Zurich Lectures in Advanced Mathematics 2004; 100 pp; softcover Volume: 2 ISBN10: 303719006X ISBN13: 9783037190067 List Price: US$28 Member Price: US$22.40 Order Code: EMSZLEC/2
 Nonlinear 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 semilinear elliptic equations in the study of Gaussian curvature and scalar curvature has been extended to a family of fully nonlinear 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, blowup analysis) in order to solve a fully nonlinear equation in prescribing the ChernGaussBonnet 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
 MoserTrudinger inequality (on the sphere)
 Polyakov formula on compact surfaces
 Conformal covariant operatorsPaneitz operator
 Functional determinant on 4manifolds
 Extremal metrics for the logdeterminant 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
