2012/10/31 by Yoel Groman, Jake P. Solomon · 1 citation
Mathematics · #math.SG #math.AG #math.DG #msc:32Q65 #msc:53D12 #msc:14P99 #msc:53D45
paper · pdf · doi:10.1007/s00039-014-0295-2
published as Geom. Funct. Anal. 24 (2014), no. 5, 1448-1515 · 70 pages, 8 figures, corrected minor errors, added application to adic convergence, updated references
arxiv created 2014/06/12 · arxiv updated 2014/09/30
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 \mathbb R Pn ⊂ \mathbb C Pn. 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.