By Ulrich Dierkes
Minimum Surfaces I is an advent to the sphere of minimum surfaces and a presentation of the classical idea in addition to of components of the fashionable improvement founded round boundary price difficulties. half II offers with the boundary behaviour of minimum surfaces. half I is especially apt for college kids who are looking to input this attention-grabbing zone of study and differential geometry which over the past 25 years of mathematical examine has been very energetic and effective. Surveys of assorted subareas will lead the scholar to the present frontiers of data and will even be important to the researcher. The lecturer can simply base classes of 1 or semesters on differential geometry on Vol. 1, as many subject matters are labored out in nice element. a number of computer-generated illustrations of outdated and new minimum surfaces are integrated to help instinct and mind's eye. half 2 leads the reader as much as the regularity idea for nonlinear elliptic boundary worth difficulties illustrated via a selected and engaging subject. there's no comparably complete remedy of the matter of boundary regularity of minimum surfaces to be had in publication shape. This long-awaited e-book is a well timed and welcome boost to the mathematical literature.
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Extra info for Minimal Surfaces II: Boundary Regularity (Grundlehren Der Mathematischen Wissenschaften)
Sample text
Denote by 93(X) the set of tangential vector fields V: 0 -* 683 which are of class C. Each V e 93(X) can be written in the form V(W) = Va(W)X,a(W), W ESL, where V', VZ e C°(S2). We can consider 93(X) as an R(X)-Modul over the function space a(X) := C°(Q), that is, if f, g a j5(X) and V, W e 93(X), then also f V + gW c-93(X) where (f V +gW)(w) = f(w)V(w) + g(w)W(w), w e 0. 5 Covariant Differentiation. 23(X) we can uniquely associate a differential operator L. P))(w) = U"(w)Vfl(w)X,"ft(w) + U"(w)VQ(w)X,0(w) for V = VPX,, e 23(X).
3. References to the Sources of Differential Geometry and to the Literature on Its History Euler's contributions to the curvature theory of surfaces can be found in his Opera Omnia, Ser. I, vol. 28 and 29 [4]. 1 H. Weyl ([2], 5. Auflage, p. 325) noted that the theory of parallel displacement is already contained in the kinematic considerations of the Treatise on Natural Philosophy by Thomson and Tait (edition 1912), Part I, sect. 135-137. 52 1. Differential Geometry of Surfaces in Three-Dimensional Euclidean Space The classical work of Monge appeared as Application de ('Analyse a la Geometrie between 1795 and 1807 [1].
The famous treatises [1] and [2] by Blaschke with their historical annotations still provide an excellent guide to classical differential geometry. We also refer to the modernized version of Blaschke [2] written by Leichtwei13, cf. Blaschke-Leichtweil3 [1]. As a reference to Riemannian geometry, we mention the lecture notes by Gromoll, Klingenberg, and Meyer [1], the monographs by Kobayashi and Nomizu [1], Spivak [1], Warner [1], Dubrovin, Fomenko, and Novikov [1], and the notes by Chem [1, 3], and Hicks [1].