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,ωʹ)1with a compatible almost complex structure J and a ball B in M, there exists a constant ħ> 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 (formula presented) where the constant ħ depends only on (M,ωʹ, J) and the radius of B. As a consequence, the number of times that any closed Jholomorphic curve in M passes through a point is bounded by a constant depending only on (M,ωʹ, J) and the symplectic area of ũ. Here J is any ωʹ−compatible smooth almost complex structure on M. In particular, we do not require J to be integrable.
Bibliographical noteFunding Information:
The author is partially supported by the research fellowship scheme of the Chinese University of Hong Kong. CUHK501100004853
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- Asymptotically cylindrical
- Gromov’s Monontonicity Theorem
- Hofer energy
- Holomorphic building
- J-holomorphic curve
- Stable hamiltonian structure