Decompositions of Scherk-Type Zero Mean Curvature Surfaces

TL;DR

This paper presents novel decompositions of Scherk-type zero mean curvature surfaces in Lorentz-Minkowski space, expressing them as superpositions of helicoids and applying these to construct complex maximal surfaces.

Contribution

It introduces new infinite and finite decompositions of Scherk-type zero mean curvature surfaces using Euler-Ramanujan identities and Wick rotation techniques.

Findings

Surfaces expressed as infinite superpositions of dilated helicoids.

Finite decompositions of zero mean curvature surfaces.

Application to constructing maximal codimension 2 surfaces in Lorentz-Minkowski 4-space.

Abstract

In this paper, by using a special Euler-Ramanujan identity and the idea of Wick rotation, we show that a one-parameter family of solutions to the zero mean curvature equation in Lorentz-Minkowski $3$-space $\mathbb E_1^3$, namely Scherk-type zero mean curvature surfaces, can be expressed as an infinite superposition of dilated helicoids. Further, we also obtain different finite decompositions for these surfaces. We end this paper with an application of these decompositions to formulate maximal codimension 2 surfaces into finite and infinite "sums" of weakly untrapped and *-surfaces in Lorentz-Minkowski 4-space.