Positive mass conjecture for asymptotically flat initial data

Prove that the ADM mass of an asymptotically flat initial data set satisfying the dominant energy condition is nonnegative, with equality only for a Cauchy hypersurface of Minkowski space, without relying on restrictive decay assumptions at infinity.

Background

In discussing asymptotically flat (AF) manifolds, the authors recall the classical positive mass conjecture for initial data that satisfy the dominant energy condition.

They note that the conjecture has been proven under suitable decay assumptions on the metric and second fundamental form (g,k), and they analyze sharpness of the decay rates by constructing examples showing that critical decay of k can lead to negative mass, thus emphasizing the role of decay in the conjecture.

References

A long-standing conjecture in general relativity states that the ADM mass of an AF initial data set satisfying the dominant condition is positive unless it is a Cauchy hypersurface of Minkowski space.

— Spherically symmetric solutions to the Einstein-scalar field conformal constraint equations  (2602.20099 - Castillon et al., 23 Feb 2026) in Section 6.3 (Sharpness of the decay rate in the Positive Mass Theorem for AF manifolds)