# Nonlinear Methods in Riemannian and Kählerian Geometry

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

• Jürgen Jost
Book

Part of the DMV Seminar book series (OWS, volume 10)

1. Front Matter
Pages 1-9
2. Jürgen Jost
Pages 11-72
3. Jürgen Jost
Pages 73-86
4. Jürgen Jost
Pages 110-124
5. Jürgen Jost
Pages 125-149
6. Back Matter
Pages 150-156

### Introduction

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 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.

### Keywords

Mathematik Minimal surface attention curvature differential geometry manifold system

#### Authors and affiliations

• Jürgen Jost
• 1
1. 1.Mathematisches InstitutRuhr-Universität BochumBochum 1Germany

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-0348-7690-2
• Copyright Information Birkhäuser Basel 1988
• Publisher Name Birkhäuser, Basel
• eBook Packages
• Print ISBN 978-3-7643-1920-5
• Online ISBN 978-3-0348-7690-2
• Buy this book on publisher's site
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