## 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 RP^{n} ⊂ CP^{n}. 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 |
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Pages (from-to) | 1448-1515 |

Number of pages | 68 |

Journal | Geometric and Functional Analysis |

Volume | 24 |

Issue number | 5 |

DOIs | |

State | Published - 1 Sep 2014 |

### Bibliographical note

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