{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:FLYRQLYPNCQS5YR5Q4ITXPKR7Y","short_pith_number":"pith:FLYRQLYP","canonical_record":{"source":{"id":"1005.1515","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-05-10T12:26:02Z","cross_cats_sorted":[],"title_canon_sha256":"b075fd943f7ce06a554d66db4c8f2dda722755e3bcf17432ccd418351c5443a7","abstract_canon_sha256":"b7c726fded52b77369defb77c4fd27374325e0b5078b6f5f7c6e87e95227a438"},"schema_version":"1.0"},"canonical_sha256":"2af1182f0f68a12ee23d87113bbd51fe1ad19206f63a429bccc0b52ef6877526","source":{"kind":"arxiv","id":"1005.1515","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.1515","created_at":"2026-05-18T03:41:40Z"},{"alias_kind":"arxiv_version","alias_value":"1005.1515v2","created_at":"2026-05-18T03:41:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.1515","created_at":"2026-05-18T03:41:40Z"},{"alias_kind":"pith_short_12","alias_value":"FLYRQLYPNCQS","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"FLYRQLYPNCQS5YR5","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"FLYRQLYP","created_at":"2026-05-18T12:26:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:FLYRQLYPNCQS5YR5Q4ITXPKR7Y","target":"record","payload":{"canonical_record":{"source":{"id":"1005.1515","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-05-10T12:26:02Z","cross_cats_sorted":[],"title_canon_sha256":"b075fd943f7ce06a554d66db4c8f2dda722755e3bcf17432ccd418351c5443a7","abstract_canon_sha256":"b7c726fded52b77369defb77c4fd27374325e0b5078b6f5f7c6e87e95227a438"},"schema_version":"1.0"},"canonical_sha256":"2af1182f0f68a12ee23d87113bbd51fe1ad19206f63a429bccc0b52ef6877526","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:40.561928Z","signature_b64":"Uvcl0slUAG5FcKa8B7BbGExTEbQ+DJATcqYkBAoXuAR2hAzYkM3+1hfwyDR2E3r6QZ5LAijm70y2A2GuxvD6Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2af1182f0f68a12ee23d87113bbd51fe1ad19206f63a429bccc0b52ef6877526","last_reissued_at":"2026-05-18T03:41:40.561019Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:40.561019Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1005.1515","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:41:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qSN+L9k/M9Zl90m3CQFwkD6cd6mUTi0zVrm7EAtcdTkB4Dz+tPwkpTAfE70qEr1xInWvTNXtc8xsiByl6rrQAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T19:26:56.076234Z"},"content_sha256":"fb57f75762350bb79f4a4ce88016b28e575a3abffe3a5d1acba6f707d4a365ea","schema_version":"1.0","event_id":"sha256:fb57f75762350bb79f4a4ce88016b28e575a3abffe3a5d1acba6f707d4a365ea"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:FLYRQLYPNCQS5YR5Q4ITXPKR7Y","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Convexity of the smallest principal curvature of the convex level sets of some quasi-linear elliptic equations with respect to the height","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kun Huang, Wei Zhang","submitted_at":"2010-05-10T12:26:02Z","abstract_excerpt":"For the $p$-harmonic function with strictly convex level sets, we find a test function which comes from the combination of the norm of gradient of the $p$-harmonic function and the smallest principal curvature of the level sets of $p$-harmonic function. We prove that this curvature function is convex with respect to the height of the $p$-harmonic function. This test function is an affine function of the height when the $p$-harmonic function is the $p$-Green function on the ball. For the minimal graph, we obtain a similar results."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.1515","kind":"arxiv","version":2},"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":"2026-05-18T03:41:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"686qiE4mSt7j5AxjXBB+J5SBGTnmPEucNrMF1WKeJ/NBH22KVzkuzsmuW7C8m2o1NMTsS5bf5+uErPjpHCcGCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T19:26:56.076971Z"},"content_sha256":"5a9ea5b9d4a8317133d42f753647b527cc61021824ae48ce91ff02c40befc121","schema_version":"1.0","event_id":"sha256:5a9ea5b9d4a8317133d42f753647b527cc61021824ae48ce91ff02c40befc121"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FLYRQLYPNCQS5YR5Q4ITXPKR7Y/bundle.json","state_url