{"paper":{"title":"A conformally invariant gap theorem characterizing $\\mathbb{CP}^2$ via the Ricci flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Matthew Gursky, Siyi Zhang, Sun-Yung A. Chang","submitted_at":"2018-09-16T17:28:00Z","abstract_excerpt":"We extend the sphere theorem of \\cite{CGY03} to give a conformally invariant characterization of $(\\mathbb{CP}^2, g_{FS})$. In particular, we introduce a conformal invariant $\\beta(M^4,[g]) \\geq 0$ defined on conformal four-manifolds satisfying a `positivity' condition; it follows from \\cite{CGY03} that if $0 \\leq \\beta(M^4,[g]) < 4$, then $M^4$ is diffeomorphic to $S^4$. Our main result of this paper is a `gap' result showing that if $b_2^{+}(M^4) > 0$ and $4 \\leq \\beta(M^4,[g]) < 4(1 + \\epsilon)$ for $\\epsilon > 0$ small enough, then $M^4$ is diffeomorphic to $\\mathbb{CP}^2$. The Ricci flow "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.05918","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"}