2014/09/28 by Cheikh Birahim Ndiaye, Ndiaye, Cheikh Birahim, Mohameden Ould Ahmedou +1
Mathematics · #35C60 #53C21 #58J60 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1409.7919
openalex publication_date 2014/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the prescribed Q-curvature problem on closed four-dimensional Riemannian manifolds when the total integral of the Q-curvature is a positive integer multiple of the one of the four-dimensional round sphere. This problem has a variational structure with a lack of compactness. Using some topological tools of the theory of "critical points at infinity" combined with a refined blow-up analysis and some dynamical arguments, we identify the accumulations points of all noncompact flow lines of a pseudogradient flow, the so called critical points at infinity of the associated variational problem, and associate to them a natural Morse index. We then prove strong Morse type inequalities, extending the full Morse theory to this noncompact variational problem. Finally, we derive from our results Poincaré-Hopf index type criteria for existence, extending known results in the literature and deriving new ones.