# Quadrics and Scherk towers

## TL;DR

This paper explores the mathematical relationship between quadrics, their Christoffel duals, and certain zero mean curvature surfaces, introducing para-holomorphic elliptic functions to analyze timelike minimal surfaces and their properties.

## Contribution

It introduces para-holomorphic elliptic functions to study the relation between timelike minimal surfaces and hyperboloids, advancing the understanding of isothermic surfaces of mixed causal type.

## Key findings

- Curves of type change align with the real curvature line net.
- Established connections between quadrics, Christoffel duals, and zero mean curvature surfaces.
- Provided new tools for analyzing timelike minimal surfaces.

## Abstract

We investigate the relation between quadrics and their Christoffel duals on the one hand, and certain zero mean curvature surfaces and their Gauss maps on the other hand. To study the relation between timelike minimal surfaces and the Christoffel duals of 1-sheeted hyperboloids we introduce para-holomorphic elliptic functions. The curves of type change for real isothermic surfaces of mixed causal type turn out to be aligned with the real curvature line net.