The Jang equation and the positive mass theorem in the asymptotically hyperbolic setting


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May 8, 2020 11:00 am - 12:00 pm
via Zoom Video Conferencing

Anna Sakovich - Uppsala University

We will be concerned with asymptotically hyperbolic 'hyperboloidal' initial data for the Einstein equations. Such initial data is modeled on the upper unit hyperboloid in Minkowski spacetime and consists of a Riemannian manifold (M, g) whose geometry at infinity approaches that of hyperbolic space, and a symmetric 2-tensor K representing the second fundamental form of the embedding into spacetime, such that K -> g at infinity. There is a notion of mass in this setting and a positive mass conjecture can be proven by spinor techniques. Other important results concern the case K = g, where the conjecture states that an asymptotically hyperbolic manifold whose scalar curvature is greater than or equal to that of hyperbolic space must have positive mass unless it is a hyperbolic space. In this talk, we will discuss how the method of Jang equation reduction, originally devised by Schoen and Yau to prove the positive mass conjecture for asymptotically Euclidean initial data sets, can be adapted to the asymptotically hyperbolic setting yielding a non-spinor proof of the respective positive mass conjecture. We will primarily focus on the case dim M = 3.

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