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 Kahler 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 Kaahler manifolds; (a) Kahler geometry; (b) Calabi's problem and Kahler-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.