On hofer energy of J-holomorphic curves for asymptotically cylindrical J

In this paper, we provide a bound for the generalized Hofer energy of punctured J- holomorphic curves in almost complex manifolds with asymptotically cylindrical ends. As an application, we prove a version of Gromov's Monotonicity Theorem with multiplicity. Namely, for a closed symplectic manifold (M, omega')(1) with a compatible almost complex structure J and a ball B in M, there exists a constant h > 0, such that any J- holomorphic curve (u) over tilde passing through the center of B for k times (counted with multiplicity) with boundary mapped to. B has symplectic area integral((u) over tilde -1(B)) (u) over tilde*omega' >kh, where the constant h depends only on (M, omega', J) and the radius of B. As a consequence, the number of times that any closed J-holomorphic curve in M passes through a point is bounded by a constant depending only on (M, omega', J) and the symplectic area of (u) over tilde. Here J is any omega'-compatible smooth almost complex structure on M. In particular, we do not require J to be integrable.