CBMS Regional Conference Series in Mathematics 1985; 55 pp; softcover Number: 57 Reprint/Revision History: reprinted with corrections 1987 ISBN10: 0821807072 ISBN13: 9780821807071 List Price: US$23 Member Price: US$18.40 All Individuals: US$18.40 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 nonexistence results. The intent of this volume is to give an uptodate 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. Readership Table of Contents  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
 CauchyRiemann manifold
 III. Ricci Curvature
 Local solvability of Ric\((g)=R_ij\)
 Local smoothness of metrics
 Global topological obstructions
 Uniqueness, nonexistence
 Einstein metrics on 3manifolds
 Käahler manifolds
 (a) Kähler geometry
 (b) Calabi's problem and KählerEinstein 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
