A new framework of zero mean curvature surfaces in the isotropic 3-space

TL;DR

This paper introduces a new class of zero mean curvature surfaces with singularities in isotropic 3-space, establishing inequalities related to their geometric properties and providing examples that attain these bounds.

Contribution

It defines ZMC-faces in isotropic 3-space and proves Osserman-type inequalities, linking surface asymptotics to geometric bounds, with illustrative examples.

Findings

Established three Osserman-type inequalities for ZMC-faces.

Derived conditions for equality related to asymptotic behaviors.

Provided examples of ZMC-faces that attain the inequalities.

Abstract

We introduce a class of zero mean curvature surfaces with singularities in the isotropic 3-space, called ZMC-faces. As a main result, we establish three Osserman-type inequalities for a ZMC-face under certain assumptions on both completeness and finiteness of the total curvature. The equality conditions of these inequalities are related to the asymptotic behaviors of the ends. Moreover, we present several examples of ZMC-faces attaining equalities in these inequalities.