{"paper":{"title":"Convergence of manifolds under some $L^p$-integral curvature conditions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Conghan Dong","submitted_at":"2018-11-25T12:01:25Z","abstract_excerpt":"Let $\\mathcal{C}(\\mathcal{R},n,p,\\Lambda,D,V_0)$ be the class of compact $n$-dimensional Riemannian manifolds with finite diameter $\\leq D$, non-collapsing volume $\\geq V_0$ and $L^p$-bounded $\\mathcal{R}$-curvature condition $\\|\\mathcal{R}\\|_{L^p}\\leq \\Lambda$ for some $p>\\frac n2$. Let $(M,g_0)$ be a compact Riemannian manifold and $\\mathcal{C}(M,g_0)$ the class of manifolds $(M,g)$ conformal to $(M,g_0)$. In this paper we use $\\varepsilon$-regularity to show a rigidity result in the conformal class $\\mathcal{C}(S^n,g_0)$ of standard sphere under $L^p$-scalar rigidity condition. Then we use "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.09994","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"}