Prescribed non-positive scalar curvature on asymptotically hyperbolic manifolds with application to the Lichnerowicz equation

We study the prescribed scalar curvature problem, namely finding which function can be obtained as the scalar curvature of a metric in a given conformal class. We deal with the case of asymptotically hyperbolic manifolds and restrict ourselves to non positive prescribed scalar curvature. Following [$\href{https://dx.doi.org/10.4310/CAG.2018.v26.n5.a5}{14}$, $\href{https://doi.org/10.1090/S0002-9947-1995-1321588-5}{26}$], we obtain a necessary and sufficient condition on the zero set of the prescribed scalar curvature so that the problem admits a (unique) solution.

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Received 17 October 2019

Accepted 2 September 2021

Published 10 August 2024