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arXiv · 1409.7919

Morse theory and the resonant Q-curvature problem

Cheikh Birahim Ndiaye·Mohameden Ould Ahmedou
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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.

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Cheikh Birahim Ndiaye, Mohameden Ould Ahmedou. 2014-09-28. Morse theory and the resonant Q-curvature problem. https://arxiv.org/abs/1409.7919

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