Jürgen Jost
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Max Planck Institute for Mathematics in the Sciences

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Nonlinear methods in Riemannian and Kählerian geometry

Birkhäuser, Basel, Boston, 1988. 2. Auflage 1991

Table of Contents

1. Geometric preliminaries

1.1. Connections
1.2. Riemannian manifolds, geodesics, harmonic maps, and Yang-Mills fields
1.3. Jacobi fields and approximate fundamental solutions
1.4. Complex manifolds and vector bundles
1.5. Kähler manifolds
1.6. The Yang-Mills equation in four dimensions

2. Some principles of analysis

2.1. The continuity method and the heat flow method
2.2. Elliptic and parabolic Schauder theory
2.3. Differential equations on Riemannian manifolds

3. The heat flow on manifolds. Existence and uniqueness of harmonic maps into nonpositively curved image manifolds

3.1. The linear case. Hodge theory by parabolic equations
3.2. Harmonic maps
3.3. The heat flow for harmonic maps
3.4. Uniqueness of harmonic maps

4. The parabolic Yang-Mills equation

4.1. The parabolic version of the Yang-Mills equation
4.2. The Hermitian Yang-Mills equation and its parabolic analogue
4.3. Global existence

5. Geometric applications of harmonic maps

5.1. The topology of Riemannian manifolds of nonpositive sectional curvature
5.2. Siu's strong rigidity theorem for strongly negatively curved Kähler manifolds

Appendix: Some remarks on notation and terminology

11.02.2013, 16:23