Analytic extensions of constant mean curvature one geometric catenoids in de Sitter 3-space

TL;DR

This paper introduces a new criterion called arc-properness to determine the analytic completeness of certain constant mean curvature surfaces in de Sitter 3-space, showing that some surfaces cannot be analytically extended.

Contribution

It defines arc-properness as a weak properness condition and applies it to analyze the analytic completeness of G-catenoids in de Sitter space.

Findings

Arc-properness effectively characterizes analytic completeness.

Certain G-catenoids are shown to be analytically complete with no extensions.

The criterion simplifies the analysis of constant mean curvature surfaces.

Abstract

We show that a certain simply-stated notion of "analytic completeness" of the image of a real analytic map implies the map admits no analytic extension. We also give a useful criterion for that notion of analytic completeness by defining arc-properness of continuous maps, which can be considered as a very weak version of properness. As an application, we judge the analytic completeness of a certain class of constant mean curvature surfaces (the so-called "G-catenoids") or their analytic extensions in the de Sitter 3-space.