Regularity of Minimal Surfaces (Grundlehren der mathematischen Wissenschaften, 340),Used

Product DescriptionRegularity of Minimal Surfaces begins with a survey of minimal surfaces with free boundaries. Following this, the basic results concerning the boundary behaviour of minimal surfaces and Hsurfaces with fixed or free boundaries are studied. In particular, the asymptotic expansions at interior and boundary branch points are derived, leading to general GaussBonnet formulas. Furthermore, gradient estimates and asymptotic expansions for minimal surfaces with only piecewise smooth boundaries are obtained. One of the main features of free boundary value problems for minimal surfaces is that, for principal reasons, it is impossible to derive a priori estimates. Therefore regularity proofs for nonminimizers have to be based on indirect reasoning using monotonicity formulas. This is followed by a long chapter discussing geometric properties of minimal and Hsurfaces such as enclosure theorems and isoperimetric inequalities, leading to the discussion of obstacle problems and of Plateaus problem for Hsurfaces in a Riemannian manifold. A natural generalization of the isoperimetric problem is the socalled thread problem, dealing with minimal surfaces whose boundary consists of a fixed arc of given length. Existence and regularity of solutions are discussed. The final chapter on branch points presents a new approach to the theorem that area minimizing solutions of Plateaus problem have no interior branch points.ReviewFrom the reviews of the second edition:The most complete and thorough record of the ongoing efforts to justify Lagranges optimism. contain a wealth of new material in the form of newly written chapters and sections . a compilation of results and proofs from a vast subject. Here were true scholars in the best sense of the word at work, creating their literary lifetime achievements. They wrote with love for detail, clarity and history, which makes them a pleasure to read. will become instantaneous classics. (Matthias Weber, The Mathematical Association of America, June, 2011)From the Back CoverRegularity of Minimal Surfaces begins with a survey of minimal surfaces with free boundaries. Following this, the basic results concerning the boundary behaviour of minimal surfaces and Hsurfaces with fixed or free boundaries are studied. In particular, the asymptotic expansions at interior and boundary branch points are derived, leading to general GaussBonnet formulas. Furthermore, gradient estimates and asymptotic expansions for minimal surfaces with only piecewise smooth boundaries are obtained. One of the main features of free boundary value problems for minimal surfaces is that, for principal reasons, it is impossible to derive a priori estimates. Therefore regularity proofs for nonminimizers have to be based on indirect reasoning using monotonicity formulas. This is followed by a long chapter discussing geometric properties of minimal and Hsurfaces such as enclosure theorems and isoperimetric inequalities, leading to the discussion of obstacle problems and of Plateaus problem for Hsurfaces in a Riemannian manifold. A natural generalization of the isoperimetric problem is the socalled thread problem, dealing with minimal surfaces whose boundary consists of a fixed arc of given length. Existence and regularity of solutions are discussed. The final chapter on branch points presents a new approach to the theorem that area minimizing solutions of Plateaus problem have no interior branch points.