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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 nonconstant harmonic mapping X: OmegaoR^3 which is conformally parametrized on OmegasubsetR^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. TOC:Introduction.- Part I. Introduction to the Geometry of Surfaces and to Minimal Surfaces.- 1.Differential Geometry of Surfaces in Three-Dimensional Euclidean Space.- 2.Minimal Surfaces.- 3.Representation Formulas and Examples of Minimal Surfaces.- Part II. Plateau`s Problem.- 4.The Plateau Problem, and its Ramifications.- 5.Stable Minimal- and H-Surfaces.- 6.Unstable Minimal Surfaces.- 7.Graphs with Prescribed Mean Curvature.- 8.Introduction to the Douglas Problem.- Problems.- 9. Appendix 1. On Relative Minimizers of Area and Energy.- Appendix 2. Minimal Surfaces in Heisenberg Groups.- Bibliography.- Index.

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Minimal Surfaces ab 87.99 EURO 2nd, revidierte and enlarged ed. 2010.