[Paper Review] A class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature

David Maxwell|ArXiv.org|Apr 5, 2008

Geometric Analysis and Curvature Flows19 references3 citations

TL;DR

This paper establishes a sufficient condition for constructing solutions to the vacuum Einstein constraint equations on compact manifolds using the conformal method, requiring only a global supersolution and no global subsolution. It proves that the HNT global supersolution suffices for existence of far-from-CMC vacuum solutions, extending prior non-vacuum results and simplifying near-CMC existence hypotheses.

ABSTRACT

We give a sufficient condition, with no restrictions on the mean curvature, under which the conformal method can be used to generate solutions of the vacuum Einstein constraint equations on compact manifolds. The condition requires a so-called global supersolution but does not require a global subsolution. As a consequence, we construct a class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature, extending a recent result of Holst, Nagy, and Tsogtgerel [HNT07] which constructed similar solutions in the presence of matter. We give a second proof of this result showing that vacuum solutions can be obtained as a limit of [HNT07] non-vacuum solutions. Our principal existence theorem is of independent interest in the near-CMC case, where it simplifies previously known hypotheses required for existence.

Motivation & Objective

• To extend the conformal method for solving the vacuum Einstein constraint equations beyond the constant or near-constant mean curvature (CMC/near-CMC) regime.
• To resolve the open question of whether the conformal method can generate far-from-CMC vacuum solutions without requiring a global subsolution.
• To simplify existing existence hypotheses in the near-CMC case by eliminating the need for a subsolution.
• To demonstrate that the HNT global supersolution is sufficient for vacuum solution construction, even in the absence of matter fields.

Proposed method

• Uses the conformal method to parametrize solutions via a conformal factor $\phi$ and a vector field $W$, with the metric $\bar{g} = \phi^4 g$ and second fundamental form $\bar{K} = \phi^{-2}(\sigma + \mathbf{L}W) + \frac{\tau}{3}\bar{g}$.
• Applies an a-priori estimate (Proposition 10) to replace the need for a global subsolution, relying solely on the existence of a global supersolution.
• Constructs a sequence of non-vacuum solutions with matter fields $\rho_n \to 0$, showing convergence to a vacuum solution via uniform bounds and weak convergence in $W^{2,p}$.
• Employs a technique from Maxwell (2005a) to build subsolutions in the limit process, ensuring uniform lower and upper bounds on the conformal factor.
• Uses elliptic regularity and compact embedding to extract a convergent subsequence of conformal factors $\phi_n$ in $W^{2,p}$, yielding a strong solution in $W^{2,p}_+$.
• Establishes that the limiting solution satisfies the vacuum constraint equations: $-8\Delta\phi + R\phi = -\frac{2}{3}\tau^2\phi^5 + |\sigma + \mathbf{L}W|^2\phi^{-7}$.

Experimental results

Research questions

• RQ1 Can the conformal method generate vacuum solutions with freely specified mean curvature without requiring a global subsolution?
• RQ2 Is the HNT global supersolution sufficient for existence of far-from-CMC vacuum solutions on Yamabe-positive compact manifolds?
• RQ3 Can vacuum solutions be obtained as a limit of non-vacuum solutions with vanishing matter fields?
• RQ4 Can the existence hypotheses in the near-CMC regime be simplified by removing the need for a subsolution?
• RQ5 What are the minimal conditions on the data (mean curvature, transverse traceless tensor) for existence of vacuum solutions using the conformal method?

Key findings

• A global supersolution alone is sufficient for existence of solutions to the vacuum Einstein constraint equations, eliminating the need for a global subsolution.
• The HNT global supersolution applies in the vacuum case, enabling construction of far-from-CMC solutions for any mean curvature $\tau$ on Yamabe-positive compact 3-manifolds.
• The near-CMC existence hypothesis is simplified: the requirement for a subsolution is no longer necessary, reducing the technical conditions on the data.
• A sequence of non-vacuum solutions with $\rho_n \to 0$ converges uniformly and in $W^{1,p}$ to a vacuum solution, providing a second proof of the main result.
• The conformal factor $\phi_n$ is uniformly bounded from below and above, ensuring convergence to a positive $W^{2,p}$ solution $\phi \in W^{2,p}_+$.
• The limiting solution satisfies the vacuum constraint equations in strong form, with $\phi$ solving $-8\Delta\phi + R\phi = -\frac{2}{3}\tau^2\phi^5 + |\sigma + \mathbf{L}W|^2\phi^{-7}$.