Conformal Invariants and Partial Differential Equations

Our goal is to study quantities in Riemannian geometry which remain invariant under the “conformal change of metrics”–that is, under changes of metrics which stretch the length of vectors but preserve the angles between any pair of vectors. We call such a quantity “conformally invariant”. In conjunction with the study of conformal invariants, we are also interested in studying “conformally covariant operators”, that is, linear differential operators defined on a manifold which prescribes the change of a geometric quantity under conformal change of metrics. The study of conformal invariants has a long history going back at least to Poincaré and Cartan. In recent years, there has been intensive study of the existence and construction of conformal covariant operators and also applications to problems in geometry and related problems in string theory and mathematical physics. In these talks, I will report some progress in this research area, with emphasis on the PDE approach to these problems. A model example is that of the Laplace operator ∆g on a compact surface M with a Riemannian metric g. In this case, under the conformal change of metric gw = eg, ∆gw = e ∆g is a familiar example of a conformally covariant operator.