Prescribing scalar curvature in conformal geometry
This book discusses the prescribing scalar curvature problem (KW-2) in conformal classes, which is about finding a metric conformal to a given metric on a Riemannian manifold \(M\), such that the scalar curvature equals a previously given function \(K: M\to{\mathbb{R}}\). A parallel version of the prescribing scalar curvature problem (KW-1) solves a similar question without conformal restriction. Since the problem (KW-1) can be solved under rather mild requirements on the prescribed curvature function \(K\) (see Chapter 3), the book is mainly devoted to the discussion of the problem (KW-2) on conformal classes. In terms of the conformal factor function, the problem is boiled down to solve a semilinear elliptic PDE with critical nonlinear exponents. For loss of compactness in general, the first chapter of the book uses subcritical problems, which can be solved by the variational method of lower semi-continuity to approximate the critical problem (see Chapter I in [\textit{M. Struwe}, Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Berlin etc.: Springer-Verlag (1990; Zbl 0746.49010)]). Some concentration phenomenon occurs. Using blow-up analysis, some known results concerning the isolated singularities of the blow-up solution are presented. In Chapter 2, some preliminaries on differential geometry and differential equations are introduced. In Chapter 3, by introducing the Yamabe quotient and analysing the first eigenvalue of the conformal Laplacian \(L_g\), the Yamabe conjecture on spheres \({\mathbb{S}}^n\) and the proof of Theorem A are discussed. In Chapter 4, after analysing the variational structure of problem (KW-2) for Riemannian manifolds of nonpositive curvature, existence and uniqueness results are derived. In the case of positive curvature, the variational approach is also used to prove Theorem B. Chapter 5 is devoted to the classification of entire solutions on Euclidean spaces. Based on the methods of moving planes and moving spheres, Liouville theorems in the subcritical cases are discussed. The main purpose of Chapter 6 is to carry out the blow-up analysis for concentration phenomena in the critical nonlinear problem. As applications, isolated and isolated-simple blow-ups are discussed. In particular, blow-up analysis is applied to the lower dimensions three and four. Applying the Lyapunov-Schmidt reduction to the subcritical approximating problem (\(E_{K,\tau}\)), the existence of blow-up solutions and analysis of blow-up behaviours are discussed for arbitrary dimensions in Chapter 7. In Chapter 8, some existence results are presented for (\(E_K\)) without a symmetry assumption, but requiring conditions on the structure of critical points of the prescribed function \(K\). The results of the previous two chapters, together with degree theory, the min-max principle, and Morse-theoretical methods are involved. As an application, Theorem C and Theorem D are proven in this Chapter. Contrarily, some nonexistence results are demonstrated in Chapter 9, which are based on the Kazdan-Warner condition for solutions. In the final Chapter 10, some results and perspectives on other related topics are discussed. Summing up, this book serves very well as a textbook on the prescribed scalar curvature problem in geometric analysis, which is based on the variational method and blow-up analysis for critical concentration phenomena.
scientific article; zbMATH DE number 7750337