Singularities of spacelike mean curvature one surfaces in de Sitter space

TL;DR

This paper investigates the singularities of spacelike constant mean curvature one surfaces in de Sitter space, establishing dualities, invariants, and classifications of singular points.

Contribution

It introduces the duality between conelike and 5/2-cuspidal edges, and defines invariants for classifying singularities on these surfaces.

Findings

Duality between conelike and 5/2-cuspidal edges.

Introduction of $eta$- and $ au$-invariants for singularity classification.

Complete classification of non-degenerate singular points.

Abstract

In this paper, we study the singularities of spacelike constant mean curvature one (CMC 1) surfaces in the de Sitter 3-space. We prove the duality between generalized conelike singular points and 5/2-cuspidal edges on spacelike CMC 1 surfaces. To describe the duality between $A_{k+3}$ singularities and cuspidal $S_k$ singularities, we introduce two invariants, called the $\alpha$-invariant and $\sigma$-invariant, of spacelike CMC 1 surfaces at their singular points. Moreover, we give a classification of non-degenerate singular points on spacelike CMC 1 surfaces.