# Non-linear Elliptic Equations in Conformal Geometry

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*Sun-Yung Alice Chang*

A publication of the European Mathematical Society

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.