Geometric surgeries of three-dimensional flag structures and non-uniformizable examples

TL;DR

This paper introduces geometric surgery techniques for 3D flag structures, providing new examples of structures that are both uniformizable and non-uniformizable, advancing understanding of their geometric properties.

Contribution

It develops a novel notion of geometric surgery for flag structures and constructs explicit examples demonstrating both uniformizable and non-uniformizable cases.

Findings

Constructed new examples of flag structures

Demonstrated existence of non-uniformizable structures

Extended the theory of geometric surgeries

Abstract

In this paper, we introduce a notion of geometric surgery for flag structures, which are geometric structures locally modelled on the three-dimensional flag space under the action of ${\mathrm{PGL}}_3(\mathbb{R})$. Using such surgeries we provide examples of flag structures, of both uniformizable and non-uniformizable type.