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Assume that $(M,g_0)$ is a complete non-compact K\\\"ahler manifold such that there is a potential function $f$ of the Ricci tensor, i.e., $$ R_{i\\bar{j}}(g_0)=f_{i\\bar{j}}. $$ Assume that the quantity $|f|_{C^0}+|\\nabla_{g_0}f|_{C^0}$ is finite and the L2 Sobolev inequality holds true on $(M,g_0)$. Then the Kahler-Ricci flow with the initial metric $g_0$ either blows up at finite time or infinite time to Ricci flat metric or exists globally with Ricci-flat limit at "},"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":"1209.5063","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-09-23T12:34:18Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"b2ed95f8d5b7c2f24fb7c90e8ecfce9bb90ba1f2f0885f03c43904bbd15efe88","abstract_canon_sha256":"30a15ee8b036ac1e787e442787d00fc9281e3573a523854b8df446842fa6e760"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:01.535832Z","signature_b64":"eBMQQHxeYg44FO262W4yQWOGZnYL0LYI4kHv6ELkkcI3qPQ58ETIdszVuvTvgfrXUz9KzuXi94TVvd1kcZpdCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab04d49fb7d7e75c4314446a278ace519aee7288b986eacc9747a01b629b0dfb","last_reissued_at":"2026-05-18T01:32:01.535203Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:01.535203Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Global Kahler-Ricci Flow on Complete Non-Compact Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Li Ma","submitted_at":"2012-09-23T12:34:18Z","abstract_excerpt":"In this paper, we study the global K\\\"ahler-Ricci flow on a complete non-compact K\\\"ahler manifold. We prove the following result. Assume that $(M,g_0)$ is a complete non-compact K\\\"ahler manifold such that there is a potential function $f$ of the Ricci tensor, i.e., $$ R_{i\\bar{j}}(g_0)=f_{i\\bar{j}}. $$ Assume that the quantity $|f|_{C^0}+|\\nabla_{g_0}f|_{C^0}$ is finite and the L2 Sobolev inequality holds true on $(M,g_0)$. Then the Kahler-Ricci flow with the initial metric $g_0$ either blows up at finite time or infinite time to Ricci flat metric or exists globally with Ricci-flat limit at "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.5063","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":"1209.5063","created_at":"2026-05-18T01:32:01.535300+00:00"},{"alias_kind":"arxiv_version","alias_value":"1209.5063v3","created_at":"2026-05-18T01:32:01.535300+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.5063","created_at":"2026-05-18T01:32:01.535300+00:00"},{"alias_kind":"pith_short_12","alias_value":"VMCNJH5X27TV","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_16","alias_value":"VMCNJH5X27TVYQYU","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_8","alias_value":"VMCNJH5X","created_at":"2026-05-18T12:27:25.539911+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/VMCNJH5X27TVYQYUIRVCPCWOKG","json":"https://pith.science/pith/VMCNJH5X27TVYQYUIRVCPCWOKG.json","graph_json":"https://pith.science/api/pith-number/VMCNJH5X27TVYQYUIRVCPCWOKG/graph.json","events_json":"https://pith.science/api/pith-number/VMCNJH5X27TVYQYUIRVCPCWOKG/events.json","paper":"https://pith.science/paper/VMCNJH5X"},"agent_actions":{"view_html":"https://pith.science/pith/VMCNJH5X27TVYQYUIRVCPCWOKG","download_json":"https://pith.science/pith/VMCNJH5X27TVYQYUIRVCPCWOKG.json","view_paper":"https://pith.science/paper/VMCNJH5X","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1209.5063&json=true","fetch_graph":"https://pith.science/api/pith-number/VMCNJH5X27TVYQYUIRVCPCWOKG/graph.json","fetch_events":"https://pith.science/api/pith-number/VMCNJH5X27TVYQYUIRVCPCWOKG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VMCNJH5X27TVYQYUIRVCPCWOKG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VMCNJH5X27TVYQYUIRVCPCWOKG/action/storage_attestation","attest_author":"https://pith.science/pith/VMCNJH5X27TVYQYUIRVCPCWOKG/action/author_attestation","sign_citation":"https://pith.science/pith/VMCNJH5X27TVYQYUIRVCPCWOKG/action/citation_signature","submit_replication":"https://pith.science/pith/VMCNJH5X27TVYQYUIRVCPCWOKG/action/replication_record"}},"created_at":"2026-05-18T01:32:01.535300+00:00","updated_at":"2026-05-18T01:32:01.535300+00:00"}