
SIGMA 3 (2007), 122, 17 pages arXiv:0712.2794
http://dx.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
SunYung 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 σ_{2}metrics in a class of conformal
4manifolds. 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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