  Inspired by work of Besson-Courtois-Gallot, we construct a flow called the
natural flow on a nonpositively curved Riemannian manifold $M$. As with the
natural map, the $k$-Jacobian of the natural flow is directly related to the
critical exponent $\delta$ of the fundamental group. There are several
applications of the natural flow that connect dynamical, geometrical, and
topological invariants of the manifold. First, we give $k$-dimensional linear
isoperimetric inequalities when $k > \delta$. This, in turn, produces lower
bounds on the Cheeger constant. We resolve a recent conjecture of Dey-Kapovich
on the non-existence of $k$-dimensional compact, complex subvarieties of
complex hyperbolic manifolds with $2k > \delta$. We also provide upper bounds
on the homological dimension, generalizing work of Kapovich and work of Farb
with the first two authors. Using different methods, we also give upper bounds
on the cohomological dimension which partially resolves a conjecture of
Kapovich. We also specialize to the setting of locally symmetric manifolds
where we further refine our work. Finally, we introduce a new growth condition
on the Bowen-Margulis measure that we call uniformly exponentially bounded that
we connect to the cohomological dimension and which could be of independent
interest.
