Zoll Metrics, Branched Covers, and Holomorphic Disks

TL;DR

This paper advances the understanding of Zoll metrics and projective structures on S^2, providing concrete conditions for their realization via embeddings and exploring their significance in Lorentzian Einstein-Weyl equations.

Contribution

It introduces an explicit open condition ensuring a totally real embedding of RP^2 in CP^2 corresponds to a unique Zoll projective structure on S^2.

Findings

Established a concrete criterion for Zoll structures from embeddings.

Linked Zoll structures to solutions of Lorentzian Einstein-Weyl equations.

Strengthened previous results on moduli spaces of Zoll metrics.

Abstract

We strengthen our previous results regarding the moduli spaces of Zoll metrics and Zoll projective structures on S^2. In particular, we describe a concrete, open condition which suffices to guarantee that a totally real embedding of RP^2 in CP_2 arises from a unique Zoll projective structure on the 2-sphere. Our methods ultimately reflect the special role such structures play in the initial value problem for the 3-dimensional Lorentzian Einstein-Weyl equations.