Wild singularities of flat surfaces

TL;DR

This paper investigates the complex singularities of flat surfaces, especially wild singularities, by analyzing their local topological and metric structures, revealing affine invariants and the need for additional data.

Contribution

It introduces a framework for understanding wild singularities on flat surfaces through topological and metric invariants, expanding the classification of surface singularities.

Findings

Homeomorphism type of L(x) is an affine invariant

Wild singularities require additional metric data for description

Classification extends beyond cone points and infinite-angle singularities

Abstract

We consider flat surfaces and the points of their metric completions, particularly the singularities to which the flat structure of the surface does not extend. The local behavior near a singular point x can be partially described by a topological space L(x) which captures all the ways that x can be "approached linearly". The homeomorphism type of L(x) is an affine invariant. When x is not a cone point or an infinite-angle singularity, we say it is wild; in this case it is necessary to add further metric data to L(x) to get a quantitative description of the surface near x.