Prescribing scalar curvature in conformal geometry
type: Item
instance of: scholarly article
title: Prescribing scalar curvature in conformal geometry
author: Andrea Malchiodi
published in: Zurich Lectures in Advanced Mathematics
publication date: 2023-10-13T00:00:00Z
review text: 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…
reviewed by: Shi-Zhong Du
Mathematics Subject Classification ID: 53-02
Mathematics Subject Classification ID: 58-02
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C18
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 58J47
Mathematics Subject Classification ID: 58J90
Mathematics Subject Classification ID: 58E30
Mathematics Subject Classification ID: 58K55
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 7750337
zbMATH Keywords: prescribed scalar curvature
zbMATH Keywords: elliptic equation
zbMATH Keywords: critical exponents
zbMATH Keywords: blow-up analysis
zbMATH Keywords: concentration phenomena
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/zlam/31
OpenAlex ID: W4386514522
DOI: 10.4171/ZLAM/31
description: scientific article; zbMATH DE number 7750337