E-grāmata: Course in Minimal Surfaces

"Minimal surfaces date back to Euler and Lagrange and the beginning of the calculus of variations. Many of the techniques developed have played key roles in geometry and partial differential equations. Examples include monotonicity and tangent cone analysis originating in the regularity theory for minimal surfaces, estimates for nonlinear equations based on the maximum principle arising in Bernstein's classical work, and even Lebesgue's definition of the integral that he developed in his thesis on the Plateau problem for minimal surfaces. This book starts with the classical theory of minimal surfaces and ends up with current research topics. Of the various ways of approaching minimal surfaces (from complex analysis, PDE, or geometric measure theory), theauthors have chosen to focus on the PDE aspects of the theory. The book also contains some of the applications of minimal surfaces to other fields including low dimensional topology, general relativity, and materials science."--

Mathematicians Colding (New York U.) and Minicozzi (Johns Hopkins U.) aspire to start with students who have no more mathematical background than Riemannian geometry and some familiarity with the maximum principle, and to leave them able to comprehend current research topics in minimal surfaces. They cover the beginning of the theory; curvature estimates and consequences; weak convergences, compactness, and applications; existence results; min-max constructions; embedded solutions of the plateau problem; minimal surfaces in three-manifolds; and the structure of embedded minimal surfaces. Annotation ©2011 Book News, Inc., Portland, OR (booknews.com)