Canonical sections

canonical cuts

A system of canonical sections is a set

$$ S = \{ a _ {1} , b _ {1} \dots a _ {g} ,\ b _ {g} , l _ {1} \dots l _ \nu \} $$

of $ 2g + \nu $ curves on a finite Riemann surface $ R $ of genus $ g $ with a boundary of $ \nu $ components such that when these curves are removed from $ R $, i.e. on cutting $ R $ along the curves of $ S $, there remains a (planar) simply-connected domain $ R ^ {*} $. More precisely, a system $ S $ is a set of canonical sections if to each closed or cyclic section $ a _ {j} $, $ j = 1 \dots g $, in $ S $( or cycle for short) there is exactly one so-called adjoint cycle $ b _ {j} $ cutting $ a _ {j} $ at exactly one fixed point $ p _ {0} \in R $ common to all the sections of $ S $. The remaining cycles $ a _ {k} , b _ {k} $, $ k \neq j $, and curves $ l _ {s} $, $ s = 1 \dots \nu $, have only the point $ p _ {0} $ in common, and do not pass from one side of the section $ a _ {j} $ to the other; each curve $ l _ {s} $ joins $ p _ {0} $ with the corresponding boundary component. On a given Riemann surface $ R $ there exists an infinite set of systems of canonical sections. In particular, for any simply-connected domain $ D \subset R $ that, together with its closure $ \overline{D}\; $, lies strictly in the interior of $ R $, a system of canonical sections can be chosen such that $ D \subset R ^ {*} $.

Furthermore, it is always possible to find a system of canonical sections $ S $ consisting entirely of analytic curves. The uniqueness of a system $ S $ of analytic curves can be ensured, for example, by the additional requirement that some functional related to $ S $ attains an extremum. In particular, one can draw cyclic canonical sections $ a _ {j} , b _ {j} $ of a system $ S $ such that the greatest value of the Robin constant in the class of systems homotopic to $ S $ is attained at a point $ p _ {0} $ in a specific domain $ D \subset R $, $ p _ {0} \in D $. Uniqueness of the curves $ l _ {s} $ 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)