Higher conformal fundamental forms and the asymptotically Poincare-Einstein Condition

Rod Gover (U of Auckland)

Sep 15. 2021, 09:30 — 10:30

Given a conformally compact manifold an important question is whether
  the metric is conformally related to a conformally compact Einstein
  metric (i.e. a Poincar\'e--Einstein metric). In general such a
  conformal rescaling is obstructed by conformal invariants of the
  boundary hypersurface embedding, the first of which is the
  trace-free second fundamental form and then, at the next order, it
  is the Fialkow tensor, or equivalently its trace-free part. We show
  that the trace-free second fundamental form and the trace-free part
  of the Fialkow tensor are the lowest order examples in a a sequence
  of conformally invariant higher fundamental forms determined by the
  data of a conformal hypersurface embedding, and we construct these
  trace-free symmetric rank two tensors here. The vanishing of these
  fundamental forms is a necessary and sufficient condition for a
  conformally compact metric to be conformally related to an
  asyptotically Poincare-Einstein metric.

This is joint work with Sam Blitz and Andrew Waldron

Further Information
ESI Boltzmann Lecture Hall
Associated Event:
Geometry for Higher Spin Gravity: Conformal Structures, PDEs, and Q-manifolds (Thematic Programme)
Xavier Bekaert (U of Tours)
Andreas Cap (U of Vienna)
Stefan Fredenhagen (U of Vienna)
Maxim Grigoriev (U of Mons)
Alexei Kotov (U Hradec Králové)