{"paper":{"title":"New Asymptotic Geometric Quantities in Riemannian Geometry and their Geometric applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"On complete noncompact Riemannian manifolds, volume entropy bounds infinity capacity, which bounds the infinity eigenvalue equal to the Maz'ya limit.","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jiabin Yin, Xiaoshang Jin","submitted_at":"2026-04-16T04:11:19Z","abstract_excerpt":"This paper studies the large $p$ asymptotics of three geometric quantities on complete noncompact Riemannian manifolds: the $p-$capacity of a compact set, the first Dirichlet $p-$eigenvalue, and the Maz'ya constant, thereby offering a new perspective on the study of such manifolds. We introduce the infinity capacity $\\mathcal{C}(\\Omega)$, the infinity eigenvalue $\\Lambda(M)$, and the Maz'ya limit $\\mathcal{M}(M)$, and establish the general inequality, for any $\\Omega\\subset M$, $$ \\mathcal{V}(M) \\ge \\mathcal{C}(\\Omega) \\ge \\Lambda(M) = \\mathcal{M}(M), $$ where $\\mathcal{V}(M)$ is the volume en"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For any compact set Ω subset M, V(M) ≥ C(Ω) ≥ Λ(M) = M(M), where V is volume entropy, C infinity capacity, Λ infinity eigenvalue, and M the Maz'ya limit.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The definitions of the infinity capacity, infinity eigenvalue, and Maz'ya limit are well-posed and the large-p limits exist on complete noncompact Riemannian manifolds.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"On complete noncompact Riemannian manifolds the volume entropy bounds the infinity capacity which bounds the infinity eigenvalue, which equals the Maz'ya limit, with equality under isoperimetric or curvature conditions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"On complete noncompact Riemannian manifolds, volume entropy bounds infinity capacity, which bounds the infinity eigenvalue equal to the Maz'ya limit.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"d05ce7e7bdfe77f262b7e52644a6960402e351ac77c490c7d5cc7bf2d09403e3"},"source":{"id":"2604.14600","kind":"arxiv","version":2},"verdict":{"id":"0a160c40-0c8f-4e46-b73f-a90cf31106cb","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T10:35:53.462837Z","strongest_claim":"For any compact set Ω subset M, V(M) ≥ C(Ω) ≥ Λ(M) = M(M), where V is volume entropy, C infinity capacity, Λ infinity eigenvalue, and M the Maz'ya limit.","one_line_summary":"On complete noncompact Riemannian manifolds the volume entropy bounds the infinity capacity which bounds the infinity eigenvalue, which equals the Maz'ya limit, with equality under isoperimetric or curvature conditions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The definitions of the infinity capacity, infinity eigenvalue, and Maz'ya limit are well-posed and the large-p limits exist on complete noncompact Riemannian manifolds.","pith_extraction_headline":"On complete noncompact Riemannian manifolds, volume entropy bounds infinity capacity, which bounds the infinity eigenvalue equal to the Maz'ya limit."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.14600/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}