Bernstein-type theorem for constant mean curvature surfaces in the isotropic 3-space

TL;DR

This paper proves a Bernstein-type theorem for complete spacelike constant mean curvature surfaces in isotropic 3-space, using a value distribution theorem for Gaussian curvature, extending classical results to this geometric setting.

Contribution

It introduces a new value distribution theorem for Gaussian curvature that leads to a Bernstein-type classification of CMC surfaces in isotropic 3-space.

Findings

Establishment of a value distribution theorem for Gaussian curvature.

Proof of a Bernstein-type theorem for CMC graphs in isotropic 3-space.

Characterization of entire spacelike CMC surfaces in the isotropic setting.

Abstract

There are many non-trivial entire spacelike graphs with constant mean curvature $H$ (CMC $H$, for short) in the isotropic 3-space $\mathbb{I}^3$. In this paper, we show a value distribution theorem of Gaussian curvature of complete spacelike constant mean curvature surfaces in $\mathbb{I}^3$, which implies a Bernstein-type theorem for CMC $H$ graphs in $\mathbb{I}^3$.