Canonical sections
canonical cuts
A system of canonical sections is a set
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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) |
Canonical sections. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Canonical_sections&oldid=17965