Past lectures

Abstract

Elliptic equations have always been an important tool in the study of problems in geometry. In the recent decades, non-linear second order elliptic equations with critical exponents have played a special role in the solutions of several important problems in conformal geometry; e.g. the problem of prescribing Gaussian curvature and the Yamabe problem. In this course, I will describe some recent effort to extend the role played by second order equations to some higher order ones. I will first summarize some basic results and tools used in the prescribing Gaussian curvature problem on compact surfaces (e.g. the Moser-Trudinger inequality); then describe properties of a class of conformal covariant operators - in particular a 4-th order operator with its leading symbol the bi-Laplace operator, discovered by Paneitz in 1983; then proceed to establish the relations of these operators to some natural functionals (e.g. the zeta-functional determinant for the Laplace operator) and elliptic equations (e.g. the Monge-Ampere equation).

I will discuss questions of existence, uniqueness and regularity of the associated nonlinear equations. As applications, I will go over some recent research work of Gursky-Viaclovsky; and work of Chang-Gursky-Yang on a problem in prescribing the Ricci curvature on a class of 4-dimensional manifolds.