A reverse isoperimetric inequality for J-holomorphic curves

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 $2\pi$ for the Lagrangian submanifold $\mathbb{R}{P}^n\subset \mathbb{C}{P}^n.$ 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.

70 pages, 8 figures, corrected minor errors, added application to adic convergence, updated references

References in corpus (3)

• Cyclic symmetry and adic convergence in LagrangianFloer theory
• Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions
• Disk enumeration on the quintic 3-fold

Cited by in corpus (5)

• Disk counting and wall-crossing phenomenon via family Floer theory
• Projective twists and the Hopf correspondence
• Family Floer mirror space for local SYZ singularities
• Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds
• Family Floer superpotential's critical values are eigenvalues of quantum product by $c_1$