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Transactions of the American Mathematical Society

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On the uniqueness of certain families of holomorphic disks


Author: Frédéric Rochon
Journal: Trans. Amer. Math. Soc. 363 (2011), 633-657
MSC (2010): Primary 53C28, 53C56
DOI: https://doi.org/10.1090/S0002-9947-2010-05159-1
Published electronically: September 20, 2010
MathSciNet review: 2728581
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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.


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Additional Information

Frédéric Rochon
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G4
Address at time of publication: Department of Mathematics, Australian National University, Canberra, ACT 0200, Australia
Email: rochon@math.utoronto.ca, frederic.rochon@anu.edu.au

DOI: https://doi.org/10.1090/S0002-9947-2010-05159-1
Received by editor(s): August 15, 2008
Published electronically: September 20, 2010
Additional Notes: The author acknowledges the support of the Fonds québécois de la recherche sur la nature et les technologies while part of this work was conducted.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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