
A Course in Minimal Surfaces
American Mathematical Society (Publisher)
Will be published approx. on 28. February 2011
Book
Paperback/Softback
313 pages
978-1-4704-7640-3 (ISBN)
Description
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), the authors 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. The only prerequisites needed for this book are a basic knowledge of Riemannian geometry and some familiarity with the maximum principle.
More details
Series
Language
English
Place of publication
Providence
United States
Target group
Professional and scholarly
ISBN-13
978-1-4704-7640-3 (9781470476403)
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Schweitzer Classification
Persons
Tobias Holck Colding, Massachusetts Institute of Technology, Cambridge, MA.
William P. Minicozzi II, Johns Hopkins University, Baltimore, MD.
William P. Minicozzi II, Johns Hopkins University, Baltimore, MD.
Content
The beginning of the theory
Curvature estimates and consequences
Weak convergence, compactness and applications
Existence results
Min-max constructions
Embedded solutions of the Plateau problem
Minimal surfaces in three-manifolds
The structure of embedded minimal surfaces
Exercises
Bibliography
Index.
Curvature estimates and consequences
Weak convergence, compactness and applications
Existence results
Min-max constructions
Embedded solutions of the Plateau problem
Minimal surfaces in three-manifolds
The structure of embedded minimal surfaces
Exercises
Bibliography
Index.