[Paper Review] Ellipticity of Bartnik boundary data for stationary vacuum spacetimes

Zhongshan An|arXiv (Cornell University)|Jul 1, 2018

Geometric Analysis and Curvature Flows15 references3 citations

TL;DR

This paper establishes that the Bartnik boundary data for stationary vacuum spacetimes form an elliptic boundary value problem by showing that the stationary vacuum equations with Bartnik boundary conditions constitute a Fredholm system. The key result is that small perturbations of flat boundary data admit a unique (up to diffeomorphism) stationary vacuum extension locally, confirming the ellipticity of Bartnik's quasi-local mass construction.

ABSTRACT

We establish a moduli space $\mathbb E$ of stationary vacuum metrics in a spacetime, and set up a well-defined boundary map $Π$ in $\mathbb E$, assigning a metric class with its Bartnik boundary data. Furthermore, we prove the boundary map $Π$ is Fredholm by showing that the stationary vacuum equations (combined with proper gauge terms) and the Bartnik boundary conditions form an elliptic boundary value problem. As an application, we show that the Bartnik boundary data near the standard flat boundary data admits a unique (up to diffeomorphism) stationary vacuum extension locally.

Motivation & Objective

• To resolve Bartnik's open question on whether Bartnik boundary data are elliptic for stationary vacuum spacetimes.
• To define a well-posed boundary map Π from the moduli space of stationary vacuum metrics to Bartnik boundary data.
• To prove that the combined system of stationary vacuum equations and Bartnik boundary conditions forms a Fredholm elliptic boundary value problem.
• To establish local uniqueness (up to diffeomorphism) of stationary vacuum extensions for boundary data near flat data.

Proposed method

• Constructs a moduli space 𝔼 of stationary vacuum metrics on a spacetime with a Cauchy surface M.
• Defines a boundary map Π: 𝔼 → 𝔹(∂M), assigning each metric its Bartnik boundary data (g_Σ, H_Σ, tr_ΣK, ω_n_Σ).
• Shows that the stationary vacuum equations (Ric_{g^{(4)}} = 0) combined with Bartnik boundary conditions form a boundary value problem.
• Establishes ellipticity by verifying the relevant symbol conditions and using the fact that the linearized system is formally self-adjoint with compact resolvent.
• Applies analytic perturbation theory to eigenvalue branches of a family of self-adjoint operators to rule out nontrivial solutions in a neighborhood of flat data.
• Uses the Bianchi operator in Minkowski spacetime to derive the linearized constraints and verify gauge conditions.

Experimental results

Research questions

• RQ1 Is the boundary value problem formed by the stationary vacuum equations and Bartnik boundary conditions elliptic?
• RQ2 Does the boundary map Π: 𝔼 → 𝔹(∂M) define a Fredholm operator?
• RQ3 Can a unique stationary vacuum extension be constructed for Bartnik boundary data near flat data?
• RQ4 Is the Bartnik quasi-local mass construction well-posed in the sense of elliptic regularity?
• RQ5 Do the constraint equations with Bartnik boundary data admit a well-defined solution space with finite-dimensional kernel and closed range?

Key findings

• The boundary map Π is Fredholm, confirming that the system of stationary vacuum equations with Bartnik boundary conditions is an elliptic boundary value problem.
• The linearized system at the flat metric is elliptic and formally self-adjoint, with compact resolvent.
• There are no nontrivial solutions to the linearized system in a neighborhood of the flat metric, implying local uniqueness of extensions.
• The space of solutions to the boundary value problem is finite-dimensional, and the kernel is trivial for small perturbations of flat data.
• The system admits a unique (up to diffeomorphism) stationary vacuum extension for boundary data sufficiently close to flat data.
• The Bianchi operator in Minkowski spacetime confirms the consistency of the linearized constraints and gauge conditions.