Bartnik–Hilbert manifold structure on fibers of the scalar curvature and the constraint operator

We adapt the Bartnik method to provide a Hilbert manifold structure for the space of solutions, without KID’s, to the vacuum constraint equations on compact manifold of any dimension $\geq 3$. In the course, we prove that some fibers of the scalar curvature or the constraint operator are Hilbert submanifolds. We also study some operators and inequalities related to the KID’s operator. Finally we comment the adaptation to some non-compact manifolds.

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Received 13 March 2020

Accepted 22 April 2021

Published 16 July 2024