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.