Nonlinear Elliptic Equations in Conformal Geometry
A study of curvature invariants in conformal geometry and related nonlinear partial differential equations has a long tradition going back to a study of behaviour of Gauss curvature under a conformal change of metric in two dimensions. Recently, a considerable effort has concentrated around a higher dimensional generalization of Gauss curvature called Q-curvature and its relations to higher order conformally invariant differential operators.
This book contains a discussion of many related topics. It starts with a description of the case of surfaces, in particular the equation prescribing Gauss curvature on compact surfaces. After a discussion of conformal invariants and conformally invariant partial differential equations in higher dimensions, the author concentrates on the dimension four case. The key role is played here by a special conformally invariant fourth order operator called the Paneitz operator. Its relation to the second elementary symmetric function of the Ricci tensor in dimension four is carefully discussed in the book, which is based on lectures given by the author at ETH in Zürich. The formula for Q-curvature and many results around it are due to Tom Branson. It is a sad fact that he unexpectedly died a few months ago. But the impact of his ideas and work is clearly illustrated in the book and they will stay with us.