":"https://pith.science/pith/FLYRQLYPNCQS5YR5Q4ITXPKR7Y/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FLYRQLYPNCQS5YR5Q4ITXPKR7Y/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":"2026-06-08T19:26:56Z","links":{"resolver":"https://pith.science/pith/FLYRQLYPNCQS5YR5Q4ITXPKR7Y","bundle":"https://pith.science/pith/FLYRQLYPNCQS5YR5Q4ITXPKR7Y/bundle.json","state":"https://pith.science/pith/FLYRQLYPNCQS5YR5Q4ITXPKR7Y/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FLYRQLYPNCQS5YR5Q4ITXPKR7Y/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:FLYRQLYPNCQS5YR5Q4ITXPKR7Y","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":"b7c726fded52b77369defb77c4fd27374325e0b5078b6f5f7c6e87e95227a438","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-05-10T12:26:02Z","title_canon_sha256":"b075fd943f7ce06a554d66db4c8f2dda722755e3bcf17432ccd418351c5443a7"},"schema_version":"1.0","source":{"id":"1005.1515","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.1515","created_at":"2026-05-18T03:41:40Z"},{"alias_kind":"arxiv_version","alias_value":"1005.1515v2","created_at":"2026-05-18T03:41:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.1515","created_at":"2026-05-18T03:41:40Z"},{"alias_kind":"pith_short_12","alias_value":"FLYRQLYPNCQS","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"FLYRQLYPNCQS5YR5","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"FLYRQLYP","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:5a9ea5b9d4a8317133d42f753647b527cc61021824ae48ce91ff02c40befc121","target":"graph","created_at":"2026-05-18T03:41:40Z","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":"For the $p$-harmonic function with strictly convex level sets, we find a test function which comes from the combination of the norm of gradient of the $p$-harmonic function and the smallest principal curvature of the level sets of $p$-harmonic function. We prove that this curvature function is convex with respect to the height of the $p$-harmonic function. This test function is an affine function of the height when the $p$-harmonic function is the $p$-Green function on the ball. For the minimal graph, we obtain a similar results.","authors_text":"Kun Huang, Wei Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-05-10T12:26:02Z","title":"Convexity of the smallest principal curvature of the convex level sets of some quasi-linear elliptic equations with respect to the height"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.1515","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:fb57f75762350bb79f4a4ce88016b28e575a3abffe3a5d1acba6f707d4a365ea","target":"record","created_at":"2026-05-18T03:41:40Z","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":"b7c726fded52b77369defb77c4fd27374325e0b5078b6f5f7c6e87e95227a438","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-05-10T12:26:02Z","title_canon_sha256":"b075fd943f7ce06a554d66db4c8f2dda722755e3bcf17432ccd418351c5443a7"},"schema_version":"1.0","source":{"id":"1005.1515","kind":"arxiv","version":2}},"canonical_sha256":"2af1182f0f68a12ee23d87113bbd51fe1ad19206f63a429bccc0b52ef6877526","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2af1182f0f68a12ee23d87113bbd51fe1ad19206f63a429bccc0b52ef6877526","first_computed_at":"2026-05-18T03:41:40.561019Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:41:40.561019Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Uvcl0slUAG5FcKa8B7BbGExTEbQ+DJATcqYkBAoXuAR2hAzYkM3+1hfwyDR2E3r6QZ5LAijm70y2A2GuxvD6Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:41:40.561928Z","signed_message":"canonical_sha256_bytes"},"source_id":"1005.1515","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fb57f75762350bb79f4a4ce88016b28e575a3abffe3a5d1acba6f707d4a365ea","sha256:5a9ea5b9d4a8317133d42f753647b527cc61021824ae48ce91ff02c40befc121"],"state_sha256":"4089f445505bd44cc01c3e285f97624061223db21a910f7ded185884b75efd38"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MioVtZB+7odtPaCzR80HNU6mQXx1o+bDmvjEQzPjKjSsdXdjFvUQMrNlMrDBGE4FF/j2O932fn7EBSYkPuShAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T19:26:56.080883Z","bundle_sha256":"3ed64eb125edecd01811ab578131c2804de2d631955f8abf1e372a5cee680396"}}