Equality in the Spacetime Positive Mass Theorem

Abstract

We affirm the rigidity conjecture of the spacetime positive mass theorem in dimensions less than eight. Namely, if an asymptotically flat initial data set satisfies the dominant energy condition and has E = |P |, then E = |P | = 0, where (E, P ) is the ADM energy-momentum vector. The dimensional restriction can be removed if we assume the positive mass inequality holds. Previously the result was only known for spin manifolds [5,6].

Initial Data Rigidity Results

“…We refer to [11] for the definition of the energy-momentum vector. The case of equality E = |P| has recently been characterized by L.-H. Huang and D. Lee [17].…”

Mass Rigidity for Hyperbolic Manifolds

“…≥ p 0 p 0 (γ) − p 2 1 + · · · + p 2 n p 1 (γ) 2 + · · · + p n (γ) 2 = p 0 p 0 (γ) − (p 1 (γ)) 2 + · · · + (p n (γ)) 2 ≥ 0 with equalities realized at γ = g. By Theorem 4, L g : C 2,α −q → C 0,α −q is surjective, so we can apply the method of Lagrange Multipliers (see, for example, [14,Theorem C.1]) to obtain λ ∈ (C 0,α −q ) * that satisfies…”

“…We remark that the approach is motivated by a constrained minimization scheme proposed by R. Bartnik [3] for his quasi-local mass program. The connection between the constrained minimization and mass rigidity was recently employed by D. Lee and the first named author in their proof to the rigidity conjecture of the spacetime positive mass theorem [14].…”

The Jang Equation and the Positive Mass Theorem in the Asymptotically Hyperbolic Setting

“…Furthermore, in the light of Schoen and Yau's recent work [SY17] it is natural to anticipate the extension of these results to dimensions n > 7. For other important developments concerning spacetime positive mass theorem in higher dimensions see [EHLS16,HL20,Loh16].…”

“…At the same time, the optimal rigidity theorem for asymptotically hyperbolic manifolds has recently been proven in[HJM20]. It is feasible that the methods of[HJM20] and[HL20] can be used to prove more general rigidity results for asymptotically hyperbolic initial data than Theorem 9.1.…”