Nonlinear Methods in Riemannian and Kählerian Geometry

Delivered at the German Mathematical Society Seminar in Düsseldorf in June, 1986

Jürgen Jost (Autor)

Buch | Softcover

156 Seiten

2013 | 2nd ed. 1991. Softcover reprint of the original 2nd ed. 1991
Springer Basel (Verlag)
978-3-0348-7708-4 (ISBN)

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In this book, I present an expanded version of the contents of my lectures at a Seminar of the DMV (Deutsche Mathematiker Vereinigung) in Düsseldorf, 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 Ieads to a nonlinearity in the equations. The simplest case is the one of geodesics which are described by a system of second ordernonlinear ODE; their linearizations are the Jacobi fields. More recently, nonlinear PDE played a more and more pro~inent röle in geometry. Let us Iist some of the most important ones: - harmonic maps between Riemannian and Kählerian manifolds - minimal surfaces in Riemannian manifolds - Monge-Ampere equations on Kähler manifolds - Yang-Mills equations in vector bundles over manifolds. While the solution of these equations usually is nontrivial, it can Iead to very signifi 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.

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.

Erscheint lt. Verlag	20.11.2013
Zusatzinfo	156 p.
Verlagsort	Basel
Sprache	englisch
Maße	155 x 235 mm
Gewicht	253 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	Curvature • Differential Geometry • manifold • minimal surface
ISBN-10	3-0348-7708-0 / 3034877080
ISBN-13	978-3-0348-7708-4 / 9783034877084
Zustand	Neuware