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.