Barycenter technique and the Riemann mapping problem of Cherrier–Escobar

Abstract

We solve in the affirmative the remaining cases of the Riemann mapping problem of Cherrier–Escobar [35][38] first raised by Cherrier [35] in 1984. Indeed, performing a suitable scheme of the barycenter technique of Bahri–Coron [14] via the Almaraz–Chen’s bubbles [3][34], we completely solve all the cases left open after the work of Chen [34]. Hence, combining our work with the ones of Almaraz [2], Chen [34], Cherrier [35], Escobar [38][40] and Marques [55][56], 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.