The Gauss-Bonnet-Chern formula via scalar conformal invariants

The Gauss-Bonnet-Chern formula is typically expressed as the integral of the Pfaffian (when stressing intrinsic geometry) or as the integral of a generalization of the Gauss-Kronecker curvature (when stressing extrinsic geometry). In this talk I will prove two version of the Gauss-Bonnet-Chern formula that are expressed in terms of scalar conformal invariants and a special quantity called Q-curvature. One benefit of this approach is that it yields computable formulas on conformally compact Einstein manifolds and minimal submanifolds therein.