{"paper":{"title":"The deformation of symplectic critical surfaces in a K\\\"ahler surface-II---Compactness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jiayu Li, Jun Sun, Xiaoli Han","submitted_at":"2016-07-06T13:16:29Z","abstract_excerpt":"In this paper we consider the compactness of $\\beta$-symplectic critical surfaces in a K\\\"ahler surface. Let $M$ be a compact K\\\"ahler surface and $\\Sigma_i\\subset M$ be a sequence of closed $\\beta_i$-symplectic critical surfaces with $\\beta_i\\to\\beta_0\\in (0,\\infty)$. Suppose the quantity $\\int_{\\Sigma_i}\\frac{1}{\\cos^q\\alpha_i}d\\mu_i$ (for some $q>4$) and the genus of $\\Sigma_{i}$ are bounded, then there exists a finite set of points ${\\mathcal S}\\subset M$ and a subsequence $\\Sigma_{i'}$ that converges uniformly in the $C^l$ topology (for any $l<\\infty$) on compact subsets of $M\\backslash {"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.01606","kind":"arxiv","version":1},"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"}