Functionals of Finite Riemann Surfaces
| Date: 16 April 2011, 00:24
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Preface
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 relations between the functionals of two Riemann surfaces one of
which is imbedded in the other are studied, and series developments
are given for the functionals of the smaller surface in terms of those
of the larger. Conditions are found in order that a local holomorphic
imbedding can be extended to an imbedding in the large of one
surface in the other. It may be remarked that the notion of imbedding
is a natural generalization of the concept of schlicht functions in a
plane domain since these functions imbed the plane domain into
the sphere.
If a surface imbedded in another converges to the larger surface,
asymptotic formulas are obtained which lead directly to the varia-
tional theory of Riemann surfaces. A systematic development of the
variational calculus is then given in which topological or conformal
type may or may not be preserved.
The variational calculus is applied to the study of relations between
the various functionals of a given Riemann surface and to extremum
problems in the imbedding of one surface into another. By speciali-
zation, applications to classical conformal mapping are obtained.
In a final chapter some aspects of the generalization of the theory
to Kahler manifolds of higher dimension are discussed.
The first three chapters contain a development of the classical
theory along historical lines, and these chapters may be omitted
by the specialist. It was felt to be desirable to include these chapters
as a means of providing a historical perspective of the field. A more
modern treatment is included in Chapter 9 as the special case of a
Kahler manifold of complex dimension 1 (a Riemann surface may
always be made into a Kahler manifold by the construction of a
Kahler metric).
The monograph is self-contained except for a few places where
references to the literature are given.
M. SCHIFFER and D. C. SPENCER,
Hebrew University, Jerusalem, and
Princeton University
December, 1951
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