# 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)
How to Cite This Entry:
Canonical sections. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Canonical_sections&oldid=46196
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article