{"paper":{"title":"Macroscopic scalar curvature and areas of cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hannah Alpert, Kei Funano","submitted_at":"2017-05-08T15:36:32Z","abstract_excerpt":"In this paper we prove the following. Let $\\Sigma$ be an $n$--dimensional closed hyperbolic manifold and let $g$ be a Riemannian metric on $\\Sigma \\times \\mathbb{S}^1$. Given an upper bound on the volumes of unit balls in the Riemannian universal cover $(\\widetilde{\\Sigma\\times \\mathbb{S}^1},\\widetilde{g})$, we get a lower bound on the area of the $\\mathbb{Z}_2$--homology class $[\\Sigma \\times \\ast]$ on $\\Sigma \\times \\mathbb{S}^1$, proportional to the hyperbolic area of $\\Sigma$. The theorem is based on a theorem of Guth and is analogous to a theorem of Kronheimer and Mrowka involving scalar "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.02923","kind":"arxiv","version":2},"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"}