On the Moduli Space of Null Curves in Klein's Quadric

TL;DR

This paper investigates the structure of the moduli space of null curves in Klein's quartic, revealing non-existence results for certain minimal surfaces and proposing new conjectures about the moduli space.

Contribution

It applies Bryant's methods to analyze the moduli space of null curves in Klein's quartic, providing new non-existence results and conjectures.

Findings

Minimal surfaces with 9 embedded planar ends do not exist

Formulation of new conjectures about the moduli space

Application of Bryant's methods to Klein's quartic

Abstract

We study the moduli space of null curves in Klein's quartic in the four-dimensional (complex) projective plane using methods developed by Robert Bryant. As a consequence, we show that minimal surfaces with $9$ embedded planar ends do not exist and formulate some conjectures about the previous moduli space.