Abstract:A Zoll metric is a metric whose geodesics are all circles of equal length. In this talk, we will first review the definition of the twistor correspondence of LeBrun and Mason for Zoll metrics on the sphere $\bbS^{2}$. It associates to a Zoll metric on $\bbS^{2}$ a family of holomorphic disks in $\cp{2}$ with boundary in a totally real submanifold $P\subset \cp{2}$. For a fixed $P\subset \cp{2}$, we will indicate how one can show that such a family is unique whenever it exists, implying that the twistor correspondence of LeBrun and Mason is in some sense injective. One of the key ingredients in the proof will be the blow-up and blow-down constructions in the sense of Melrose.