{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:EU4IPV2RXPDQBWL6B22LJZ74IF","short_pith_number":"pith:EU4IPV2R","canonical_record":{"source":{"id":"2410.16384","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-10-21T18:01:02Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"21a450fb7e57852523151e924354012a8e437fa21e436bd060537681b192c59f","abstract_canon_sha256":"a89c94347f63aa57df5336621142fb97bb6fa8231672ec4ad5ee03348e3b5b08"},"schema_version":"1.0"},"canonical_sha256":"253887d751bbc700d97e0eb4b4e7fc41408a9bb474b527010fab8a3d22144cd3","source":{"kind":"arxiv","id":"2410.16384","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2410.16384","created_at":"2026-07-05T09:23:35Z"},{"alias_kind":"arxiv_version","alias_value":"2410.16384v1","created_at":"2026-07-05T09:23:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.16384","created_at":"2026-07-05T09:23:35Z"},{"alias_kind":"pith_short_12","alias_value":"EU4IPV2RXPDQ","created_at":"2026-07-05T09:23:35Z"},{"alias_kind":"pith_short_16","alias_value":"EU4IPV2RXPDQBWL6","created_at":"2026-07-05T09:23:35Z"},{"alias_kind":"pith_short_8","alias_value":"EU4IPV2R","created_at":"2026-07-05T09:23:35Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:EU4IPV2RXPDQBWL6B22LJZ74IF","target":"record","payload":{"canonical_record":{"source":{"id":"2410.16384","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-10-21T18:01:02Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"21a450fb7e57852523151e924354012a8e437fa21e436bd060537681b192c59f","abstract_canon_sha256":"a89c94347f63aa57df5336621142fb97bb6fa8231672ec4ad5ee03348e3b5b08"},"schema_version":"1.0"},"canonical_sha256":"253887d751bbc700d97e0eb4b4e7fc41408a9bb474b527010fab8a3d22144cd3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:23:35.275353Z","signature_b64":"wKxQ1W7JfI/lcHnAEhLgzox9sY3taokq/YDATTNYuTihqEk/9YSBMOUk3EswFrXlMCQ+tNUhqfjKcZRrDsMiDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"253887d751bbc700d97e0eb4b4e7fc41408a9bb474b527010fab8a3d22144cd3","last_reissued_at":"2026-07-05T09:23:35.274862Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:23:35.274862Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2410.16384","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:23:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IwlyA9sSya7SgK5X214Cf1YGOohrjCwiIzeqv1BE2XHLYNzd+WiDqrQlUasn48Nn0IN4FDTcK9orU8oZxAVqAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T18:55:08.602287Z"},"content_sha256":"dc7078c0fc70c9590eba1538386bc51727e500e50a1d323f70de3c33279c5170","schema_version":"1.0","event_id":"sha256:dc7078c0fc70c9590eba1538386bc51727e500e50a1d323f70de3c33279c5170"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:EU4IPV2RXPDQBWL6B22LJZ74IF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Liouville Theorem with Boundary Conditions from Chern--Gauss--Bonnet Formula","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"BaoZhi Chu, Yanyan Li, Zongyuan Li","submitted_at":"2024-10-21T18:01:02Z","abstract_excerpt":"The $\\sigma_k(A_g)$ curvature and the boundary $\\mathcal{B}^k_g$ curvature arise naturally from the Chern--Gauss--Bonnet formula for manifolds with boundary. In this paper, we prove a Liouville theorem for the equation $\\sigma_k(A_g)=1$ in $\\overline{\\mathbb{R}^n_+}$ with the boundary condition $\\mathcal{B}^k_g=c$ on $\\partial\\mathbb{R}^n_+$, where $g=e^{2v}|dx|^2$ and $c$ is some nonnegative constant. This extends an earlier result of Wei, which assumes the existence of $\\lim_{|x|\\to\\infty}(v(x)+2\\log|x|)$. In addition, we establish a local gradient estimate for solutions of such equations, a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.16384","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2410.16384/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T09:23:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ok4+in2sQ8cf98TCHifjI6Yjl24JQYnuitG+YtGPx3jFJDWs2yAQFfybKwenA5p1CTtupIGV+u8lDIM7dMxyBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T18:55:08.603010Z"},"content_sha256":"c10335afb8cb5512e16db1ef84b1cf6a8e9ba6644973a41fe9c6691a4f982375","schema_version":"1.0","event_id":"sha256:c10335afb8cb5512e16db1ef84b1cf6a8e9ba6644973a41fe9c6691a4f982375"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EU4IPV2RXPDQBWL6B22LJZ74IF/bundle.json","state_url":"https://pith.science/pith