{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:5A76VCF26MB7PSORQURDZ2FF4K","short_pith_number":"pith:5A76VCF2","canonical_record":{"source":{"id":"1410.3619","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-10-14T09:00:13Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"16e4db52e213080719af4daa2302616eb6acf98d27034449b448759ee83d6ae2","abstract_canon_sha256":"2162a6ed085a0a02d718df561f5d4d5345ba0b1d1a8ab01bef943a50c5feeff9"},"schema_version":"1.0"},"canonical_sha256":"e83fea88baf303f7c9d185223ce8a5e29ba0e72c0ab45bddc0dcd5836c3350b7","source":{"kind":"arxiv","id":"1410.3619","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.3619","created_at":"2026-05-18T01:35:00Z"},{"alias_kind":"arxiv_version","alias_value":"1410.3619v3","created_at":"2026-05-18T01:35:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.3619","created_at":"2026-05-18T01:35:00Z"},{"alias_kind":"pith_short_12","alias_value":"5A76VCF26MB7","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"5A76VCF26MB7PSOR","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"5A76VCF2","created_at":"2026-05-18T12:28:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:5A76VCF26MB7PSORQURDZ2FF4K","target":"record","payload":{"canonical_record":{"source":{"id":"1410.3619","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-10-14T09:00:13Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"16e4db52e213080719af4daa2302616eb6acf98d27034449b448759ee83d6ae2","abstract_canon_sha256":"2162a6ed085a0a02d718df561f5d4d5345ba0b1d1a8ab01bef943a50c5feeff9"},"schema_version":"1.0"},"canonical_sha256":"e83fea88baf303f7c9d185223ce8a5e29ba0e72c0ab45bddc0dcd5836c3350b7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:00.893998Z","signature_b64":"6fU2+0puveyCmCyYsyPrWzM1idlahYAXWEnx916Qiun06Ak5hL3xibh4JkS1/xbUUjAqvoIKfFwx/6fM5uBzAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e83fea88baf303f7c9d185223ce8a5e29ba0e72c0ab45bddc0dcd5836c3350b7","last_reissued_at":"2026-05-18T01:35:00.893287Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:00.893287Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.3619","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:35:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3sDWngkhDXuuZKmQqylPWVGuo1zBZs/8zAi+miX/zs6833ZkBL5N97l8ttJ19W4N49jtbjAkmeeVdyQrnDQ7Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T22:37:53.888377Z"},"content_sha256":"91382159cde2d10ae708c743d63e61316e05d370bcc074747d6b87077d05062e","schema_version":"1.0","event_id":"sha256:91382159cde2d10ae708c743d63e61316e05d370bcc074747d6b87077d05062e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:5A76VCF26MB7PSORQURDZ2FF4K","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Area-stationary and stable surfaces of class $C^1$ in the sub-Riemannian Heisenberg group ${\\mathbb H}^1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Manuel Ritor\\'e, Matteo Galli","submitted_at":"2014-10-14T09:00:13Z","abstract_excerpt":"We consider surfaces of class $C^1$ in the $3$-dimensional sub-Riemannian Heisenberg group ${\\mathbb H}^1$. Assuming the surface is area-stationary, i.e., a critical point of the sub-Riemannian perimeter under compactly supported variations, we show that its regular part is foliated by horizontal straight lines. In case the surface is complete and oriented, without singular points, and stable, i.e., a second order minimum of perimeter, we prove that the surface must be a vertical plane. This implies the following Bernstein type result: a complete locally area-minimizing intrinsic graph of a $C"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.3619","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:35:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5X/XP9g/JrttGihb3+mFH77uhQltzsQ/U85SkGaEwdu3BAYhy34Vb5WX5TZibNtCTtYPe5apWd3V6aNc7t3zAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T22:37:53.888722Z"},"content_sha256":"6437228a2fdf2a0cd7189089ccf28bc1e6a83be9106e94a2bc23ee3e7b9a8ec5","schema_version":"1.0","event_id":"sha256:6437228a2fdf2a0cd7189089ccf28bc1e6a83be9106e94a2bc23ee3e7b9a8ec5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5A76VCF26MB7PSORQURDZ2FF4K/bundle.json","state_url":"https://pith.science/pith/5