Topological invariants and Holomorphic Mappings

TL;DR

This paper introduces new topological invariants for Riemann surfaces and hyperbolic manifolds based on minimal measure mappings, exploring their monotonicity under holomorphic maps and applications in complex analysis.

Contribution

It develops novel invariants involving minimal measure over homotopy classes and demonstrates their monotonicity properties under holomorphic mappings, with applications to complex regions and submanifolds.

Findings

Invariants are monotonic under holomorphic maps.

Strict monotonicity under certain conditions.

Applications to annular regions and real submanifolds in complex spaces.

Abstract

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.