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

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.