Abstract

We introduce a flow on nonpositively curved manifolds inspired by the natural maps of Besson, Courtois and Gallot for which the Morse theoretic data can be computed in terms of the geometric structure of the manifold. We present several applications of this flow, including conditions for the nonexistence of complex subvarieties and estimates of the Cheeger constant on such manifolds. Most importantly, we show the vanishing of the homology of nonpositively curved manifolds above a certain threshold which is computable from the geometry of its universal cover and the critical exponent of the representation of the fundamental group. We will present some examples including those arising from Anosov representations in higher rank lie groups. This is joint work with Shi Wang and Ben McReynolds.