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 $\mathbb {S}^{2}$ corresponds to a family of holomorphic disks in $\mathbb {CP}_{2}$ with boundary in a totally real submanifold $P\subset \mathbb {CP}_{2}$. In this paper, we show that for a fixed $P\subset \mathbb {CP}_{2}$, 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.