On the uniqueness of certain families of holomorphic disks

Abstract

A Zoll metric is a Riemannian metric whose geodesics are all circles of equal length. Via the twistor correspondence of LeBrun and Mason, a Zoll metric on the sphere S² corresponds to a family of holomorphic disks in CP₂ with boundary in a totally real submanifold P ⊂ CP₂. In this paper, we show that for a fixed P ⊂ CP₂, such a family is unique if it exists, implying that the twistor correspondence of LeBrun and Mason is injective. One of the key ingredients in the proof is the blow-up and blow-down constructions in the sense of Melrose.