# Small angle limits of Hamilton's footballs

## TL;DR

This paper studies the degeneration of Hamilton's footballs, a class of compact Ricci solitons with cone points, showing their limits as cone angles approach zero, including the cigar soliton as a limit.

## Contribution

It provides a complete description of how Hamilton's footballs degenerate when cone angles vanish, linking compact conical solitons to the cigar soliton.

## Key findings

- Hamilton's footballs degenerate into simpler geometries as cone angles tend to zero.
- The cigar soliton emerges as the Gromov--Hausdorff limit of conical teardrop solitons.
- Degeneration patterns are fully characterized for these Ricci solitons.

## Abstract

Compact Ricci solitons on surfaces have at most two cone points, and are known as Hamilton's footballs. In this note we completely describe the degenerations of these footballs as one or both of the cone angles approaches zero. In particular, we show that Hamilton's famous non-compact cigar soliton is the Gromov--Hausdorff limit of Hamilton's compact conical teardrop solitons.