{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:T3RE6R2LYAWD4YUGG6M7KXLHTU","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"368688b3638e140012e74825e88d1779731e56b2175c330573c2371516c1ae18","cross_cats_sorted":["math.GT","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-09-28T07:43:36Z","title_canon_sha256":"0c93b47ed05f6b2d93248284a57e4c304000a352e7da82868dfbbd39bf9ba9cb"},"schema_version":"1.0","source":{"id":"0909.4981","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0909.4981","created_at":"2026-05-18T00:13:14Z"},{"alias_kind":"arxiv_version","alias_value":"0909.4981v3","created_at":"2026-05-18T00:13:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.4981","created_at":"2026-05-18T00:13:14Z"},{"alias_kind":"pith_short_12","alias_value":"T3RE6R2LYAWD","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"T3RE6R2LYAWD4YUG","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"T3RE6R2L","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:f150377bf9949b46e0a1f90c89a1838ba4ec2230eebbbf68dfdc5eb7983340db","target":"graph","created_at":"2026-05-18T00:13:14Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper, we show a local energy convexity of $W^{1,2}$ maps into $CAT(K)$ spaces. This energy convexity allows us to extend Colding and Minicozzi's width-sweepout construction to produce closed geodesics in any closed Alexandrov space of curvature bounded from above, which also provides a generalized version of the Birkhoff-Lyusternik theorem on the existence of non-trivial closed geodesics in the Alexandrov setting.","authors_text":"Longzhi Lin","cross_cats":["math.GT","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-09-28T07:43:36Z","title":"Closed geodesics in Alexandrov spaces of curvature bounded from above"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.4981","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:e6368e1bb0c415dfaa8324e5bb1ec8b714a5c6873da220190b2df9ba57ab76cb","target":"record","created_at":"2026-05-18T00:13:14Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"368688b3638e140012e74825e88d1779731e56b2175c330573c2371516c1ae18","cross_cats_sorted":["math.GT","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-09-28T07:43:36Z","title_canon_sha256":"0c93b47ed05f6b2d93248284a57e4c304000a352e7da82868dfbbd39bf9ba9cb"},"schema_version":"1.0","source":{"id":"0909.4981","kind":"arxiv","version":3}},"canonical_sha256":"9ee24f474bc02c3e62863799f55d679d33771bfaa36d7247ef4f15afbc56848b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9ee24f474bc02c3e62863799f55d679d33771bfaa36d7247ef4f15afbc56848b","first_computed_at":"2026-05-18T00:13:14.288959Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:13:14.288959Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NbBn2UtOvW5MQPSF38emH1sFRKL/AIA/9GfA8hRR/Ym0EPOPcIjl6RE0skD5Wb/bTKLA2x2EBcm7Qm+MfcOGCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:13:14.289597Z","signed_message":"canonical_sha256_bytes"},"source_id":"0909.4981","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e6368e1bb0c415dfaa8324e5bb1ec8b714a5c6873da220190b2df9ba57ab76cb","sha256:f150377bf9949b46e0a1f90c89a1838ba4ec2230eebbbf68dfdc5eb7983340db"],"state_sha256":"dbe1826567b8b33270887f3a78a3c140a77ce07540ad0f568e20d0ff0f17f5da"}