Mohameden Ould Ahmedou, Gießen
Conformal metrics of prescribed Q-curvature on 4-manifolds: A Morse
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
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.
The Borel-Weil-Bott theorem via the equivariant McKean-Singer formula
Borel-Weil-Bott theorem states that irreducible representations of a compact connected Lie group G can be constructed by calculating the sheaf cohomologies of holomorphic sections of complex line bundles over the flag variety G/T associated to various T-representations. This result can be understood as an equivariant index theorem for a Dirac operator on G/T. In this talk we shall explain how this index theorem can be easily reached using Kostant's cubic Dirac operator and the equivariant McKean-Singer formula.
Bernd Ammann, 4.11.2014 oder später