/EU4IPV2RXPDQBWL6B22LJZ74IF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EU4IPV2RXPDQBWL6B22LJZ74IF/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":"2026-08-08T18:55:08Z","links":{"resolver":"https://pith.science/pith/EU4IPV2RXPDQBWL6B22LJZ74IF","bundle":"https://pith.science/pith/EU4IPV2RXPDQBWL6B22LJZ74IF/bundle.json","state":"https://pith.science/pith/EU4IPV2RXPDQBWL6B22LJZ74IF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EU4IPV2RXPDQBWL6B22LJZ74IF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:EU4IPV2RXPDQBWL6B22LJZ74IF","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":"a89c94347f63aa57df5336621142fb97bb6fa8231672ec4ad5ee03348e3b5b08","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-10-21T18:01:02Z","title_canon_sha256":"21a450fb7e57852523151e924354012a8e437fa21e436bd060537681b192c59f"},"schema_version":"1.0","source":{"id":"2410.16384","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2410.16384","created_at":"2026-07-05T09:23:35Z"},{"alias_kind":"arxiv_version","alias_value":"2410.16384v1","created_at":"2026-07-05T09:23:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.16384","created_at":"2026-07-05T09:23:35Z"},{"alias_kind":"pith_short_12","alias_value":"EU4IPV2RXPDQ","created_at":"2026-07-05T09:23:35Z"},{"alias_kind":"pith_short_16","alias_value":"EU4IPV2RXPDQBWL6","created_at":"2026-07-05T09:23:35Z"},{"alias_kind":"pith_short_8","alias_value":"EU4IPV2R","created_at":"2026-07-05T09:23:35Z"}],"graph_snapshots":[{"event_id":"sha256:c10335afb8cb5512e16db1ef84b1cf6a8e9ba6644973a41fe9c6691a4f982375","target":"graph","created_at":"2026-07-05T09:23:35Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2410.16384/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The $\\sigma_k(A_g)$ curvature and the boundary $\\mathcal{B}^k_g$ curvature arise naturally from the Chern--Gauss--Bonnet formula for manifolds with boundary. In this paper, we prove a Liouville theorem for the equation $\\sigma_k(A_g)=1$ in $\\overline{\\mathbb{R}^n_+}$ with the boundary condition $\\mathcal{B}^k_g=c$ on $\\partial\\mathbb{R}^n_+$, where $g=e^{2v}|dx|^2$ and $c$ is some nonnegative constant. This extends an earlier result of Wei, which assumes the existence of $\\lim_{|x|\\to\\infty}(v(x)+2\\log|x|)$. In addition, we establish a local gradient estimate for solutions of such equations, a","authors_text":"BaoZhi Chu, Yanyan Li, Zongyuan Li","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-10-21T18:01:02Z","title":"Liouville Theorem with Boundary Conditions from Chern--Gauss--Bonnet Formula"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.16384","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:dc7078c0fc70c9590eba1538386bc51727e500e50a1d323f70de3c33279c5170","target":"record","created_at":"2026-07-05T09:23:35Z","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":"a89c94347f63aa57df5336621142fb97bb6fa8231672ec4ad5ee03348e3b5b08","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-10-21T18:01:02Z","title_canon_sha256":"21a450fb7e57852523151e924354012a8e437fa21e436bd060537681b192c59f"},"schema_version":"1.0","source":{"id":"2410.16384","kind":"arxiv","version":1}},"canonical_sha256":"253887d751bbc700d97e0eb4b4e7fc41408a9bb474b527010fab8a3d22144cd3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"253887d751bbc700d97e0eb4b4e7fc41408a9bb474b527010fab8a3d22144cd3","first_computed_at":"2026-07-05T09:23:35.274862Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:23:35.274862Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wKxQ1W7JfI/lcHnAEhLgzox9sY3taokq/YDATTNYuTihqEk/9YSBMOUk3EswFrXlMCQ+tNUhqfjKcZRrDsMiDw==","signature_status":"signed_v1","signed_at":"2026-07-05T09:23:35.275353Z","signed_message":"canonical_sha256_bytes"},"source_id":"2410.16384","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dc7078c0fc70c9590eba1538386bc51727e500e50a1d323f70de3c33279c5170","sha256:c10335afb8cb5512e16db1ef84b1cf6a8e9ba6644973a41fe9c6691a4f982375"],"state_sha256":"5a6e50f35074f5a34b17563b78ef2e82cff01764f2c55dc351de585924d3f3d0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VURM33bMtUYdvyGdOw65YF2xkkRxYCEpE+306rwdcTmCOymOyrbrbbRMAQfEcxgkjsSnTYxK7v/ObdwlDxEvDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T18:55:08.606237Z","bundle_sha256":"057b1b92b343922582db68889a5b084b39e8302069d82dc3e69eda91ac7ae7c6"}}