Geometry Seminar: R. Avalos (Tübingen)
Geometry Seminar (Geometric Analysis)
Speaker: Rodrigo Avalos (Tübingen)
Title: A Q-curvature positive energy theorem and rigidity of Q-singular manifolds.
Abstract: In this talk we will present recent results related to a notion of energy which is associated with fourth-order gravitational theories, where it plays an analogous role to that of the classical ADM energy in the context of general relativity. We shall show that this quantity obeys a positive energy theorem with natural rigidity in the critical case of zero energy. Furthermore, we will comment on how the resulting notion of energy underlies rigidity phenomena in geometry associated with Q-curvature, in particular, in the case of asymptotically Euclidean (AE) Q-singular spaces. These spaces contain Riemannian manifolds which satisfy a fourth-order analogue of the Einstein condition, which we refer to as J-Einstein manifolds. The analysis of J-Einstein manifolds is analytically more challenging, involving fourth order geometric partial differential equations, but several interesting properties of Einstein manifolds are still retained by this wider family. In this talk we shall show that the J-tensor retains optimal controls on the decay of the metric tensor at infinity, and also that J-flat Yamabe positive AE manifolds exhibit the same rigidity properties as Ricci-flat AE manifolds do.
Speaker: Rodrigo Avalos (Tübingen)
Title: A Q-curvature positive energy theorem and rigidity of Q-singular manifolds.
Abstract: In this talk we will present recent results related to a notion of energy which is associated with fourth-order gravitational theories, where it plays an analogous role to that of the classical ADM energy in the context of general relativity. We shall show that this quantity obeys a positive energy theorem with natural rigidity in the critical case of zero energy. Furthermore, we will comment on how the resulting notion of energy underlies rigidity phenomena in geometry associated with Q-curvature, in particular, in the case of asymptotically Euclidean (AE) Q-singular spaces. These spaces contain Riemannian manifolds which satisfy a fourth-order analogue of the Einstein condition, which we refer to as J-Einstein manifolds. The analysis of J-Einstein manifolds is analytically more challenging, involving fourth order geometric partial differential equations, but several interesting properties of Einstein manifolds are still retained by this wider family. In this talk we shall show that the J-tensor retains optimal controls on the decay of the metric tensor at infinity, and also that J-flat Yamabe positive AE manifolds exhibit the same rigidity properties as Ricci-flat AE manifolds do.