[Paper Review] Hilbert manifold structure for asymptotically hyperbolic relativistic initial data

Erwann Delay, Jérémie Fougeirol|arXiv (Cornell University)|Jul 19, 2016

Geometric Analysis and Curvature Flows14 references3 citations

TL;DR

This paper establishes a Hilbert manifold structure for the space of asymptotically hyperbolic initial data solving the vacuum constraint equations in general relativity, using novel weighted Poincaré and Korn-type inequalities on manifolds with inner boundary and weakly regular metrics. The key contribution is proving the triviality of the adjoint kernel of the linearized constraint operator, enabling the application of the inverse function theorem to establish the submanifold structure.

ABSTRACT

We provide a Hilbert manifold structure {à} la Bartnik for the space of asymptotically hyperbolic initial data for the vacuum constraint equations. The adaptation led us to prove new weighted Poincar{é} and Korn type inequalities for AH manifolds with inner boundary and weakly regular metric.

Motivation & Objective

• To extend Bartnik's Hilbert manifold framework for asymptotically flat initial data to the asymptotically hyperbolic setting.
• To address the lack of suitable elliptic estimates and kernel triviality results in the asymptotically hyperbolic case with weak regularity.
• To prove the existence of a Hilbert manifold structure on the space of solutions to the vacuum constraint equations under $L^2$-weighted regularity conditions.
• To establish the necessary functional analytic tools—specifically weighted Poincaré and Korn-type inequalities—for the constraint operator on asymptotically hyperbolic manifolds with boundary.
• To ensure compatibility with the definition of mass in asymptotically hyperbolic spacetimes under the same regularity assumptions.

Proposed method

• Introduce a Hessian-type operator $\mathring{T}$ and a second-order differential operator $\mathring{U}$ derived from the Killing operator $\mathring{S}$ to handle the asymptotically hyperbolic geometry.
• Prove new weighted Poincaré and Korn-type inequalities for $n=3$ on asymptotically hyperbolic manifolds with inner boundary and weakly regular metrics.
• Analyze the linearized constraint operator $\mathbf{\Phi}$ and its adjoint using weighted Sobolev and Hölder spaces with decay rate $\delta \in (-(n+1)/2, 0]$.
• Establish semi-Fredholm properties for the operators $F$ and $\widetilde{F^*}$ via elliptic estimates and Ehrling-type inequalities in weighted $L^2$ spaces.
• Use the triviality of the kernel of the adjoint operator $F^*$ to apply the inverse function theorem and deduce that the level sets of $\mathbf{\Phi}$ are smooth submanifolds.
• Apply the inverse function theorem to the constraint map $\mathbf{\Phi}$, proving that the space of solutions forms a smooth Hilbert submanifold of the ambient function space.

Experimental results

Research questions

• RQ1 Can a Hilbert manifold structure be established for the space of asymptotically hyperbolic initial data under weak $L^2$-weighted regularity assumptions?
• RQ2 What weighted elliptic estimates are necessary to control the linearized constraint operator on asymptotically hyperbolic manifolds with boundary?
• RQ3 Is the kernel of the adjoint of the linearized constraint operator trivial under these regularity conditions?
• RQ4 How can the Hessian-type operator $\mathring{T}$ and the second-order operator $\mathring{U}$ be constructed to facilitate the analysis of the constraint equations?
• RQ5 Does the resulting manifold structure remain compatible with the definition of mass in asymptotically hyperbolic spacetimes?

Key findings

• The space of asymptotically hyperbolic initial data satisfying the vacuum constraint equations admits a Hilbert manifold structure under $L^2$-weighted regularity assumptions.
• New weighted Poincaré and Korn-type inequalities were proven for $n=3$ on asymptotically hyperbolic manifolds with inner boundary and weakly regular metrics.
• The adjoint of the linearized constraint operator has trivial kernel, which is essential for the inverse function theorem to apply.
• The linearized constraint operator $D\mathbf{\Phi}$ has closed range and finite codimension, ensuring the regular level sets are smooth submanifolds.
• The operator $F$, representing the linearized constraint, is semi-Fredholm, with finite-dimensional kernel and closed range, enabling the manifold structure proof.
• The construction of $\mathring{T}$ and $\mathring{U}$ enabled the derivation of the required elliptic estimates and kernel triviality in the asymptotically hyperbolic setting.