Conformal invariants and partial differential equations
Sun‐Yung A. Chang · Bulletin of the American Mathematical Society · 2005
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 2 with a Riemannian metric g.In this case, under the conformal change of metric g w = e 2w g, ∆ g w = e -2w ∆ g is a familiar example of a conformally covariant operator.