  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.
