Morse theory and the resonant Q-curvature problem

TL;DR

This paper applies Morse theory and topological methods to analyze the prescribed Q-curvature problem on four-dimensional manifolds, addressing noncompactness and establishing existence criteria through Morse inequalities.

Contribution

It extends Morse theory to a noncompact variational problem involving Q-curvature, identifying critical points at infinity and deriving new existence criteria.

Findings

Identification of critical points at infinity and their Morse indices

Extension of Morse inequalities to noncompact variational problems

New Poincaré-Hopf index criteria for existence results

Abstract

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 Poincar\'e-Hopf index type criteria for existence, extending known results in the literature and deriving new ones.