Extensions of spacetime Bartnik data and estimates for the Bartnik mass outside of time-symmetry

TL;DR

The paper advances the understanding of Bartnik mass beyond time-symmetry by constructing non-time-symmetric extensions of Bartnik data via a collar-and-gluing framework that preserves the dominant energy condition and avoids generalized horizons. By gluing collar data to spherically symmetric Schwarzschild graphs, it produces admissible initial data sets that realize the given boundary data and embed into a Schwarzschild exterior with controlled mass, enabling explicit upper bounds on the spacetime Bartnik mass in terms of Hawking mass and geometric constants. A key innovation is the introduction of explicit collar constructions with quantitative DEC and MOTS-free criteria, along with gluing and bending lemmas adapted to non-time-symmetric settings. The work also provides a reduction mechanism to time-symmetric estimates, showing how non-time-symmetric Bartnik mass bounds can be inferred from established time-symmetric results, thereby broadening the applicability and computability of Bartnik-type quasi-local mass in general relativity.

Abstract

Bartnik's quasi-local mass is a functional on Bartnik data $(\mathbb S^2,γ,H,P,ω^\perp)$, consisting of a metric $γ$, scalar functions $H$ and $P$, and a 1-form $ω^\perp$ on the $2$-sphere $\mathbb S^2$. We construct initial data $(M,g,K)$ for the Einstein equations with boundary $Σ\cong\mathbb S^2$, and boundary conditions for $g$ and $K$ determined by Bartnik data with $H,P$ constant and $ω^\perp\equiv0$. Furthermore this initial data agrees with spherically symmetric initial data for a Schwarzschild spacetime outside of a compact set with controlled mass. As an application, we obtain estimates for the Bartnik mass for such Bartnik data, outside of the time-symmetric setting. We also construct initial data on the cylinder $\mathbb S^2\times[0,1]$ connecting this same class of Bartnik data to time-symmetric data so that estimates for the Bartnik mass outside of time-symmetry can be obtained from prior estimates for time-symmetric data.

Theorems & Definitions (45)

• Remark 1.1
• Definition 2.1
• Remark 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Corollary 3.4
• Lemma 4.1
• proof