Hardcover ISBN:  9780821853238 
Product Code:  GSM/121 
313 pp 
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Electronic ISBN:  9781470411824 
Product Code:  GSM/121.E 
313 pp 
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Book DetailsGraduate Studies in MathematicsVolume: 121; 2011MSC: Primary 49; 53; 58; 57; 35; 83;
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.ReadershipGraduate students and research mathematicians interested in the theory of minimal surfaces.

Table of Contents

Chapters

Chapter 1. The beginning of the theory

Chapter 2. Curvature estimates and consequences

Chapter 3. Weak convergence, compactness and applications

Chapter 4. Existence results

Chapter 5. Minmax constructions

Chapter 6. Embedded solutions of the Plateau problem

Chapter 7. Minimal surfaces in threemanifolds

Chapter 8. The structure of embedded minimal surfaces

Exercises


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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.
Graduate students and research mathematicians interested in the theory of minimal surfaces.

Chapters

Chapter 1. The beginning of the theory

Chapter 2. Curvature estimates and consequences

Chapter 3. Weak convergence, compactness and applications

Chapter 4. Existence results

Chapter 5. Minmax constructions

Chapter 6. Embedded solutions of the Plateau problem

Chapter 7. Minimal surfaces in threemanifolds

Chapter 8. The structure of embedded minimal surfaces

Exercises