  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.
