Reflection principles for zero mean curvature surfaces in the simply isotropic 3-space

TL;DR

This paper explores reflection principles for zero mean curvature surfaces in simply isotropic 3-space, revealing how these surfaces relate to minimal and maximal surfaces and establishing a reflection principle along singular isotropic line segments.

Contribution

It introduces reflection principles specific to zero mean curvature surfaces in isotropic space, extending classical geometric reflection concepts to this intermediate geometry.

Findings

Reflection principle for isotropic line segments on zero mean curvature surfaces.

Identification of singular induced metrics along these line segments.

Connection between isotropic surfaces and classical minimal/maximal surface theories.

Abstract

Zero mean curvature surfaces in the simply isotropic 3-space $\mathbb{I}^3$ naturally appear as intermediate geometry between geometry of minimal surfaces in $\mathbb{E}^3$ and that of maximal surfaces in $\mathbb{L}^3$. In this paper, we investigate reflection principles for zero mean curvature surfaces in $\mathbb{I}^3$ as with the above surfaces in $\mathbb{E}^3$ and $\mathbb{L}^3$. In particular, we show a reflection principle for isotropic line segments on such zero mean curvature surfaces in $\mathbb{I}^3$, along which the induced metrics become singular.