Nonlinear Methods in Riemannian and Kรคhlerian Geometry [electronic resource] : Delivered at the German Mathematical Society Seminar in Dรผsseldorf in June, 1986 / by Jรผrgen Jost
In this book, I present an expanded version of the contents of my lectures at a Seminar of the DMV (Deutsche Mathematiker Vereinigung) in Diisseldorf, June, 1986. The title "Nonlinear methods in complex geometry" already indicates a combination of techniques from nonlinear partial differential equations and geometric concepts. In older geometric investigations, usually the local aspects attracted more attention than the global ones as differential geometry in its foundations provides approximations of local phenomena through infinitesimal or differential constructions. Here, all equations are linear. If one wants to consider global aspects, however, usually the presence of curvature leads to a nonlinearity in the equations. The simplest case is the one of geodesics which are described by a system of second order nonlinear ODE; their linearizations are the Jacobi fields. More recently, nonlinear PDE played a more and more prominent role in geometry. Let us list some of the most important ones: - harmonic maps between Riemannian and Kahlerian manifolds - minimal surfaces in Riemannian manifolds - Monge-Ampere equations on Kahler manifolds - Yang-Mills equations in vector bundles over manifolds. While the solution of these equations usually is nontrivial, it can lead to very signifiยญ cant results in geometry, as solutions provide maps, submanifolds, metrics, or connections which are distinguished by geometric properties in a given context. All these equations are elliptic, but often parabolic equations are used as an auxiliary tool to solve the elliptic ones
CONTENT
1. Geometric preliminaries -- 2. Some principles of analysis -- 3. The heat flow on manifolds. Existence and uniqueness of harmonic maps into nonpositively curved image manifolds -- 4. The parabolic Yang-Mills equation -- 5. Geometric applications of harmonic maps -- Appendix: Some remarks on notation and terminology
