  Invariants for Riemann surfaces covered by the disc and for hyperbolic
manifolds in general involving minimizing the measure of the image over the
homotopy and homology classes of closed curves and maps of the $k$-sphere into
the manifold are investigated. The invariants are monotonic under holomorphic
mappings and strictly monotonic under certain circumstances. Applications to
holomorphic maps of annular regions in $\mathbb{C}$ and tubular neighborhoods
of compact totally real submanifolds in general in $\mathbb{C}^n$, $n \geq 2$,
are given. The contractibility of a hyperbolic domain with contracting
holomorphic mapping is explained.
