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.
Bibliographical noteFunding 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.
© 2014, Springer Basel.