The Jang Equation Reduction of the Spacetime Positive Energy Theorem in Dimensions Less Than Eight

Abstract

Abstract. We extend the Jang equation proof of the positive energy theorem due to R. Schoen and S.-T. Yau [29] from dimension n = 3 to dimensions 3 ≤ n < 8. This requires us to address several technical difficulties that are not present when n = 3. The regularity and decay assumptions for the initial data sets to which our argument applies are weaker than those in [29].

Geometric Inequalities for Quasi-Local Masses

“…Finally by appealing to the Riemannian version of the positive mass theorem [54] we find thatm ≥ 0, from which the desired result follows. The rigidity statement of the theorem is established in the typical manner [23,53].…”

Jenkins–Serrin-type results for the Jang equation

“…It follows from the construction in the proof of Theorem 2.2 in [6, Chapter 4] that the MOTSs Σ ε ⊂ Ω spanning Γ ε are (strictly) ordered. To see this, recall that the open subset of Ω whose relative boundary is Σ ε is the geometric limit as t ց 0 of downward translations of the regions lying above the graphs u ε t ∈ C ∞ loc (Ω) constructed in [6,Lemma 4.2]. Given t > 0 and 0 < ε < ε ′ small, we have that S u ε ′ t ⊂ S u ε t (in the notation of [6]) and hence u ε ′ t ≤ u ε t .…”

“…To see this, recall that the open subset of Ω whose relative boundary is Σ ε is the geometric limit as t ց 0 of downward translations of the regions lying above the graphs u ε t ∈ C ∞ loc (Ω) constructed in [6,Lemma 4.2]. Given t > 0 and 0 < ε < ε ′ small, we have that S u ε ′ t ⊂ S u ε t (in the notation of [6]) and hence u ε ′ t ≤ u ε t . It follows that the regions above the graphs are ordered so that Σ ε ′ lies to one side (towards ∂ + Ω) of Σ ε .…”

“…This result is subtle in view of the non-variational nature of these surfaces. The proof is based on the fact that using the existence theory in [6], we can construct ordered families of solutions of the Plateau problem. This result is applied in the recent proof of the spacetime positive mass theorem by L.-H. Huang, D. Lee, R. Schoen, and the first named author in [9].…”

“…In Section 6 we employ techniques from the geometric theory of the Jang equation, in particular the capillarity regularization and the geometric blow up analysis from [34] and ideas from the solution of the non-variational Plateau problem for marginally outer trapped surfaces in [6], to the classical Jenkins-Serrin-Spruck problem [19,20,40] of finding necessary and sufficient conditions for a domain in a Riemannian surface to support a Scherk-type constant mean curvature graph. In the case of positive mean curvature, we are able to dispense with the a priori assumption that the domain admit a sub solution which is required in the foundational paper by J. Spruck [40] (in R 2 ) and its recent extension to domains in S 2 and H 2 by L. Hauswirth, H. Rosenberg, and J. Spruck [18], and to domains in Hadamard manifolds [11] by A. Folha and H. Rosenberg.…”