Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 122, 17 pages      arXiv:0712.2794
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

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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