×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] M. M. Schiffer and D. C. Spencer, Functionals of Finite Riemann Surfaces, Princeton University Press, Princeton (1954). · Zbl 0059.06901
[2] I. M. Krichever and S. P. Novikov, Funkts. Anal. Prilozhen.,21, No. 2, 46-65 (1987).
[3] A. Yu. Orlov and E. I. Schulman, Lett. Math. Phys.,12, 171-179 (1986). · Zbl 0618.35107
[4] I. M. Krichever, Usp. Mat. Nauk,44, No. 2, 121-184 (1989).
[5] I. M. Krichever and S. P. Novikov, Funkts. Anal. Prikozhen.,23, No. 1, 24-40 (1989).
[6] G. Segal and G. Wilson, Loop Groups and Equations of KdV-Type, Oxford (1985). · Zbl 0592.35112
[7] B. L. Feigin and D. B. Fuks, Funkts. Anal. Prilozhen.,16, No. 2, 47-63 (1982). · Zbl 0493.46061
[8] M. L. Kontsevich, Funkts. Anal. Prilozhen.,21, No. 2, 78-79 (1987).
[9] A. A. Beilinson and V. V. Schechtman, Determinant Bundles and Virasoro Algebras, Preprint No. 497, University Utrecht (1988). · Zbl 0665.17010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.