**CBMS Regional Conference Series in Mathematics**

Volume: 57;
1985;
55 pp;
Softcover

MSC: Primary 53;
Secondary 58

Print ISBN: 978-0-8218-0707-1

Product Code: CBMS/57

List Price: $25.00

Individual Price: $20.00

**Electronic ISBN: 978-1-4704-2419-0
Product Code: CBMS/57.E**

List Price: $25.00

Individual Price: $20.00

# Prescribing the Curvature of a Riemannian Manifold

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*Jerry L. Kazdan*

A co-publication of the AMS and CBMS

These notes were the basis for a series of ten lectures given in
January 1984 at Polytechnic Institute of New York under the
sponsorship of the Conference Board of the Mathematical Sciences and
the National Science Foundation. The lectures were aimed at
mathematicians who knew either some differential geometry or partial
differential equations, although others could understand the
lectures.

Author's Summary:Given a Riemannian Manifold \((M,g)\) one can compute
the sectional, Ricci, and scalar curvatures. In other special
circumstances one also has mean curvatures, holomorphic curvatures,
etc. The inverse problem is, given a candidate for some curvature,
to determine if there is some metric \(g\) with that as its curvature. One
may also restrict ones attention to a special class of metrics, such
as Kähler or conformal metrics, or those coming from an embedding.
These problems lead one to (try to) solve nonlinear partial
differential equations. However, there may be topological or analytic
obstructions to solving these equations. A discussion of these
problems thus requires a balanced understanding between various
existence and non-existence results.

The intent of this volume is to give an up-to-date survey of these
questions, including enough background, so that the current research
literature is accessible to mathematicians who are not necessarily
experts in PDE or differential geometry.

The intended audience is mathematicians and graduate students who know
either PDE or differential geometry at roughly the level of an
intermediate graduate course.

#### Table of Contents

# Table of Contents

## Prescribing the Curvature of a Riemannian Manifold

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents iii4 free
- Preface v6 free
- Notation vii8 free
- I. Gaussian Curvature 110 free
- II. Scalar Curvature 918
- III. Ricci Curvature 1928
- IV. Boundary Value Problems 3342
- Some Open Problems 4756
- References 5160
- Back Cover Back Cover165