A Correspondence Between Maximal Surfaces and Timelike Minimal Surfaces in \mathbbL³

2019/09/17 by Patel, Aryaman
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1909.07744

Abstract

We show that to every maximal surface with conelike singularities in Lorentz-Minkowski space \mathbbL³ that can be locally represented as the graph of a smooth function, there exists a corresponding timelike minimal surface in \mathbbL³. 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 \mathbbL³. Finally, we derive some non-trivial identities using existing Euler-Ramanujan identities, and some familiar timelike minimal surfaces in parametric form.

Related

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