We solve in the affirmative the remaining cases of the Riemann mapping problem of Cherrier–Escobar  first raised by Cherrier  in 1984. Indeed, performing a suitable scheme of the barycenter technique of Bahri–Coron  via the Almaraz–Chen’s bubbles , we completely solve all the cases left open after the work of Chen . Hence, combining our work with the ones of Almaraz , Chen , Cherrier , Escobar  and Marques , we have that every compact Riemannian manifold with boundary, of dimension greater or equal than three, and with finite Sobolev quotient, carries a conformal scalar flat metric with constant mean curvature on the boundary.
"Barycenter technique and the Riemann mapping problem of Cherrier–Escobar." J. Differential Geom. 107 (3) 519 - 560, November 2017. https://doi.org/10.4310/jdg/1508551224