Deformations of Riemann Surfaces

TL;DR

This paper demonstrates that all Riemann surfaces except the sphere can undergo infinitesimal complex structure deformations, using geodesic length analysis in the Kobayashi/Poincare metric.

Contribution

It introduces a new approach to deformation theory of Riemann surfaces by linking geodesic lengths to complex structure changes.

Findings

All non-spherical Riemann surfaces admit infinitesimal deformations.

Geodesic length analysis is effective in studying complex structure deformations.

The method applies the Kobayashi/Poincare metric to deformation problems.

Abstract

We prove that every Riemann surface not isomorphic to the Riemann sphere admits an infinitesimal deformation of the complex structure. The proof is based in an investigation of the length of geodesics for the Kobayashi/Poincare metric.