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Two-Dimensional Geometric Variational Problems

Wiley-Interscience, Chichester, 1991

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Table of Contents

1. Examples, definitions, and elementary results

1.1. Plateau's problem
1.2. Two-dimensional conformally invariant variational problems
1.3. Harmonie maps, conformal maps, and holomorphic quadratic differentials
1.4. Some applications of holomorphic quadratic differentials. Surfaces in R3. The Gauss map

2. Regularity and uniqueness results

2.1. Harmonic coordinates
2.2. Uniqueness of harmonic maps
2.3. Continuity of weak solutions
2.4. Removability of isolated singularities
2.5. Higher regularity
2.6. The Hartmann-Wintner Lemma and some of its consequences. Asymptotic expansions at branch points
2.7. Estimates from below for the functional determinant of univalent harmonic mappings

3. Conformal representation

3.1. Conformal representation of surfaces homeomorphic to S2
3.2. Conformal representation of surfaces homeomorphic to circular domains
3.3. Conformal representation of closed surfaces of higher genus

4. Existence results

4.1. The local existence theorem for harmonic maps. An easy proof of the existence of energy-minimizing maps
4.2. The general existence theorem. First part of the proof
4.3. Completion of the proof of Theorem 4.2.1
4.4. Corollaries and consequences of the general existence theorem. Boundary conditions
4.5. Non-existence results. Existence of maps with holomorphic quadratic differentials
4.6. Another proof of the existence of unstable minimal surfaces
4.7. The Plateau-Douglas problem in Riemannian manifolds

5. Harmonic maps between surfaces

5.1. The existence of harmonic diffeomorphisms
5.2. Local computations. Consequences for non-positively curved image metrics. Harmonic diffeomorphisms. Kneser's Theorem
5.3. Miscellaneous results about harmonic branched coverings and harmonic diffeomorphisms

6. Harmonic maps and Teichmüller spaces

6.1. The basic definitions
6.2. The topological and differentiable structure of Tp.Teichmüller's Theorem
6.3. The complex structure
6.4. The energy as a function of the domain metric
6.5. The metric structure. The Weil-Petersson metric. Kähler property. The curvature


01.10.2020, 14:49