{"paper":{"title":"On Hofer Energy of J-holomorphic Curves for Asymptotically Cylindrical J","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SG","authors_text":"Erkao Bao","submitted_at":"2013-03-18T21:31:01Z","abstract_excerpt":"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 $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.4430","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}