Chang, Sun-Yung A.; Qing, Jie; Yang, Paul

Some progress in conformal geometry

SIGMA, Symmetry Integrability Geom. Methods Appl. 3, Paper 122, 17 p., electronic only (2007)


Summary: This is a survey paper of our current research on the theory of partial differential equations in conformal geometry. Our intention is to describe some of our current works in a rather brief and expository fashion. We are not giving a comprehensive survey on the subject and references cited here are not intended to be complete. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bounds on compact manifolds of dimension 4. As applications, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and diameter bound of the $\sigma _{2}$-metrics in a class of conformal 4-manifolds. For conformally compact Einstein metrics we introduce an eigenfunction compactification. As a consequence we obtain some topological constraints in terms of renormalized volumes.

Mathematics Subject Classification

53A30, 53C20, 35J60


bach flat metrics, bubble tree structure, degeneration of metrics, conformally compact, Einstein, renormalized volume