Zoll metrics, branched covers, and holomorphic disks

We strengthen our previous results [12] regarding the moduli spaces of Zoll metrics and Zoll projective structures on S2. In particular, we describe a concrete, open condition which suffices to guarantee that a totally real embedding ℝℙ²→ ℂℙ₂ arises from a unique Zoll projective structure on the two-sphere. Our methods ultimately reflect the special role such structures play in the initial value problem for the three-dimensional Lorentzian Einstein-Weyl equations.