Limit polygons of convex domains in the projective plane

TL;DR

This paper characterizes the limits of sequences of convex domains in the projective plane under certain conditions, and applies these results to domains generated by projective triangular reflection groups.

Contribution

It provides a complete description of Hausdorff limits of convex domains with bounded Pick differential zeros, advancing understanding of geometric limits in projective geometry.

Findings

Identified all possible Hausdorff limit domains under specified conditions.

Connected limit domain behavior to properties of Pick differentials.

Applied results to convex domains from projective triangular reflection groups.

Abstract

For any sequence of properly convex domains in the real projective plane such that the zeros of Pick differentials have bounded multiplicity and get further and further apart, we determined all Hausdorff limit domains that one can obtain after normalizing each member of the sequence by a projective transformation. We then show that the result can be applied to convex domains generated by projective triangular reflection groups.