Deformation and singularities of maximal surfaces with planar curvature lines

TL;DR

This paper refines the classification of maximal surfaces with planar curvature lines in Lorentz-Minkowski space, introduces a deformation connecting all such surfaces, and studies their singularities and conjugates.

Contribution

It provides an alternative classification method, demonstrates a continuous deformation among these surfaces, and explores their singularities and conjugate counterparts.

Findings

Existence of a deformation connecting all maximal surfaces with planar curvature lines.

Characterization of singularities on these maximal surfaces.

Results on conjugate surfaces that are affine minimal.

Abstract

Minimal surfaces with planar curvature lines in the Euclidean space have been studied since the late 19th century. On the other hand, the classification of maximal surfaces with planar curvature lines in the Lorentz-Minkowski space has only recently been given. In this paper, we use an alternative method not only to refine the classification of maximal surfaces with planar curvature lines, but also to show that there exists a deformation consisting exactly of all such surfaces. Furthermore, we investigate the types of singularities that occur on maximal surfaces with planar curvature lines. Finally, by considering the conjugate of maximal surfaces with planar curvature lines, we obtain analogous results for maximal surfaces that are also affine minimal surfaces.