Conformal metrics of prescribed $Q-$curvature on 4-manifolds
Mohameden Ould Ahmedou (Giessen)
In this talk we address the question of existence, on a four
dimensional riemannian manifold, of conformal metrics of prescribed
Q-curvature. The Q-curvature is a generalization on four manifolds of
the two dimensional Gauß curvature. This scalar quantity turns out be
helpful in the understanding of the topology and geometry of four
manifolds.
This problem amounts to solve a fourth order nonlinear PDE involving
the Paneitz Operator. This PDE enjoys a variational formulation,
however the corresponding Euler-Lagrange functional does not satisfy
the Palais-Smale condition.
In this talk we will report on recent existence results obtained
through a Morse theoretical approach to this noncompact variational
problem combined with a refined analysis of the singularities of the
corresponding gradient flow.