{"product_id":"minimal-surfaces-339-grundlehren-der-mathematischen-wissenschaften","title":"Minimal Surfaces: 339 (Grundlehren der mathematischen Wissenschaften)","description":"\u003cp\u003eMinimal 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 \u0026amp; 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: \\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 Bjrlings 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 Plateaus 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 Nitsches uniqueness theorem and Tomis finiteness result. In addition, a theory of unstable solutions of Plateaus problems is developed which is based on Courants mountain pass lemma. Furthermore, Dirichlets problem for nonparametric H-surfaces is solved, using the solution of Plateaus problem for H-surfaces and the pertinent estimates.\u003c\/p\u003e","brand":"Springer","offers":[{"title":"Default Title","offer_id":45884445688006,"sku":"DADAX3642116973","price":209.2,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0695\/9389\/1014\/files\/51mRiLQ480L.jpg?v=1779822495","url":"https:\/\/ergodemedia.com\/products\/minimal-surfaces-339-grundlehren-der-mathematischen-wissenschaften","provider":"Ergodemedia","version":"1.0","type":"link"}