A generalization of Gromov's almost flat manifold theorem

Esther Cabezas Rivas (Muenster)

Taking as an starting point a question by John Lott about the vanishing of the $\hat A$-genus 
for spin almost-non-negatively curved manifolds, we conjecture that an almost-non-negatively 
curved manifold is either conformally equivalent to a manifold with positive scalar curvature or 
it is finitely covered by a Nilmanifold. In the way to prove such a claim, we found a generalization 
of Gromov's almost flat manifold theorem where $L^\infty$ bounds for the curvature are relaxed 
to mixed curvature bounds. During the talk, we will give the precise statement of our theorem 
and a detailed sketch of the proof.

This is a joint work with Burkhard Wilking.