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Canonical sections

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canonical cuts

A system of canonical sections is a set

of curves on a finite Riemann surface of genus with a boundary of components such that when these curves are removed from , i.e. on cutting along the curves of , there remains a (planar) simply-connected domain . More precisely, a system is a set of canonical sections if to each closed or cyclic section , , in (or cycle for short) there is exactly one so-called adjoint cycle cutting at exactly one fixed point common to all the sections of . The remaining cycles , , and curves , , have only the point in common, and do not pass from one side of the section to the other; each curve joins with the corresponding boundary component. On a given Riemann surface there exists an infinite set of systems of canonical sections. In particular, for any simply-connected domain that, together with its closure , lies strictly in the interior of , a system of canonical sections can be chosen such that .

Furthermore, it is always possible to find a system of canonical sections consisting entirely of analytic curves. The uniqueness of a system of analytic curves can be ensured, for example, by the additional requirement that some functional related to attains an extremum. In particular, one can draw cyclic canonical sections of a system such that the greatest value of the Robin constant in the class of systems homotopic to is attained at a point in a specific domain , . Uniqueness of the curves can also be ensured by requiring that the Robin constants are maximized at a specified pair of points (see [2]).

References

[1] A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 1 , Springer (1964) pp. Chapt.8
[2] M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954)
How to Cite This Entry:
Canonical sections. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Canonical_sections&oldid=17965
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098