SIGMA 3 (2007), 122, 17 pages      arXiv:0712.2794      https://doi.org/10.3842/SIGMA.2007.122
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

Some Progress in Conformal Geometry

Sun-Yung A. Chang ^a, Jie Qing ^b and Paul Yang ^a
a) Department of Mathematics, Princeton University, Princeton, NJ 08540, USA
^b) Department of Mathematics, University of California, Santa Cruz, Santa Cruz, CA 95064, USA

Received August 30, 2007, in final form December 07, 2007; Published online December 17, 2007

Abstract
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 σ₂-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.

Key words: Bach flat metrics; bubble tree structure; degeneration of metrics; conformally compact; Einstein; renormalized volume.

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