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M. Schiffer, Donald C. Spencer, ""

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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