A Correspondence Between Maximal Surfaces and Timelike Minimal Surfaces in $\mathbb{L}^3$

TL;DR

This paper establishes a one-to-one correspondence between maximal surfaces with conelike singularities and timelike minimal surfaces in Lorentz-Minkowski space, revealing structural similarities and preserving key geometric properties.

Contribution

It introduces a linear transformation linking maximal and timelike minimal surfaces, extending the understanding of their geometric relationship and leading to a Kobayashi type theorem.

Findings

Linear transformation maps singularities between surfaces

Establishes a one-to-one correspondence preserving Gauss map properties

Derives identities using Euler-Ramanujan identities

Abstract

We show that to every maximal surface with conelike singularities in Lorentz-Minkowski space $\mathbb{L}^3$ that can be locally represented as the graph of a smooth function, there exists a corresponding timelike minimal surface in $\mathbb{L}^3$. There exists a linear transformation between such a maximal surface and its corresponding timelike minimal surface and it maps the singularities of one to the singularities of the other. Moreover, this transformation establishes a one-one correspondence between such maximal surfaces and timelike minimal surfaces and also preserves the one-one property of the Gauss map. This leads to a Kobayashi type theorem for timelike minimal surfaces in $\mathbb{L}^3$. Finally, we derive some non-trivial identities using existing Euler-Ramanujan identities, and some familiar timelike minimal surfaces in parametric form.