Regularity of Minimal Surfaces

1. Front Matter

   Pages I-XVII

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2. Boundary Behaviour of Minimal Surfaces

   1. Front Matter

      Pages 1-1

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3. Boundary Behaviour of Minimal Surfaces

   1. Minimal Surfaces with Free Boundaries

      • Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba

      Pages 3-73

   2. The Boundary Behaviour of Minimal Surfaces

      • Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba

      Pages 75-212

   3. Singular Boundary Points of Minimal Surfaces

      • Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba

      Pages 213-276

4. Geometric Properties of Minimal Surfaces and H-Surfaces

   1. Front Matter

      Pages 277-277

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5. Geometric Properties of Minimal Surfaces

   1. Enclosure and Existence Theorems for Minimal Surfaces and H-Surfaces. Isoperimetric Inequalities

      • Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba

      Pages 279-439

   2. The Thread Problem

      • Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba

      Pages 441-485

   3. Branch Points

      • Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba

      Pages 487-560

6. Back Matter

   Pages 561-623

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Regularity 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 H-surfaces with fixed or free boundaries are studied. In particular, the asymptotic expansions at interior and boundary branch points are derived, leading to general Gauss-Bonnet 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 non-minimizers have to be based on indirect reasoning using monotonicity formulas. This is followed by a long chapter discussing geometric properties of minimal and H-surfaces such as enclosure theorems and isoperimetric inequalities, leading to the discussion of obstacle problems and of Plateau´s problem for H-surfaces in a Riemannian manifold. A natural generalization of the isoperimetric problem is the so-called 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 Plateau´s problem have no interior branch points.

• Boundary value problem
• Minimal surface
• Riemannian manifold
• calculus of variations
• conformal mappings
• differential geometry
• manifold
• minimal surfaces
• minimum
• regularity theory
• partial differential equations

From the reviews of the second edition:

“The most complete and thorough record of the ongoing efforts to justify Lagrange’s 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)

• Faculty of Mathematics, University of Duisburg, Duisburg, Germany

  Ulrich Dierkes

• Mathematical Institute, University of Bonn, Bonn, Germany

  Stefan Hildebrandt

• Baskin 621B, Department of Mathematics, University of California at Santa Cruz, Santa Cruz, USA

  Anthony J. Tromba

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