Hilbert Manifold Structure of The Set of Solutions of Constraint Equations For Coupled Einstein and Scalar Fields

TL;DR

This paper demonstrates that the set of solutions to the coupled Einstein-scalar field constraint equations forms a Hilbert manifold, using advanced mathematical tools like weighted Sobolev spaces and the Implicit Function Theorem.

Contribution

It establishes the Hilbert manifold structure of the solution space for Einstein-scalar field constraints, extending previous mathematical frameworks in general relativity.

Findings

Solution space has Hilbert manifold structure

Utilizes weighted Sobolev spaces and Implicit Function Theorem

Builds on and extends previous work by R. Bartnik

Abstract

In this paper, we prove that the set of solutions of constraint equations for coupled Einstein and scalar fields in classical general relativity possesses Hilbert manifold structure. We follow the work of R. Bartnik [2] and use weighted Sobolev spaces and Implicit Function Theorem to prove our results.