Some progress in conformal geometry

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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 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.

Original languageEnglish (US)
Article number122
JournalSymmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume3
DOIs
StatePublished - 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

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