# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1903.06131