minimal surfaces is the first volume of a three volume treatise on minimal surfaces (grundlehren nr. 339-341). each volume which can be read and studied independently of the others. the central theme is boundary value problems for minimal surfaces. the treatise is a greatly 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 surfacesof zero mean curvature. the final definition of a minimal surface is that of a non-constant harmónic mapping x: o-> r3 which is conformally parametrized on o-> r2 and may have branch points. thereafter the classical theroy 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 theorem 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 suurfaces 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 dor h-surfaces and the pertinent estimates.
