Constant Mean Curvature Surfaces with Boundary

1. Front Matter

   Pages I-XIV

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2. A Brief Historical Introduction and Motivations

   • Rafael López

   Pages 1-12

3. Surfaces with Constant Mean Curvature

   • Rafael López

   Pages 13-36

4. The Comparison and Tangency Principles

   • Rafael López

   Pages 37-54

5. Constant Mean Curvature Embedded Surfaces

   • Rafael López

   Pages 55-79

6. The Flux Formula for Constant Mean Curvature Surfaces

   • Rafael López

   Pages 81-101

7. The Area and the Volume of a Constant Mean Curvature Surface

   • Rafael López

   Pages 103-121

8. Constant Mean Curvature Disks with Circular Boundary

   • Rafael López

   Pages 123-132

9. The Dirichlet Problem of the cmc Equation

   • Rafael López

   Pages 133-172

10. The Dirichlet Problem in Unbounded Domains

    • Rafael López

    Pages 173-187

11. Constant Mean Curvature Surfaces in Hyperbolic Space

    • Rafael López

    Pages 189-217

12. The Dirichlet Problem in Hyperbolic Space

    • Rafael López

    Pages 219-234

13. Constant Mean Curvature Spacelike Surfaces in Lorentz-Minkowski Space

    • Rafael López

    Pages 235-259

14. Back Matter

    Pages 261-292

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The study of surfaces with constant mean curvature (CMC) is one of the main topics in classical differential geometry. Moreover, CMC surfaces are important mathematical models for the physics of interfaces in the absence of gravity, where they separate two different media or for capillary phenomena. Further, as most techniques used in the theory of CMC surfaces not only involve geometric methods but also PDE and complex analysis, the theory is also of great interest for many other mathematical fields.

While minimal surfaces and CMC surfaces in general have already been treated in the literature, the present work is the first to present a comprehensive study of “compact surfaces with boundaries,” narrowing its focus to a geometric view. Basic issues include the discussion whether the symmetries of the curve inherit to the surface; the possible values of the mean curvature, area and volume; stability; the circular boundary case and the existence of the Plateau problem in the non-parametric case. The exposition provides an outlook on recent research but also a set of techniques that allows the results to be expanded to other ambient spaces. Throughout the text, numerous illustrations clarify the results and their proofs.

The book is intended for graduate students and researchers in the field of differential geometry and especially theory of surfaces, including geometric analysis and geometric PDEs. It guides readers up to the state-of-the-art of the theory and introduces them to interesting open problems.

• 53, 53A, 53A10, 53C42, 35J, 35J60
• Dirichlet problem
• flux formula
• mean curvature
• tangency principle
• partial differential equations

• Department of Geometry and Topology, University of Granada, Granada, Spain

  Rafael López

R. López, Professor at the University of Granada’s Department of Geometry and Topology has published more than 70 articles, many of them in journals with high Impact Factors (Duke Math. J., J. Diff. Eq. Calculus of Var., Comm. Math. Phys. SIAM J. Math. Anal.) and has served as a visiting professor at Paris VII, Idaho St. Univ., Sao Paulo, Toledo, Iasi and KIAS. His main research area is ‘surfaces with prescribed mean curvature,’ including elliptic equations and general relativity.

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