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Harmonic Maps Between Surfaces

Springer Berlin & New York, 1984
Number 1062 in LNM.

Table of Contents

1. Introduction

1.1. A short history of variational principles
1.2. The concept of geodesics
1.3. Definition of harmonic maps
1.4. Mathematical problems arising from the concept of harmonic maps
1.5. Physical significance
1.6. Some remarks about notation and terminology

2. Geometric considerations

2.1. Convexity, existence of geodesics arcs and conjugate points
2.2. Convexity of the squared distance function
2.3. Uniqueness of geodesic arcs in convex discs
2.4. Remark: The higher dimensional analogue of 2.3
2.5. Curvature of parallel curves
2.6. Local coordinates with curvature controlled Christoffel symbols

3. Conformal mappings

3.1. Statement of Thm. 3.1 concerning conformal representations of compact surfaces homeomorphic to plane domains
3.2. The Courant-Lebesgue Lemma
3.3. - 3.6. Proof of Theorem 3.1
3.7. Uniqueness of conformal representations
3.8. Applications of Theorem 3.1
3.9. The Hartman-Wintner Lemma

4. Existence theorems for harmonic maps between surfaces

4.1. A maximum principle for energy minimizing maps
4.2. The Dirichlet problem, if the image is contained in a convex disc
4.3. Remarks about the higher-dimensional situation
4.4. The Theorem of Lemaire and Sacks-Uhlenbeck
4.5 The Dirichlet problem, if the image is homeomorphic to S2 Two different solutions, if the boundary values are nonconstant
4.6. Nonexistence for constant boundary values
4.7. Existence results in arbitrary dimensions

5. Uniqueness theorems

5.1. Composition of harmonic maps with convex functions
5.2. The uniqueness theorem of Jäger and Kaul
5.3. Uniqueness for the Dirichlet problem if the image has nonpositive curvature
5.4. Uniqueness results for closed solutions, if the image has nonpositive curvature
5.5. Uniqueness and nonuniqueness for harmonic maps between closed surfaces

6. A-priori C1,α-estimates

6.1. Composition of harmonic maps with conformal maps
6.2. A maximum principle
6.3. Interior modulus of continuity
6.4. Interior estimates for the energy
6.5. Boundary continuity
6.6. Interior C1-estimates
6.7. Interior C1,α-estimates
6.8. C1- and C1,α-estimates at the boundary

7. A-priori estimates from below for the functional determinant of harmonic diffeomorphisms

7.1. A Harnack inequality of E. Heinz
7.2. Interior estimates
7.3. Boundary estimates
7.4. Discussion of the situation in higher dimensions

8. The existence of harmonic diffeomorphisms which solve a Dirichlet problem

8.1. Proof of the existence theorem in case the image is contained in a convex ball and bounded by a convex curve
8.2. Approximation arguments
8.3. Remarks: Plane domains, necessity of the hypotheses of Theorem 8.1

9. C1,α-a-priori estimates for arbitrary domains. Non-variational existence proofs

9.1. C1,α-estimates on arbitrary surfaces
9.2. Estimates for the functional determinant on arbitrary surfaces
9.3. A non-variational proof of Theorem 4.1
9.4. A non -variational proof of Theorem 8.1

10. Harmonic coordinates. C2,α-a-priori-estimates for harmonic maps

10.1. Existence of harmonic coordinates. C1,α-estimates
10.2. C2,α-estimates for harmonic coordinates
10.3. Bounds on the Christoffel symbols. Conformal coordinates
10.4. Higher regularity of harmonic coordinates
10.5. C2,α-estimates for harmonic maps
10.6. Higher regularity of harmonic maps

11. The existence of harmonic diffeomorphisms between surfaces

11.1. Harmonic diffeomorphisms between closed surfaces (Theorem 11.1)
11.2. Proof of Theorem 11.1
11.3. Extension of Theorem 8.1
11.4 Remarks about the situation in higher dimensions

12. Applications of harmonic maps between surfaces

12.1. Holomorphicity of certain harmonic maps and an analyti proof of Kneser's Theorem
12.2. Proof of Theorem 12.1
12.3. Contractibility of Teichmüller space and the diffeomorphism group
12.4. Tromba's proof that Teichmüller space is a cell
12.5. The approach of Gerstenhaber-Rauch
12.6. Harmonic Gauss maps and Bernstein theorems
12.7. Surfaces of constant Gauss curvature in 3- space

01.10.2020, 14:49