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We show that the K\\\"ahler angle $\\alpha(t)$ satisfies the evolution equation: $$ (\\frac{\\partial}{\\partial t}-\\Delta)\\cos\\alpha=|\\overline\\nabla J_{\\Sigma_t}|^2\\cos\\alpha+R\\sin^2\\alpha\\cos\\alpha, $$ where $R$ is the scalar curvature of $(M, \\overline{g}(t))$.\n  The equation implies that, if"},"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":"1105.1200","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-05-06T01:06:41Z","cross_cats_sorted":[],"title_canon_sha256":"4722db9e0d5073c6b1372d6cefc1b0d87ae695e721f5ff62b9dc48a374827d0a","abstract_canon_sha256":"32f1d222d3bfc44374742999333cfa8c763b4b992e2b858ef439c63b8b48f15a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:22:40.896903Z","signature_b64":"ad63f7U4ye/ygh9MsVkpokWfeGEAY9A5/aXtsGKNd98N0mo/BwbhO6jbPT6KpBiMcOf7sz4fnyy9wucvV9uxCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"49eaac3793a0a7218810ee87a3fe5229d8d1221410c4d77e709a30994d55f99f","last_reissued_at":"2026-05-18T04:22:40.896487Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:22:40.896487Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The mean curvature flow along the K\\\"ahler-Ricci flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jiayu Li, Xiaoli Han","submitted_at":"2011-05-06T01:06:41Z","abstract_excerpt":"Let $(M,\\overline{g})$ be a K\\\"ahler surface, and $\\Sigma$ an immersed surface in $M$. The K\\\"ahler angle of $\\Sigma$ in $M$ is introduced by Chern-Wolfson \\cite{CW}. Let $(M,\\overline{g}(t))$ evolve along the K\\\"ahler-Ricci flow, and $\\Sigma_t$ in $(M,\\overline{g}(t))$ evolve along the mean curvature flow. We show that the K\\\"ahler angle $\\alpha(t)$ satisfies the evolution equation: $$ (\\frac{\\partial}{\\partial t}-\\Delta)\\cos\\alpha=|\\overline\\nabla J_{\\Sigma_t}|^2\\cos\\alpha+R\\sin^2\\alpha\\cos\\alpha, $$ where $R$ is the scalar curvature of $(M, \\overline{g}(t))$.\n  The equation implies that, if"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.1200","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":"1105.1200","created_at":"2026-05-18T04:22:40.896543+00:00"},{"alias_kind":"arxiv_version","alias_value":"1105.1200v1","created_at":"2026-05-18T04:22:40.896543+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.1200","created_at":"2026-05-18T04:22:40.896543+00:00"},{"alias_kind":"pith_short_12","alias_value":"JHVKYN4TUCTS","created_at":"2026-05-18T12:26:32.869790+00:00"},{"alias_kind":"pith_short_16","alias_value":"JHVKYN4TUCTSDCAQ","created_at":"2026-05-18T12:26:32.869790+00:00"},{"alias_kind":"pith_short_8","alias_value":"JHVKYN4T","created_at":"2026-05-18T12:26:32.869790+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/JHVKYN4TUCTSDCAQ52D2H7SSFH","json":"https://pith.science/pith/JHVKYN4TUCTSDCAQ52D2H7SSFH.json","graph_json":"https://pith.science/api/pith-number/JHVKYN4TUCTSDCAQ52D2H7SSFH/graph.json","events_json":"https://pith.science/api/pith-number/JHVKYN4TUCTSDCAQ52D2H7SSFH/events.json","paper":"https://pith.science/paper/JHVKYN4T"},"agent_actions":{"view_html":"https://pith.science/pith/JHVKYN4TUCTSDCAQ52D2H7SSFH","download_json":"https://pith.science/pith/JHVKYN4TUCTSDCAQ52D2H7SSFH.json","view_paper":"https://pith.science/paper/JHVKYN4T","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1105.1200&json=true","fetch_graph":"https://pith.science/api/pith-number/JHVKYN4TUCTSDCAQ52D2H7SSFH/graph.json","fetch_events":"https://pith.science/api/pith-number/JHVKYN4TUCTSDCAQ52D2H7SSFH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JHVKYN4TUCTSDCAQ52D2H7SSFH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JHVKYN4TUCTSDCAQ52D2H7SSFH/action/storage_attestation","attest_author":"https://pith.science/pith/JHVKYN4TUCTSDCAQ52D2H7SSFH/action/author_attestation","sign_citation":"https://pith.science/pith/JHVKYN4TUCTSDCAQ52D2H7SSFH/action/citation_signature","submit_replication":"https://pith.science/pith/JHVKYN4TUCTSDCAQ52D2H7SSFH/action/replication_record"}},"created_at":"2026-05-18T04:22:40.896543+00:00","updated_at":"2026-05-18T04:22:40.896543+00:00"}