Causal characters of zero mean curvature surfaces of Riemann type in the Lorentz-Minkowski 3-space

TL;DR

This paper classifies zero mean curvature surfaces of Riemann-type in Lorentz-Minkowski 3-space based on their causal characters, revealing that surfaces with two causal characters have lightlike parts forming straight lines.

Contribution

It provides a complete classification of Riemann-type zero mean curvature surfaces by their causal characters, including a new result on lightlike parts when two causal characters are present.

Findings

Surfaces with exactly two causal characters have lightlike parts that are straight lines.

Classification of Riemann-type zero mean curvature surfaces by causal character.

Identification of the structure of lightlike parts in these surfaces.

Abstract

A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann-type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann-type according to their causal characters, and as a corollary, we prove that if a zero mean curvature surface of Riemann-type has exactly two causal characters, then the lightlike part of the surface is a part of a straight line.