This book deals with nonlinear problems in geometry such as the problems of Yamabe, Calabi, Nirenberg and some other problems, Monge-Ampere equations on compact Kahler manifolds and on a bounded domain. The book presents a number of methods to deal with these problems, some basic results, and some open problems. The book is intended as an introduction to research. It consists of 8 chapters, a bibliography (280 entries), and a subject index. The first 4 chapters (1. Riemannian geometry, 2. Sobolev spaces, 3. Background material, 4. Green's function for Riemannian manifolds) present some background material. Chapters 5–8 (5. The methods, 6. The scalar curvature, 7. Complex Monge-Ampere equation on compact Kahler manifolds, 8. Monge-Ampere equations) deal with the nonlinear problems mentioned above. For example the Yamabe problem is studied: Let (Mn,g) be a C∞ compact Riemannian manifold of dimension n≥3, R its scalar curvature, g its metric. Does there exist a metric g′, conformal to g, such that the corresponding scalar curvature R′=const? The problem is reduced to the equation (1) 4((n−1)/(n−2))ΔΦ+RΦ=R′Φ(n+2)/(n−2), Φ>0, Φ∈C∞, with R′=const. Consider the equation (2) ΔΦ+h(x)Φ=λf(x)ΦN−1, with h,f∈C∞, f>0, N=2n/(n−2). Does there exist a Φ∈C∞, Φ>0, and a real number λ such that (2) holds? In Chapters 5 and 6 a variational method to study equations (1), (2) is given, and the results of Yamabe, Kazdan and Warner, the author and others are reported. In particular, Kazdan and Warner described the set of scalar curvature functions R∈C∞(Mn) associated with Riemannian metrics on compact manifolds Mn, n≥3. They proved that a given function R∈C∞(Mn) is the scalar curvature of some Riemannian metric on Mn if (i) R is negative somewhere on Mn, or (ii) R≥0 on Mn and there exists a metric with constant positive scalar curvature on Mn. Thus, every R∈C∞(Mn) is a scalar curvature if and only if Mn admits a metric of positive constant scalar curvature. Calabi's conjecture asserts that every form representing the first Chern class c1(M) is the Ricci form of some Kahler metric on a compact Kahler manifold (M,g). The conjecture is proved in Section 7.8 among other results. In particular, the following existence theorem due to the author is proved: A compact Kahler manifold with negative first Chern class has an Einstein-Kahler metric. The Dirichlet problem for real Monge-Ampere equations is studied in Chapter 8. Let B be the ball of radius 1 in Rn, I be a closed interval of R, f(x,t)∈C∞(B¯¯¯¯×I), g∈C∞ be a Riemannian metric on B¯¯¯¯, u∈C∞(∂B), u:S→I, S=∂B, u is a restriction to S of a C∞ function γ on B¯¯¯¯. The problem is to prove the existence of a function u∈C∞(B¯¯¯¯) such that (3) logdet(∇ijΦ+aij)=f(x,Φ), Φ=u on S, where aij=aji, 1≤i,j≤n are C∞ functions on B¯¯¯¯. The a priori estimates of solutions to (3) are established and the solvability of (3) is proved. The results of Nirenberg, Pogorelov, Cheng and Yau, Bedford and Taylor, P. Lions , Cafarelli, Nirenberg and Spruck are reported. The book is clearly and accurately written. Much of the material in Chapters 5–8 was not available earlier in book form. The book is a valuable contribution to the literature by one of the active researchers in the field. {Remark: In Sections 10-11 Kondrakov should be changed to Kondrashov.}
Reviewer: Ramm, A. G. [form MathSciNet]