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We prove that there exists a positive $\\varepsilon$, depending on $n$ and upper bounds on the area and the $C^2$-regularity of $S$, such that if $osc(H) \\leq \\varepsilon$ then there exist two concentric balls $B_{r_i}$ and $B_{r_e}$ such that $S \\subset \\overline{B}_{r_e} \\setminus B_{r_i}$ and $r_e -r_i \\leq C \\, osc(H)$, with $C$ depending only on $n$ and "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1501.07845","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-01-30T17:05:25Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"11dacaa969051f270a8020dbc758533786026cd5f7598ef8cc84efbbfd6ef57e","abstract_canon_sha256":"2aefb2a4b48445c0d1058f04269d884469ae574cba9bc0c23e37362505370a60"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:04.315143Z","signature_b64":"gfKE8nVatEDi23rmFbwmbgGpNCkm69B8H3eLibkDnf6l3EQXYsmGkjenOa6LNa9vTsNzp1Dfs5IdCmrYZjWnCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e9080e49c8954f7ea1d3b74dd94708408d2bdd67a0fc998bd77cb227bf2ce4d6","last_reissued_at":"2026-05-18T01:23:04.314493Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:04.314493Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A sharp quantitative version of Alexandrov's theorem via the method of moving planes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Giulio Ciraolo, Luigi Vezzoni","submitted_at":"2015-01-30T17:05:25Z","abstract_excerpt":"We prove the following quantitative version of the celebrated Soap Bubble Theorem of Alexandrov. Let $S$ be a $C^2$ closed embedded hypersurface of $\\mathbb{R}^{n+1}$, $n\\geq1$, and denote by $osc(H)$ the oscillation of its mean curvature. We prove that there exists a positive $\\varepsilon$, depending on $n$ and upper bounds on the area and the $C^2$-regularity of $S$, such that if $osc(H) \\leq \\varepsilon$ then there exist two concentric balls $B_{r_i}$ and $B_{r_e}$ such that $S \\subset \\overline{B}_{r_e} \\setminus B_{r_i}$ and $r_e -r_i \\leq C \\, osc(H)$, with $C$ depending only on $n$ and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.07845","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1501.07845","created_at":"2026-05-18T01:23:04.314586+00:00"},{"alias_kind":"arxiv_version","alias_value":"1501.07845v3","created_at":"2026-05-18T01:23:04.314586+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.07845","created_at":"2026-05-18T01:23:04.314586+00:00"},{"alias_kind":"pith_short_12","alias_value":"5EEA4SOISVHX","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_16","alias_value":"5EEA4SOISVHX5IOT","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_8","alias_value":"5EEA4SOI","created_at":"2026-05-18T12:29:05.191682+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5EEA4SOISVHX5IOTW5G5SRYIIC","json":"https://pith.science/pith/5EEA4SOISVHX5IOTW5G5SRYIIC.json","graph_json":"https://pith.science/api/pith-number/5EEA4SOISVHX5IOTW5G5SRYIIC/graph.json","events_json":"https://pith.science/api/pith-number/5EEA4SOISVHX5IOTW5G5SRYIIC/events.json","paper":"https://pith.science/paper/5EEA4SOI"},"agent_actions":{"view_html":"https://pith.science/pith/5EEA4SOISVHX5IOTW5G5SRYIIC","download_json":"https://pith.science/pith/5EEA4SOISVHX5IOTW5G5SRYIIC.json","view_paper":"https://pith.science/paper/5EEA4SOI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1501.07845&json=true","fetch_graph":"https://pith.science/api/pith-number/5EEA4SOISVHX5IOTW5G5SRYIIC/graph.json","fetch_events":"https://pith.science/api/pith-number/5EEA4SOISVHX5IOTW5G5SRYIIC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5EEA4SOISVHX5IOTW5G5SRYIIC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5EEA4SOISVHX5IOTW5G5SRYIIC/action/storage_attestation","attest_author":"https://pith.science/pith/5EEA4SOISVHX5IOTW5G5SRYIIC/action/author_attestation","sign_citation":"https://pith.science/pith/5EEA4SOISVHX5IOTW5G5SRYIIC/action/citation_signature","submit_replication":"https://pith.science/pith/5EEA4SOISVHX5IOTW5G5SRYIIC/action/replication_record"}},"created_at":"2026-05-18T01:23:04.314586+00:00","updated_at":"2026-05-18T01:23:04.314586+00:00"}