Ashwin Rod Gover, University of Auckland
If a manifold with boundary has on its interior a conformally compact Einstein metric, then what restrictions does that place on the geometry of the boundary embedding? It is well known that it is necessarily totally umbilic, meaning that the trace-free part of the second fundamental form must vanish. We show that the trace-free second fundamental form is the lowest order example in a series of trace-free 2-tensor conformal invariants that provide the order-by-order obstructions to the Poincare-Einstein condition. In the case of even dimensional manifolds, probing further yields a conformal invariant that captures the image of the non-linear Poincare-Einstein Dirichlet-Neumann map — meaning that it is the higher Neumann-type data for the Poincare-Einstein problem with Dirichlet data a boundary conformal structure.