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

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,!0) with a compatible almost complex structure J and a ball B in M, there exists a constant ~ > 0, such that any J-holomorphic curve ũ passing through the center of B for k times (counted with multiplicity) with boundary mapped to @B has symplectic area ũ 1(B) ũ ⇤!0 > k~, where the constant ~ depends only on (M,!0, 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,!0, J)1 and the symplectic area of ũ. Here J is any !0 compatible smooth almost complex structure on M . In particular, we do not require J to be integrable.