A differential form approach to the genus of Open Riemann surfaces

TL;DR

This paper introduces a differential form method to determine the finite genus of open Riemann surfaces and identifies the minimal genus for their holomorphic embedding.

Contribution

It presents a novel differential form approach to classify the genus of open Riemann surfaces and establish biholomorphic equivalences with subsets of compact surfaces.

Findings

Finite genus surfaces are biholomorphic to open subsets of compact Riemann surfaces.

A quotient space of forms characterizes the finite genus property.

The minimal genus for holomorphic embedding is determined.

Abstract

We will show that any open Riemann surface $M$ of finite genus is biholomorphic to an open set of a compact Riemann surface. Moreover, we will introduce a quotient space of forms in $M$ that determines if $M$ has finite genus and also the minimal genus where $M$ can be holomorphically embedded.