Dierkes, Ulrich; Hildebrandt, Stefan; Sauvigny, Friedrich [Küster, A.; Jakob, R.] Minimal surfaces. With assistance and contributions by A. Küster and R. Jakob. 2nd revised and enlarged ed. (English) Zbl 1213.53002 Grundlehren der Mathematischen Wissenschaften 339. Dordrecht: Springer (ISBN 978-3-642-11697-1/hbk; 978-3-642-11698-8/e-book; 978-3-642-11715-2/3-vol.set). xv, 688 p. (2010). This book is the first volume of a treatise on minimal surfaces consisting of three volumes which deal with boundary value problems, mainly for two-dimensional surfaces in Euclidean space. It is a greatly expanded version of the monograph “Minimal surfaces I, II” [Minimal surfaces I. Boundary value problems. Grundlehren der Mathematischen Wissenschaften. 295. Berlin: Springer-Verlag. xiii, 507 p. (1992; Zbl 0777.53012) and Minimal surfaces II. Boundary regularity. Grundlehren der Mathematischen Wissenschaften. 296. Berlin: Springer-Verlag. xi, 421 p. (1992; Zbl 0777.53013)]. It contains 139 figures, 9 color plates, 2 appendices, a bibliography of 81 pages and an index of names and subjects.Like the earlier monograph, this book is divided into two parts: 1) Introduction to the Geometry of Surfaces and to Minimal Surfaces, and 2) the Plateau Problem. Compared to the first edition, it is the second part which has been substantially enlarged. There are now five chapters instead of the earlier three, and results are presented which have only been obtained within the past few years.The book presents an integral view of its subject. The analytical side with definitions and formulation of results as well as predominantly detailed proofs is supplemented with numerous figures to explain the concepts and geometric interpretations. The logic of the subject is easy to follow thanks to explanations why the results are presented as they are in the particular context and sometimes repeatedly. The scholia at the end of each chapter give a good overview of the historical developments and of some additional problems in the particular area under discussion. References across all three volumes assist in the formation of an integral picture.The book has much to offer to students and researchers working in areas of differential geometry and/or calculus of variations. lmportant from their viewpoint are also the problems not yet solved which are presented at the end of the book, as well as problems which arose around the turn of the century and are not being dealt with in these volumes. Reviewer: Kaarin Riives (Tartu) Cited in 85 Documents MSC: 53-02 Research exposition (monographs, survey articles) pertaining to differential geometry 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 35J60 Nonlinear elliptic equations 49Q05 Minimal surfaces and optimization 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations 49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control Keywords:minimal surfaces; surfaces with prescribed mean curvature; nonlinear elliptic equations; minimal surfaces (calculus of variations); research monographs (differential geometric, partial differential equations, calculus of variations) Citations:Zbl 0777.53012; Zbl 0777.53013 × Cite Format Result Cite Review PDF Full Text: DOI