Semi-discrete linear Weingarten surfaces with Weierstrass-type representations and their singularities

TL;DR

This paper classifies semi-discrete linear Weingarten surfaces with Weierstrass representations in various spaceforms, analyzes their curvature properties, and studies their singularities, including those of constant Gaussian and mean curvature surfaces.

Contribution

It provides a comprehensive classification of semi-discrete linear Weingarten surfaces with Weierstrass-type representations and analyzes their singularities in different spaceforms.

Findings

Classified semi-discrete linear Weingarten surfaces in Riemannian and Lorentzian spaceforms.

Analyzed singularities of semi-discrete surfaces with various constant curvatures.

Compared new singularity definitions with existing ones.

Abstract

We establish what semi-discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms are, confirming their required properties regarding curvatures and parallel surfaces, and then classify them. We then define and analyze their singularities. In particular, we discuss singularities of (1) semi-discrete surfaces with non-zero constant Gaussian curvature, (2) parallel surfaces of semi-discrete minimal and maximal surfaces, and (3) semi-discrete constant mean curvature $1$ surfaces in de Sitter $3$-space. We include comparisons with different previously known definitions of such singularities.