Abstract: We review basics in nonpositive curvature.

Abstract: For a complete Riemannian manifold of nonpositive curvature, we introduce a flow. We give an upper bound on the k-Jacobian of the flow in terms of the critical exponent of the fundamental group. We also give several applications connecting the geometry and topology of the manifold, which includes the linear isoperimetric inequality, the homological vanishing theorem, and the non-existence of compact complex subvarieties in certain complex hyperbolic manifolds. This is joint work with Chris Connell and Ben McReynolds.