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

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.

12 pages, 6 figures