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arXiv · 0712.2794

Some Progress in Conformal Geometry

Sun-Yung A. Chang·Jie Qing·Paul Yang
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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 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.

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Sun-Yung A. Chang, Jie Qing, Paul Yang. 2007-12-17. Some Progress in Conformal Geometry. https://doi.org/10.3842/sigma.2007.122

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