{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:DBV2FVIIHL3LNN2M455R4EGFQQ","short_pith_number":"pith:DBV2FVII","schema_version":"1.0","canonical_sha256":"186ba2d5083af6b6b74ce77b1e10c58405048de8f4669eaf438e7cb4f42c6c3f","source":{"kind":"arxiv","id":"1210.8076","version":1},"attestation_state":"computed","paper":{"title":"Stability and compactness for complete $f$-minimal surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Detang Zhou, Tito Mejia, Xu Cheng","submitted_at":"2012-10-30T16:42:09Z","abstract_excerpt":"Let $(M,\\bar{g}, e^{-f}d\\mu)$ be a complete metric measure space with Bakry-\\'Emery Ricci curvature bounded below by a positive constant. We prove that, in $M$, there is no complete two-sided $L_f$-stable immersed $f$-minimal hypersurface with finite weighted volume. Further, if $M$ is a 3-manifold, we prove a smooth compactness theorem for the space of complete embedded $f$-minimal surfaces in $M$ with the uniform upper bounds of genus and weighted volume, which generalizes the compactness theorem for complete self-shrinkers in $\\mathbb{R}^3$ by Colding-Minicozzi."},"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":"1210.8076","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-10-30T16:42:09Z","cross_cats_sorted":[],"title_canon_sha256":"e807e01d49a9ad5049c69e7fba9f8e4e68ea42ce9e5b73d3ea622f5b34d73eed","abstract_canon_sha256":"59768df1f1e6f4ac6798b4acf679ffd3d6b8559cce9c14cdea97c8f019261eaf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:57.124707Z","signature_b64":"pRFLnUad0m5DFDDM05ScN8NsVLxcmHBVaR4Des20Eilm2vLJOh+xQdkTaHI27bpHX9EJ2UR6WORe0itwttgkDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"186ba2d5083af6b6b74ce77b1e10c58405048de8f4669eaf438e7cb4f42c6c3f","last_reissued_at":"2026-05-18T03:41:57.123964Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:57.123964Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stability and compactness for complete $f$-minimal surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Detang Zhou, Tito Mejia, Xu Cheng","submitted_at":"2012-10-30T16:42:09Z","abstract_excerpt":"Let $(M,\\bar{g}, e^{-f}d\\mu)$ be a complete metric measure space with Bakry-\\'Emery Ricci curvature bounded below by a positive constant. We prove that, in $M$, there is no complete two-sided $L_f$-stable immersed $f$-minimal hypersurface with finite weighted volume. Further, if $M$ is a 3-manifold, we prove a smooth compactness theorem for the space of complete embedded $f$-minimal surfaces in $M$ with the uniform upper bounds of genus and weighted volume, which generalizes the compactness theorem for complete self-shrinkers in $\\mathbb{R}^3$ by Colding-Minicozzi."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.8076","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1210.8076","created_at":"2026-05-18T03:41:57.124085+00:00"},{"alias_kind":"arxiv_version","alias_value":"1210.8076v1","created_at":"2026-05-18T03:41:57.124085+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.8076","created_at":"2026-05-18T03:41:57.124085+00:00"},{"alias_kind":"pith_short_12","alias_value":"DBV2FVIIHL3L","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_16","alias_value":"DBV2FVIIHL3LNN2M","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_8","alias_value":"DBV2FVII","created_at":"2026-05-18T12:27:01.376967+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/DBV2FVIIHL3LNN2M455R4EGFQQ","json":"https://pith.science/pith/DBV2FVIIHL3LNN2M455R4EGFQQ.json","graph_json":"https://pith.science/api/pith-number/DBV2FVIIHL3LNN2M455R4EGFQQ/graph.json","events_json":"https://pith.science/api/pith-number/DBV2FVIIHL3LNN2M455R4EGFQQ/events.json","paper":"https://pith.science/paper/DBV2FVII"},"agent_actions":{"view_html":"https://pith.science/pith/DBV2FVIIHL3LNN2M455R4EGFQQ","download_json":"https://pith.science/pith/DBV2FVIIHL3LNN2M455R4EGFQQ.json","view_paper":"https://pith.science/paper/DBV2FVII","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1210.8076&json=true","fetch_graph":"https://pith.science/api/pith-number/DBV2FVIIHL3LNN2M455R4EGFQQ/graph.json","fetch_events":"https://pith.science/api/pith-number/DBV2FVIIHL3LNN2M455R4EGFQQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DBV2FVIIHL3LNN2M455R4EGFQQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DBV2FVIIHL3LNN2M455R4EGFQQ/action/storage_attestation","attest_author":"https://pith.science/pith/DBV2FVIIHL3LNN2M455R4EGFQQ/action/author_attestation","sign_citation":"https://pith.science/pith/DBV2FVIIHL3LNN2M455R4EGFQQ/action/citation_signature","submit_replication":"https://pith.science/pith/DBV2FVIIHL3LNN2M455R4EGFQQ/action/replication_record"}},"created_at":"2026-05-18T03:41:57.124085+00:00","updated_at":"2026-05-18T03:41:57.124085+00:00"}