Minimal Surfaces is the first volume of a three volume treatise on
minimal surfaces (Grundlehren Nr. 339-341). Each volume can be read and
studied independently of the others. The central theme is boundary value
problems for minimal surfaces. The treatise is a substantially revised
and extended version of the monograph Minimal Surfaces I, II
(Grundlehren Nr. 295 & 296). The first volume begins with an exposition
of basic ideas of the theory of surfaces in three-dimensional Euclidean
space, followed by an introduction of minimal surfaces as stationary
points of area, or equivalently, as surfaces of zero mean curvature. The
final definition of a minimal surface is that of a non-constant harmonic
mapping X: \Omega\to\R^3 which is conformally parametrized on
\Omega\subset\R^2 and may have branch points. Thereafter the classical
theory of minimal surfaces is surveyed, comprising many examples, a
treatment of Björling´s initial value problem, reflection principles, a
formula of the second variation of area, the theorems of Bernstein,
Heinz, Osserman, and Fujimoto. The second part of this volume begins
with a survey of Plateau´s problem and of some of its modifications. One
of the main features is a new, completely elementary proof of the fact
that area A and Dirichlet integral D have the same infimum in the class
C(G) of admissible surfaces spanning a prescribed contour G. This leads
to a new, simplified solution of the simultaneous problem of minimizing
A and D in C(G), as well as to new proofs of the mapping theorems of
Riemann and Korn-Lichtenstein, and to a new solution of the simultaneous
Douglas problem for A and D where G consists of several closed
components. Then basic facts of stable minimal surfaces are derived;
this is done in the context of stable H-surfaces (i.e. of stable
surfaces of prescribed mean curvature H), especially of cmc-surfaces (H
= const), and leads to curvature estimates for stable, immersed
cmc-surfaces and to Nitsche´s uniqueness theorem and Tomi´s finiteness
result. In addition, a theory of unstable solutions of Plateau´s
problems is developed which is based on Courant´s mountain pass lemma.
Furthermore, Dirichlet´s problem for nonparametric H-surfaces is solved,
using the solution of Plateau´s problem for H-surfaces and the pertinent
estimates.
