We will be concerned with asymptotically hyperbolic 'hyperboloidal' initial data for the Einstein equations. Such initial data is modelled on the upper unit hyperboloid in Minkowski spacetime, meaning that both the induced metric and the second fundamental form are asymptotic to the hyperbolic metric. One of the motivations to study this type of initial data sets comes from the necessity to prove the positivity of Bondi mass, measuring the total mass of an isolated physical system after the loss due to the gravitational radiation. A specific proposal how to address this problem was put forward by Schoen and Yau in 1982, building upon their celebrated proof of positive mass theorem for asymptotically Euclidean initial data sets from 1981. The key idea is to use a solution of a certain prescribed mean curvature equation (the so-called Jang equation) to deform a spacelike hypersurface suitably asymptotic to the null cone and thereby capturing the Bondi mass of the spacetime into an asymptotically Euclidean spacelike hypersurface satisfying the conditions of the respective positive mass theorem. In this talk we will describe how the above method of Schoen and Yau applies to prove the positivity of mass of asymptotically hyperbolic initial data sets in the non-radiating regime.
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