Functionals of finite Riemann surfaces - Schiffer, Menahem; Spencer, D. C. (Donald Clayton), 1912-2001 (Princeton mathematical series, 1954)
This monograph is an outgrowth of lectures given by the authors at Princeton University during the academic year 1949-1950, and it is concerned with finite Riemann surfaces- that is to say with Riemann surfaces of finite genus which have a finite number of non-degenerate boundary components. The main purpose of the monograph is the investigation of finite Riemann surfaces from the point of view of functional analysis, that is, the study of the various Abelian differentials of the surface in their dependence on the surface itself. Riemann surfaces with boundary are closed by the doubling process and their theory is thus reduced to that of closed surfaces. Attention is centered on the differentials of the third kind in terms of which the other differentials may be expressed. The monograph is self-contained except for a few places where references to the literature are given.