Topological invariants and Holomorphic Mappings

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.