The Ricci flow does not preserve the set of Zoll metrics

TL;DR

This paper investigates the limitations of Ricci flow in preserving Zoll metrics on the 2-sphere, highlighting that naive Ricci flow applications cannot resolve the connectivity question of Zoll metrics.

Contribution

It demonstrates that straightforward Ricci flow approaches are inadequate for analyzing the topology of Zoll metrics on the 2-sphere.

Findings

Naive Ricci flow does not preserve Zoll metrics

Ricci flow cannot determine the connectedness of Zoll metrics

Highlights limitations of Ricci flow in geometric topology

Abstract

The question of whether or not the set of Zoll metrics on the 2-sphere is connected is still open. Here we show that a naive application of the Ricci flow is not sufficient to answer this problem.