By Ulrich Dierkes,Stefan Hildebrandt,Friedrich Sauvigny,Ruben Jakob,Albrecht Küster
The treatise is a considerably revised and prolonged model of the monograph minimum Surfaces I, II (Grundlehren Nr. 295 & 296).
The first quantity starts off with an exposition of easy rules of the speculation of surfaces in 3-dimensional Euclidean area, via an creation of minimum surfaces as desk bound issues of quarter, or equivalently, as surfaces of 0 suggest curvature. the ultimate definition of a minimum floor is that of a nonconstant harmonic mapping X: OmegaoR^3 that is conformally parametrized on OmegasubsetR^2 and will have department issues. Thereafter the classical conception of minimum surfaces is surveyed, comprising many examples, a remedy of Björling´s preliminary worth challenge, mirrored image rules, a formulation of the second one version of quarter, the theorems of Bernstein, Heinz, Osserman, and Fujimoto.
The moment a part of this quantity starts with a survey of Plateau´s challenge and of a few of its changes. one of many major good points is a brand new, thoroughly easy evidence of the truth that quarter A and Dirichlet quintessential D have an analogous infimum within the category C(G) of admissible surfaces spanning a prescribed contour G. This results in a brand new, simplified answer of the simultaneous challenge of minimizing A and D in C(G), in addition to to new proofs of the mapping theorems of Riemann and Korn-Lichtenstein, and to a brand new answer of the simultaneous Douglas challenge for A and D the place G involves numerous closed parts.
Then easy proof of sturdy minimum surfaces are derived; this is often performed within the context of strong H-surfaces (i.e. of solid surfaces of prescribed suggest curvature H), in particular of cmc-surfaces (H = const), and results in curvature estimates for sturdy, immersed cmc-surfaces and to Nitsche´s distinctiveness theorem and Tomi´s finiteness result.
In addition, a concept of risky options of Plateau´s difficulties is built that is in keeping with Courant´s mountain move lemma. moreover, Dirichlet´s challenge for nonparametric H-surfaces is solved, utilizing the answer of Plateau´s challenge for H-surfaces and the pertinent estimates.