A76VCF26MB7PSORQURDZ2FF4K/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5A76VCF26MB7PSORQURDZ2FF4K/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-22T22:37:53Z","links":{"resolver":"https://pith.science/pith/5A76VCF26MB7PSORQURDZ2FF4K","bundle":"https://pith.science/pith/5A76VCF26MB7PSORQURDZ2FF4K/bundle.json","state":"https://pith.science/pith/5A76VCF26MB7PSORQURDZ2FF4K/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5A76VCF26MB7PSORQURDZ2FF4K/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:5A76VCF26MB7PSORQURDZ2FF4K","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":"2162a6ed085a0a02d718df561f5d4d5345ba0b1d1a8ab01bef943a50c5feeff9","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-10-14T09:00:13Z","title_canon_sha256":"16e4db52e213080719af4daa2302616eb6acf98d27034449b448759ee83d6ae2"},"schema_version":"1.0","source":{"id":"1410.3619","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.3619","created_at":"2026-05-18T01:35:00Z"},{"alias_kind":"arxiv_version","alias_value":"1410.3619v3","created_at":"2026-05-18T01:35:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.3619","created_at":"2026-05-18T01:35:00Z"},{"alias_kind":"pith_short_12","alias_value":"5A76VCF26MB7","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"5A76VCF26MB7PSOR","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"5A76VCF2","created_at":"2026-05-18T12:28:14Z"}],"graph_snapshots":[{"event_id":"sha256:6437228a2fdf2a0cd7189089ccf28bc1e6a83be9106e94a2bc23ee3e7b9a8ec5","target":"graph","created_at":"2026-05-18T01:35:00Z","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 consider surfaces of class $C^1$ in the $3$-dimensional sub-Riemannian Heisenberg group ${\\mathbb H}^1$. Assuming the surface is area-stationary, i.e., a critical point of the sub-Riemannian perimeter under compactly supported variations, we show that its regular part is foliated by horizontal straight lines. In case the surface is complete and oriented, without singular points, and stable, i.e., a second order minimum of perimeter, we prove that the surface must be a vertical plane. This implies the following Bernstein type result: a complete locally area-minimizing intrinsic graph of a $C","authors_text":"Manuel Ritor\\'e, Matteo Galli","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-10-14T09:00:13Z","title":"Area-stationary and stable surfaces of class $C^1$ in the sub-Riemannian Heisenberg group ${\\mathbb H}^1$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.3619","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":"sha256:91382159cde2d10ae708c743d63e61316e05d370bcc074747d6b87077d05062e","target":"record","created_at":"2026-05-18T01:35:00Z","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":"2162a6ed085a0a02d718df561f5d4d5345ba0b1d1a8ab01bef943a50c5feeff9","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-10-14T09:00:13Z","title_canon_sha256":"16e4db52e213080719af4daa2302616eb6acf98d27034449b448759ee83d6ae2"},"schema_version":"1.0","source":{"id":"1410.3619","kind":"arxiv","version":3}},"canonical_sha256":"e83fea88baf303f7c9d185223ce8a5e29ba0e72c0ab45bddc0dcd5836c3350b7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e83fea88baf303f7c9d185223ce8a5e29ba0e72c0ab45bddc0dcd5836c3350b7","first_computed_at":"2026-05-18T01:35:00.893287Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:35:00.893287Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6fU2+0puveyCmCyYsyPrWzM1idlahYAXWEnx916Qiun06Ak5hL3xibh4JkS1/xbUUjAqvoIKfFwx/6fM5uBzAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:35:00.893998Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.3619","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:91382159cde2d10ae708c743d63e61316e05d370bcc074747d6b87077d05062e","sha256:6437228a2fdf2a0cd7189089ccf28bc1e6a83be9106e94a2bc23ee3e7b9a8ec5"],"state_sha256":"66f1985f36afa8b29bdc2b384ebdc3cad2e3a93b6e220fad270cd194021f9c64"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"npn3yWea8vLBx8NDM6GmxiI8mWKnBjTY0EhDs5Ep6auj96pmQUU+FPH9ggh2IJY3TiHpuJHNdzhgnYT/HQsWAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T22:37:53.890603Z","bundle_sha256":"d673974f4ae57c76cefa6cdd8c5d130861cdf8302993c1b609271e9865eae189"}}