What Uniqueness for the Holst-Nagy-Tsogtgerel--Maxwell Solutions to the Einstein Conformal Constraint Equations?

TL;DR

The paper tackles the global uniqueness issue for the Holst–Nagy–Tsogtgerel–Maxwell conformal method applied to Einstein constraint equations with non-constant mean curvature. It builds a Schauder-fixed-point framework around seed data with positive Yamabe invariant and a small TT-tensor to obtain existence of solutions to the coupled Lichnerowicz and vector equations, yielding regularity $\varphi\in W^{2,p}(M)$ and $W\in W^{2,p}(M)$. The main contribution is a volume-bound-based uniqueness result: for a fixed $V_{\max}$ and sufficiently small $\|\sigma\|_{L^{2p}}$ (along with a lower bound condition on $|\sigma|$), there exists a unique solution with $\mathrm{Vol}_{\widehat{g}}(M)\le V_{\max}$, proven via a priori estimates and a contraction-type argument. This establishes a robust, numerically friendly regime in which the non-CMC conformal method yields a unique initial data set, and it clarifies how volume control can substitute for more restrictive mean-curvature conditions in ensuring uniqueness.

Abstract

This paper addresses the issue of uniqueness of solutions in the conformal method for solving the constraint equations in general relativity with arbitrary mean curvature as developed initially by Holst, Nagy, Tsogtegerel and Maxwell. We show that the solution they construct is unique amongst those having volume below a certain threshold.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Lemma 2.2
• Proposition 2.3: Lemma 2.11 and Proposition 2.12 of Pailleron
• Proposition 2.4
• Claim 1
• proof
• Claim 2
• proof