Abstract
We solve the Jang equation with respect to asymptotically hyperbolic “hyperboloidal” initial data. The results are applied to give a non-spinor proof of the positive mass theorem in the asymptotically hyperbolic setting. This work focuses on the case when the spatial dimension is equal to three.
Highlights
The classical positive mass theorem has its roots in general relativity and asserts that for a nontrivial isolated physical system, the energy of the gravitational field is nonnegative
A manifold (M, g) is asymptotically Euclidean if outside some compact set it consists of a finite number of components Mk such that each Mk is diffeomorphic to a complement of a compact set in Euclidean space
Schoen and Yau observed that, as long as the dominant energy condition is satisfied, can be equipped with an asymptotically Euclidean metric such that its scalar curvature vanishes and its ADM mass does not exceed the ADM mass of (M, g, K ). It follows from the Riemannian positive mass theorem that the ADM mass of (M, g, K ) is nonnegative, and in the case when the mass is zero the function f provides the graphical embedding into the Minkowski spacetime
Summary
The classical positive mass theorem has its roots in general relativity and asserts that for a nontrivial isolated physical system, the energy of the gravitational field is nonnegative. Schoen and Yau observed that, as long as the dominant energy condition is satisfied, can be equipped with an asymptotically Euclidean metric such that its scalar curvature vanishes and its ADM mass does not exceed the ADM mass of (M, g, K ) All in all, it follows from the Riemannian positive mass theorem that the ADM mass of (M, g, K ) is nonnegative, and in the case when the mass is zero the function f provides the graphical embedding into the Minkowski spacetime. In this work we apply Schoen and Yau’s reduction argument using the Jang equation to deform an asymptotically hyperbolic initial data set satisfying the dominant energy condition to an asymptotically Euclidean manifold with “almost nonnegative” scalar curvature which in particular yields a proof of the positive mass conjecture in the “hyperboloidal” setting. We prove Theorem 1.1 in Sects. 8 and 9
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