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