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

2017/09/21 by Masashi Yasumoto, Yasumoto, Masashi, Wayne Rossman +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities

paper · doi:10.48550/arxiv.1709.07373

openalex publication_date 2017/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Discrete linear Weingarten surfaces with singularities in Riemannian and Lorentzian spaceforms
• Weierstrass representations of discrete constant mean curvature surfaces in isotropic space
• Marginally trapped surfaces in \mathbbL⁴ and three Weierstrass representations
• Linear Weingarten surfaces in Euclidean and hyperbolic space
• A note on the generalized Weierstrass representation