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# Journal of Symplectic Geometry

## Volume 14 (2016)

### Number 1

### On Hofer energy of $J$-holomorphic curves for asymptotically cylindrical $J$

Pages: 97 – 118

DOI: http://dx.doi.org/10.4310/JSG.2016.v14.n1.a4

#### Author

#### Abstract

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 $(M, \omega^{\prime})$ with a compatible almost complex structure $J$ and a ball $B$ in $M$, there exists a constant $\hbar \gt 0$, 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 $\partial B$ has symplectic area $\int_{\tilde{u}^{-1}(B)} \tilde{u}^{*} \omega^{\prime} \gt k \hbar$, where the constant $\hbar$ depends only on $(M, \omega^{\prime}, J)$ 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 $(M, \omega^{\prime}, J)$ and the symplectic area of $\tilde{u}$. Here J is any $\omega^{\prime}$-compatible smooth almost complex structure on $M$. In particular, we do not require $J$ to be integrable.

#### Keywords

asymptotically cylindrical, stable hamiltonian structure, $J$-holomorphic curve, Hofer energy, Gromov’s Monontonicity Theorem, holomorphic building