A reverse isoperimetric inequality for J-holomorphic curves

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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 languageAmerican English
Pages (from-to)1448-1515
Number of pages68
JournalGeometric and Functional Analysis
Issue number5
StatePublished - 1 Sep 2014

All Science Journal Classification (ASJC) codes

  • Analysis
  • Geometry and Topology


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