On the uniqueness of certain families of holomorphic disks

TL;DR

This paper proves the uniqueness of certain families of holomorphic disks associated with Zoll metrics on the 2-sphere, establishing the injectivity of a twistor correspondence via complex geometric techniques.

Contribution

It demonstrates that for a fixed totally real submanifold, the family of holomorphic disks corresponding to Zoll metrics is unique, confirming the injectivity of the LeBrun-Mason twistor correspondence.

Findings

Uniqueness of holomorphic disk families for fixed totally real submanifolds

Injectivity of the LeBrun-Mason twistor correspondence

Application of Melrose's blow-up and blow-down techniques

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 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.