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Prescribing the Curvature of a Riemannian Manifold
A co-publication of the AMS and CBMS.
 SEARCH THIS BOOK:
CBMS Regional Conference Series in Mathematics
1985; 55 pp; softcover
Number: 57
Reprint/Revision History:
reprinted with corrections 1987
ISBN-10: 0-8218-0707-2
ISBN-13: 978-0-8218-0707-1
List Price: US$25 Member Price: US$20
All Individuals: US\$20
Order Code: CBMS/57

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.

• I. Gaussian Curvature
• Surfaces in $$R^3$$
• Prescribing the curvature form on a surface
• Prescribing the Gaussian curvature on a surface
• (a) Compact surfaces
• (b) Noncompact surfaces
• II. Scalar Curvature
• Topological obstructions
• Pointwise conformal deformations and the Yamabe problem
• (a) $$M^n$$ compact
• (b) $$M^n$$ noncompact
• Prescribing scalar curvature
• Cauchy-Riemann manifold
• III. Ricci Curvature
• Local solvability of Ric$$(g)=R_ij$$
• Local smoothness of metrics
• Global topological obstructions
• Uniqueness, nonexistence
• Einstein metrics on 3-manifolds
• Käahler manifolds
• (a) Kähler geometry
• (b) Calabi's problem and Kähler-Einstein metrics
• (c) Another variational problem
• IV. Boundary Value Problems
• Surfaces with constant mean curvature
• (a) Rellich's problem
• Some other boundary value problems
• (a) Graphs with prescribed mean curvature
• (b) Graphs with prescribed Gauss curvature
• The $$C^2+\alpha$$ estimate at the boundary
• Some Open Problems