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 2 dimensional sphere corresponds to a family of holomorphic disks in CP_2 with boundary in a totally real submanifold P. In this paper, we show that for a fixed totally real submanifold P, 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.