{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:O4ASSSWN2XSG46AXWKA7LZQEU4","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":"38abc895d4f29c7d1fd4d7132f7106cba4e726ed99576b61aba88a592e5c9117","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-05-29T08:37:08Z","title_canon_sha256":"424f1626c3763436d230364a3d2c5707a7a584c83ad2d0d5cd06723105d162b8"},"schema_version":"1.0","source":{"id":"1705.10071","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.10071","created_at":"2026-05-18T00:40:57Z"},{"alias_kind":"arxiv_version","alias_value":"1705.10071v2","created_at":"2026-05-18T00:40:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.10071","created_at":"2026-05-18T00:40:57Z"},{"alias_kind":"pith_short_12","alias_value":"O4ASSSWN2XSG","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_16","alias_value":"O4ASSSWN2XSG46AX","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_8","alias_value":"O4ASSSWN","created_at":"2026-05-18T12:31:34Z"}],"graph_snapshots":[{"event_id":"sha256:51d928ed7d78190042190a7f717ad2ce5e00fa27540901b08c7386afa0b9c9a0","target":"graph","created_at":"2026-05-18T00:40:57Z","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":"Under the usual condition that the volume of a geodesic ball is close to the Euclidean one or the injectivity radii is bounded from below, we prove a lower bound of the $C^{\\alpha} W^{1, q}$ harmonic radius for manifolds with bounded Bakry-\\'Emery Ricci curvature when the gradient of the potential is bounded. Under these conditions, the regularity that can be imposed on the metrics under harmonic coordinates is only $C^\\alpha W^{1,q}$, where $q>2n$ and $n$ is the dimension of the manifolds. This is almost 1 order lower than that in the classical $C^{1,\\alpha} W^{2, p}$ harmonic coordinates und","authors_text":"Meng Zhu, Qi S Zhang","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-05-29T08:37:08Z","title":"Bounds on harmonic radius and limits of manifolds with bounded Bakry-\\'Emery Ricci curvature"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.10071","kind":"arxiv","version":2},"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":"sha256:9cb8fbea509d18c55ef0af929546ec4c4c8c036bb767b52f4d1044b78278458b","target":"record","created_at":"2026-05-18T00:40:57Z","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":"38abc895d4f29c7d1fd4d7132f7106cba4e726ed99576b61aba88a592e5c9117","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-05-29T08:37:08Z","title_canon_sha256":"424f1626c3763436d230364a3d2c5707a7a584c83ad2d0d5cd06723105d162b8"},"schema_version":"1.0","source":{"id":"1705.10071","kind":"arxiv","version":2}},"canonical_sha256":"7701294acdd5e46e7817b281f5e604a73953322d889db0ed99935a6d25183ce6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7701294acdd5e46e7817b281f5e604a73953322d889db0ed99935a6d25183ce6","first_computed_at":"2026-05-18T00:40:57.527329Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:40:57.527329Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FoDVG8ode7JMF/2en4Ko9IJ1anJzwhC3JNw3qSiNvJuNo1Vh+YN/tiwYrAHsPBDysl2rJGtcSgFFlNO+DSP7BA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:40:57.527743Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.10071","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9cb8fbea509d18c55ef0af929546ec4c4c8c036bb767b52f4d1044b78278458b","sha256:51d928ed7d78190042190a7f717ad2ce5e00fa27540901b08c7386afa0b9c9a0"],"state_sha256":"573c4b37b1528f4acf37aaf0240586791efa1651b083e072d8abafda62cb5b46"}