Prescribing Morse Scalar Curvatures: Blow-Up Analysis

Abstract

Abstract We study finite-energy blow-ups for prescribed Morse scalar curvatures in both the subcritical and the critical regime. After general considerations on Palais–Smale sequences, we determine precise blow-up rates for subcritical solutions: in particular the possibility of tower bubbles is excluded in all dimensions. In subsequent papers, we aim to establish the sharpness of this result, proving a converse existence statement, together with a one-to-one correspondence of blowing-up subcritical solutions and critical points at infinity. This analysis will be then applied to deduce new existence results for the geometric problem.