.30037
Funct. Anal. Appl. 24, No. 1, 61-63 (1990); translation from Funkts. Anal. Prilozh. 24, No. 1, 72-73 (1990).
Let \(\Gamma\) be a type g Riemann surface with a distinguished point \(\infty\) with local parameter \(z=1/\lambda\) and the divisor \(\gamma_ 1,...,\gamma_ g\). Let further S be a small contour on \(\Gamma\) encircling \(\infty\), U(S) a small neighbourhood of S and \(\Gamma =\Gamma_+\cup \Gamma_ -\) with \(\infty \in \Gamma_ -\) and \(\Gamma_+\), \(\Gamma_ -\) intersecting on U(S). If v is a vector field on S which can be analytically continued to the whole U(S) we can define a deformation of \(\Gamma\) as follows: to any \(\gamma_+\in \Gamma_+\) there corresponds, instead of \(\gamma_ -\), the point \(\exp (\beta v)\gamma_ -\), where \(\beta\) is some real parameter. The change in transition map causes, in general, the change of complex structure. We obtain the natural mapping E from the old Riemann surface to the new one, with discontinuity on S. The following theorem is proven: Let the deformation of the Riemann surface \(\Gamma\) be given by the vector field on S, the divisor \(\gamma_ 1,...,\gamma_ g\) and the local parameter \(z=1/\lambda\) being transported with the help of the mapping E. Then the change of the potential \(\phi\) is given by \[ \partial \phi /\partial \beta =(2\pi i)^{-1}\oint (v\psi)\psi^*, \] where \[ (\partial^ 2_ y-\partial^ 2_ x-2\phi_ x)\psi =0,\quad (\partial^ 2_ y+\partial^ 2_ x+2\phi_ x)\psi^*=0. \]

30F99 Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14H15 Families, moduli of curves (analytic)
37B99 Topological dynamics
Full Text: DOI
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