The Weierstrass representation of discrete isotropic surfaces in $R^{2,1}$, $R^{3,1}$ and $R^{2,2}$

TL;DR

This paper develops a discrete Weierstrass representation for time-like isotropic surfaces in various semi-Riemannian spaces using integrable discrete Dirac operators, broadening the understanding of discrete differential geometry.

Contribution

It introduces a novel discrete Weierstrass representation for isotropic surfaces in $R^{2,1}$, $R^{3,1}$, and $R^{2,2}$ using integrable discrete Dirac operators, applicable under a monotonicity condition.

Findings

Discrete surfaces with isotropic edges can be represented using the new method.

The representation applies to surfaces in multiple semi-Riemannian spaces.

Any discrete surface with isotropic edges satisfying the monotonicity condition admits this representation.

Abstract

Using an integrable discrete Dirac operator, we construct a discrete version of the Weierstrass representation of time-like surfaces parametrized along isotropic directions in $R^{2,1}$, $R^{3,1}$ and $R^{2,2}$. The corresponding discrete surfaces have isotropic edges. We show that any discrete surface satisfying a general monotonicity condition and having isotropic edges admits such a representation.