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Minimal submanifolds and related topics /
The Bernstein problem and the Plateau problem are central topics in the theory of minimal submanifolds. This important book presents the Douglas-Rado solution to the Plateau problem, but the main emphasis is on the Bernstein problem and its new developments in various directions: the value distribut...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Singapore ; London :
World Scientific,
©2003.
|
| Colección: | Nankai tracts in mathematics ;
v. 8. |
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Ch. I. Introduction. 1.1. The second fundamental form. 1.2. The first variational formula. 1.3. Minimal submanifolds in euclidean space. 1.4. Minimal submanifolds in the sphere. 1.5. Examples. 1.6. Rigidity theorems. 1.7. exercises
- ch. II. Bernstein's theorem and its generalizations. 2.1. Gauss map. 2.2. The Weierstrass representation. 2.3. The value distribution of the image under the Gauss map. 2.4. Exercises
- ch. III. Weierstrass type representations. 3.1. The representation for surfaces of prescribed mean curvature. 3.2. The representation for CMC-1 surfaces in [symbol]. 3.3. Exercises
- ch. IV. Plateau's problem and Douglas-Rado solution. 4.1. Mathematical formulation. 4.2. The Dirichlet principle. 4.3. Proof of the main theorem
- ch. V. Minimal submanifolds of higher codimension. 5.1. Kähler geometry and Wirtinger's inequality. 5.2. Special Lagrangian submanifolds. 5.3. Exercises
- ch. VI. Stable minimal hypersurfaces. 6.1. The second variational formula. 6.2. Stable minimal hypersurfaces and applications. 6.3. A classification of certain stable minimal surfaces. 6.4. Stable cone and Bernstein's problem. 6.5. Curvature estimates for minimal hypersurfaces. 6.6. Exercises
- ch. VII. Bernstein type theorems for higher codimension. 7.1. Geometry of Grassmannian manifolds. 7.2. Harmonic Gauss maps. 7.3. Bernstein type theorems
- ch. VIII. Entire space-like submanifolds. 8.1. A Bochner type formula. 8.2. The Gauss image. 8.3. Estimates of the second fundamental form. 8.4. Completeness. 8.5. Bernstein's problem. 8.6. Final remarks.