Zero mean curvature surfaces in isotropic space with planar curvature lines

Abstract

We give a comprehensive account of zero mean curvature surfaces with planar curvature lines in isotropic $3$-space. After providing a complete classification of all such surfaces, we show that they belong to a $1$-parameter family of surfaces. We then investigate their relationship to Thomsen-type surfaces in isotropic $3$-space, those zero mean curvature surfaces in isotropic $3$-space that are also affine minimal.