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This talk reports on my joint work arXiv:2108.01373 with Rod Gover that is motivated by the concept of mass for asymptotically hyperbolic metrics. We work in the setting of conformally compact metrics on arbitrary manifolds with boundary (at infinity) and a class of metrics that are asymptotically hyperbolic in a weak sense and asymptotic
to each other to appropriate (higher) order. This makes sure that all the metrics in the class induce the same conformal infinity on the boundary. We then associate to two metrics in the class a two-parameter family of "relative masses", that are top degree forms on the boundary with values in the standard tractor bundle of the conformal infinity.

The construction is manifestly independent of coordinates and it is easily seen to be equivariant with respect to a natural family of diffeomorphisms. Much more subtle considerations shows that a one-parameter subfamily satisfies an additional invariance conditions with respect to diffeomorphism asymptotic to the identity. Assuming that the given class contains hyperbolic metrics, this allows us to associate an invariant to a single metric in the class. In the case of hyperbolic space and the class of metrics determined by the Poincar\'e metric, the forms we construct can be integrated to parallel sections of the tractor bundle and this recovers that mass of asymptotically hyperbolic metrics introduced by Wang and Chrusciel-Herzlich.

https://ucdavis.zoom.us/j/94506910050