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Nonlinear elliptic equations in conformal geometry. (English) Zbl 1064.53018

Zurich Lectures in Advanced Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 3-03719-006-X/pbk). viii, 92 p. (2004).
The book contains a discussion of some nonlinear partial differential equations related to curvature invariants in conformal geometry. Let \((M,g)\) be a \(4\)-dimensional Riemannian manifold. One studies the partial differential equation which describes the curvature invariant \[ \sigma_2(A):=\tfrac12 \{\operatorname{Tr}(A)^2-| A| ^2\} \] where \(A_{ij}:=R_{ij}-{1\over6}Rg_{ij}\) is the Weyl-Schouten tensor. The Moser-Trudinger inequalities are described – they form the main analytic tool. The connection between these inequalities and the Polyakov formula for the functional determinant of the Laplacian is given. General conformal invariants, the connection of conformal invariants to conformal covariant operators on manifolds of dimension \(3\) and higher, and the Paneitz operator in dimension \(4\) are discussed. The work of Chang-Gursky-Yang on the existence of manifolds with \(\sigma_2>0\) is given. In more detail, the contents of the book are:
1) Gaussian curvature equation.
2) Moser-Trudinger inequality on the sphere.
3) Polyakov formula on compact surfaces.
4) Conformal convariant operators - Paneitz operator.
5) Functional determinant on 4 manifolds.
6) Extremal metrics for the log determinant functional.
7) Elementary symmetric functions.
8) A-priori estimates for the regularized equation \((*)_\delta\).
9) Smoothing via the Yamabe flow.
10) Deforming \(\sigma_2\) to a constant function.
This book is a lovely introduction to the subject and contains an excellent bibliographic introduction to the subject comprising approximately 100 entries.

MSC:

53C20 Global Riemannian geometry, including pinching
53A30 Conformal differential geometry (MSC2010)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
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