On a potential-theoretically parabolic Riemann surface
, there exists a so-called Evans-Selberg potential, whose existence is equivalent to the parabolicity condition (see [E, K, Na62, S]).
Along the lines of other important contributions to the development of topology--and in particular that of Felix Klein--Hermann Weyl has pointed out this feature of Riemann's work, by arguing "that it is always the Riemann surface
, not the analytic form, which is regarded as the given object" (2010: 157).
He also uses Quillen's holomorphic theorem to evaluate the dependence of the determinants of the Laplace operations on the boundary conditions on the Riemann surface
At the first glance, the idea of a Riemann surface
looks as a cheap trick: we replace the domain of a given multi-valued function by a new more complex domain on which the function becomes single-valued.
Hence if, conversely, a map A from a simply connected Riemann surface
[summation] into ([h.
Another interesting instance is that of a Riemann surface
endowed with the Poincare metric.
lambda]] and its natural domain of definition is the two-sheeted Riemann surface
For any Riemann surface
S of genus g [less than or equal to] 2,
s]/[partial]z are discontinuous on crossing the screen, and is taken into account in Sommerfeld's theory by "wrapping" the diffracting half plane in a semi-infinite, two-sided Riemann surface
so that its positive and negative sides are distinguished by the values 2[pi] and 0 of the polar angle [PHI] in Fig.
The set [gamma]\[gamma] is a Riemann surface
, which can be seen to have infinite genus because of the presence of infinitely many branch points.
We numerically solved the equations, which involved path integrals of meromorphic functions on a genus-one Riemann surface
They discuss topics related to Kleinian groups, classical Riemann surface
theory, mapping class groups, geometric group theory, and statistical mechanics.