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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 $(\partial^ 2_ y-\partial^ 2_ x-2\phi_ x)\psi =0,\quad (\partial^ 2_ y+\partial^ 2_ x+2\phi_ x)\psi^*=0.$

##### MSC:
 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
##### Keywords:
Baker-Akhiezer function; Krichever-Novikov problem
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##### References:
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