On the Uniqueness of Certain Families of Holomorphic Disks

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 S 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 in some sense injective. One of the key ingredients in the proof is the blow-up and blow-down constructions in the sense of Melrose. We are also able to obtain some partial results concerning the existence of such families.