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Non-linear Elliptic Equations in Conformal Geometry
Sun-Yung Alice Chang, Princeton University, NJ
A publication of the European Mathematical Society.
cover
Zurich Lectures in Advanced Mathematics
2004; 100 pp; softcover
Volume: 2
ISBN-10: 3-03719-006-X
ISBN-13: 978-3-03719-006-7
List Price: US$28
Member Price: US$22.40
Order Code: EMSZLEC/2
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Non-linear elliptic partial differential equations are an important tool in the study of Riemannian metrics in differential geometry, in particular for problems concerning the conformal change of metrics in Riemannian geometry. In recent years the role played by the second order semi-linear elliptic equations in the study of Gaussian curvature and scalar curvature has been extended to a family of fully non-linear elliptic equations associated with other symmetric functions of the Ricci tensor. A case of particular interest is the second symmetric function of the Ricci tensor in dimension four closely related to the Pfaffian.

In these lectures, starting from the background material, the author reviews the problem of prescribing Gaussian curvature on compact surfaces. She then develops the analytic tools (e.g., higher order conformal invariant operators, Sobolev inequalities, blow-up analysis) in order to solve a fully nonlinear equation in prescribing the Chern-Gauss-Bonnet integrand on compact manifolds of dimension four.

The material is suitable for graduate students and research mathematicians interested in geometry, topology, and differential equations.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Readership

Graduate students and research mathematicians interested in geometry, topology, and differential equations.

Table of Contents

  • Gaussian curvature equation
  • Moser-Trudinger inequality (on the sphere)
  • Polyakov formula on compact surfaces
  • Conformal covariant operators--Paneitz operator
  • Functional determinant on 4-manifolds
  • Extremal metrics for the log-determinant functional
  • Elementary symmetric functions
  • A priori estimates for the regularized equation \((\ast)_\delta\)
  • Smoothing via the Yamabe flow
  • Deforming \(\sigma_2\) to a constant function
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