Event

Talk
The structure of conformally covariant powers of the Laplacian
Prof. Dr. Andreas Juhl
12 Jan 2010, 16:00 – 17:00

The sum of the Laplace-Beltrami operator of a Riemannian manifold and a certain constant multiple of the scalar curvature is covariant under conformal changes of the metric. This operator is known as the Yamabe operator. It plays a central role in conformal geometric analysis. In 1983, Paneitz discovered that suitable modications of the square of the Laplacian by lower order terms (involving the Ricci tensor and the scalar curvature) yield a conformally covariant operator (of order four). In large parts of conformal geometric analysis, the Paneitz-operator now plays a role similar to that of the Yamabe operator. More generally, Graham, Jenne, Mason and Sparling (1992) constructed higher order conformally covariant powers of the Laplacian by using the Feerman-Graham ambient metric. In turn, these operators led to the notion of Branson's Q-curvature. In recent years, the advent of the AdS/CFT- correspondence additionally stimulated the deeper study of these constructions. In particular, the systematic elaboration of the idea of bulk-boundary correspondences led to substantially new insights into the structure of these operators. The lecture will give a moderate introduction into the web of ideas in this area, and will describe some very recent results concerning the intriguing recursive structure of conformally covariant powers of the Laplacian.

Related to:

• Andreas Juhl
• Winter Term 2009/10