A reverse isoperimetric inequality for J-holomorphic curves

Yoel Groman; Jake P. Solomon

doi:10.1007/s00039-014-0295-2

A reverse isoperimetric inequality for J-holomorphic curves

Yoel Groman^*, Jake P. Solomon

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

We prove that the length of the boundary of a J-holomorphic curve with Lagrangian boundary conditions is dominated by a constant times its area. The constant depends on the symplectic form, the almost complex structure, the Lagrangian boundary conditions and the genus. A similar result holds for the length of the real part of a real J-holomorphic curve. The infimum over J of the constant properly normalized gives an invariant of Lagrangian submanifolds. We calculate this invariant to be 2π for the Lagrangian submanifold RPⁿ ⊂ CPⁿ. We apply our result to prove compactness of moduli of J-holomorphic maps to non-compact target spaces that are asymptotically exact. In a different direction, our result implies the adic convergence of the superpotential.

Original language	American English
Pages (from-to)	1448-1515
Number of pages	68
Journal	Geometric and Functional Analysis
Volume	24
Issue number	5
DOIs	https://doi.org/10.1007/s00039-014-0295-2
State	Published - 1 Sep 2014

Bibliographical note

Funding Information:
The authors would like to thank Kenji Fukaya, Asaf Horev, Mikhail Katz, David Kazhdan, Melissa Liu, and Ran Tessler, for helpful conversations. The authors were partially supported by Israel Science Foundation grant 1321/2009 and Marie Curie International Reintegration Grant No. 239381.

Publisher Copyright:
© 2014, Springer Basel.

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abstract = "We prove that the length of the boundary of a J-holomorphic curve with Lagrangian boundary conditions is dominated by a constant times its area. The constant depends on the symplectic form, the almost complex structure, the Lagrangian boundary conditions and the genus. A similar result holds for the length of the real part of a real J-holomorphic curve. The infimum over J of the constant properly normalized gives an invariant of Lagrangian submanifolds. We calculate this invariant to be 2π for the Lagrangian submanifold RPn ⊂ CPn. We apply our result to prove compactness of moduli of J-holomorphic maps to non-compact target spaces that are asymptotically exact. In a different direction, our result implies the adic convergence of the superpotential.",

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N2 - We prove that the length of the boundary of a J-holomorphic curve with Lagrangian boundary conditions is dominated by a constant times its area. The constant depends on the symplectic form, the almost complex structure, the Lagrangian boundary conditions and the genus. A similar result holds for the length of the real part of a real J-holomorphic curve. The infimum over J of the constant properly normalized gives an invariant of Lagrangian submanifolds. We calculate this invariant to be 2π for the Lagrangian submanifold RPn ⊂ CPn. We apply our result to prove compactness of moduli of J-holomorphic maps to non-compact target spaces that are asymptotically exact. In a different direction, our result implies the adic convergence of the superpotential.

AB - We prove that the length of the boundary of a J-holomorphic curve with Lagrangian boundary conditions is dominated by a constant times its area. The constant depends on the symplectic form, the almost complex structure, the Lagrangian boundary conditions and the genus. A similar result holds for the length of the real part of a real J-holomorphic curve. The infimum over J of the constant properly normalized gives an invariant of Lagrangian submanifolds. We calculate this invariant to be 2π for the Lagrangian submanifold RPn ⊂ CPn. We apply our result to prove compactness of moduli of J-holomorphic maps to non-compact target spaces that are asymptotically exact. In a different direction, our result implies the adic convergence of the superpotential.

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A reverse isoperimetric inequality for J-holomorphic curves

Abstract

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