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.