  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 $(M,\omega)$ with a compatible almost complex structure $J$
and a ball $B$ in $M,$ there exists a constant $\hbar>0,$ such that any
$J$-holomorphic curve $\tilde{u}$ passing through the center of $B$ for $k$
times (counted with multiplicity) with boundary mapped to $\partial B$ has
symplectic area $\int_{\tilde{u}^{-1}(B)}\tilde{u}^{*}\omega>k\hbar,$ where the
constant $\hbar$ 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 $\tilde{u}$. Here $J$ is any
$\omega-$compatible smooth almost complex structure on $M$. In particular, we
do not require $J$ to be integrable.
