{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:X5H5LUKBT66H5UHN6SIQX3LZOT","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":"c8d42ae66d7af404821e5194fd96a0e5c91e00def7499dea2e96e50d97281892","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-12-18T06:27:38Z","title_canon_sha256":"1c760f150dbc636ea92f40315ca8cbbb123a490bfa1e8950419ad7fb0561d764"},"schema_version":"1.0","source":{"id":"1512.05859","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.05859","created_at":"2026-05-18T01:24:06Z"},{"alias_kind":"arxiv_version","alias_value":"1512.05859v1","created_at":"2026-05-18T01:24:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.05859","created_at":"2026-05-18T01:24:06Z"},{"alias_kind":"pith_short_12","alias_value":"X5H5LUKBT66H","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"X5H5LUKBT66H5UHN","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"X5H5LUKB","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:c89f76038fb349adb3579c2b5651c966ef423200536b7b5e7edad7e056401a14","target":"graph","created_at":"2026-05-18T01:24:06Z","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":"We study immersed, connected, umbilic hypersurfaces in the Heisenberg group $H_{n}$ with $n$ $\\geq $ $2.$ We show that such a hypersurface, if closed, must be rotationally invariant up to a Heisenberg translation. Moreover, we prove that, among others, Pansu spheres are the only such spheres with positive constant sigma-k curvature up to Heisenberg translations.","authors_text":"Hung-Lin Chiu, Jenn-Fang Hwang, Jih-Hsin Cheng, Paul Yang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-12-18T06:27:38Z","title":"Umbilic hypersurfaces of constant sigma-k curvature in the Heisenberg group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.05859","kind":"arxiv","version":1},"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":"sha256:29757f0a646bf7635ed9a82ff36971e5262defbf9296c816deddfba4ac8ee6b0","target":"record","created_at":"2026-05-18T01:24:06Z","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":"c8d42ae66d7af404821e5194fd96a0e5c91e00def7499dea2e96e50d97281892","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-12-18T06:27:38Z","title_canon_sha256":"1c760f150dbc636ea92f40315ca8cbbb123a490bfa1e8950419ad7fb0561d764"},"schema_version":"1.0","source":{"id":"1512.05859","kind":"arxiv","version":1}},"canonical_sha256":"bf4fd5d1419fbc7ed0edf4910bed7974cec431c00b13bb41d63da4224cc51c67","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bf4fd5d1419fbc7ed0edf4910bed7974cec431c00b13bb41d63da4224cc51c67","first_computed_at":"2026-05-18T01:24:06.047188Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:24:06.047188Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2dXCZWFx4jY4aY1zH1vuEoN8QKGezdKN9MF5ZsvteQM0W7rK6u2bpB4SUdAfm3mcvYf75tCbgeNxHAGRkYuNDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:24:06.047811Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.05859","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:29757f0a646bf7635ed9a82ff36971e5262defbf9296c816deddfba4ac8ee6b0","sha256:c89f76038fb349adb3579c2b5651c966ef423200536b7b5e7edad7e056401a14"],"state_sha256":"ef21f5abb7a2852e2544255e3123a721450c928b18b8982d2ea644f13a5068c4"}