### 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.

Original language | English (US) |
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Article number | 122 |

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

Volume | 3 |

DOIs | |

State | Published - Jan 1 2007 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Mathematical Physics
- Geometry and Topology

### Keywords

- Bach flat metrics
- Bubble tree structure
- Conformally compact
- Degeneration of metrics
- Einstein
- Renormalized volume