{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:Z4OD27ZO7WW3FWV56UKNW2LZFM","short_pith_number":"pith:Z4OD27ZO","canonical_record":{"source":{"id":"1612.08512","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-12-27T06:47:05Z","cross_cats_sorted":[],"title_canon_sha256":"3d12335e42184bd047d4a53f04955930bba9328c75c6ee95e11ee8e4f67a2543","abstract_canon_sha256":"426ee3d14efafc8743b98b25f26608baf112b06a6adcd152b528e789e79c0522"},"schema_version":"1.0"},"canonical_sha256":"cf1c3d7f2efdadb2dabdf514db69792b042320c59401a161afe5a45e6b45c1d4","source":{"kind":"arxiv","id":"1612.08512","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.08512","created_at":"2026-05-18T00:49:36Z"},{"alias_kind":"arxiv_version","alias_value":"1612.08512v2","created_at":"2026-05-18T00:49:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.08512","created_at":"2026-05-18T00:49:36Z"},{"alias_kind":"pith_short_12","alias_value":"Z4OD27ZO7WW3","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_16","alias_value":"Z4OD27ZO7WW3FWV5","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_8","alias_value":"Z4OD27ZO","created_at":"2026-05-18T12:30:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:Z4OD27ZO7WW3FWV56UKNW2LZFM","target":"record","payload":{"canonical_record":{"source":{"id":"1612.08512","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-12-27T06:47:05Z","cross_cats_sorted":[],"title_canon_sha256":"3d12335e42184bd047d4a53f04955930bba9328c75c6ee95e11ee8e4f67a2543","abstract_canon_sha256":"426ee3d14efafc8743b98b25f26608baf112b06a6adcd152b528e789e79c0522"},"schema_version":"1.0"},"canonical_sha256":"cf1c3d7f2efdadb2dabdf514db69792b042320c59401a161afe5a45e6b45c1d4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:36.323146Z","signature_b64":"LkE3O6RAG6SJT0zB/t+NJ//XKcUEM8pMjuK7rkHzF91+X1d6VNwjM2EWk751U0CYcKMoDKA32prvu1yAH6mFDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cf1c3d7f2efdadb2dabdf514db69792b042320c59401a161afe5a45e6b45c1d4","last_reissued_at":"2026-05-18T00:49:36.322475Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:36.322475Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1612.08512","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":"2026-05-18T00:49:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bAEnaKwVCfI6I5NK6urYWz4ASpyd/72yOLzQZ/v/JOnfSbvA2VbQvC6JuV0w1mRgN0m9uDYBlwuoodyCtYIAAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T20:29:45.264660Z"},"content_sha256":"3fd7a3d3232236f0da743b64fa795865b7bce7e7df72d3a5c681a6b38573ce2b","schema_version":"1.0","event_id":"sha256:3fd7a3d3232236f0da743b64fa795865b7bce7e7df72d3a5c681a6b38573ce2b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:Z4OD27ZO7WW3FWV56UKNW2LZFM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Integral pinched gradient shrinking $\\rho$-Einstein solitons","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Guangyue Huang","submitted_at":"2016-12-27T06:47:05Z","abstract_excerpt":"The gradient shrinking $\\rho$-Einstein soliton is a triple $(M^n,g,f)$ such that $$R_{ij}+f_{ij}=(\\rho R+\\lambda) g_{ij},$$ where $(M^n,g)$ is a Riemannian manifold, $\\lambda>0, \\rho\\in\\mathbb{R}\\setminus\\{0\\}$ and $f$ is the potential function on $M^n$. In this paper, using algebraic curvature estimates and the Yamabe-Sobolev inequality, we prove some integral pinching rigidity results for compact gradient shrinking $\\rho$-Einstein solitons."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.08512","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":"2026-05-18T00:49:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kUx9euPGXflQhanlFk6R1gC94YDMSGASG2lIH489gdaUECDP/6NYWd7zPH+BKW/fH59JvLbYAhOTAxPmwSWFCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T20:29:45.265001Z"},"content_sha256":"cd1eab0d96226065480edcfd294a648a148e5e01c8baea926aeeedf9f99c7c5d","schema_version":"1.0","event_id":"sha256:cd1eab0d96226065480edcfd294a648a148e5e01c8baea926aeeedf9f99c7c5d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Z4OD27ZO7WW3FWV56UKNW2LZFM/bundle.json","state_url":"https://pith.science/pith/Z4OD27ZO7WW3FWV56UKNW2LZFM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Z4OD27ZO7WW3FWV56UKNW2LZFM/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":"2026-06-25T20:29:45Z","links":{"resolver":"https://pith.science/pith/Z4OD27ZO7WW3FWV56UKNW2LZFM","bundle":"https://pith.science/pith/Z4OD27ZO7WW3FWV56UKNW2LZFM/bundle.json","state":"https://pith.science/pith/Z4OD27ZO7WW3FWV56UKNW2LZFM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Z4OD27ZO7WW3FWV56UKNW2LZFM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:Z4OD27ZO7WW3FWV56UKNW2LZFM","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":"426ee3d14efafc8743b98b25f26608baf112b06a6adcd152b528e789e79c0522","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-12-27T06:47:05Z","title_canon_sha256":"3d12335e42184bd047d4a53f04955930bba9328c75c6ee95e11ee8e4f67a2543"},"schema_version":"1.0","source":{"id":"1612.08512","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.08512","created_at":"2026-05-18T00:49:36Z"},{"alias_kind":"arxiv_version","alias_value":"1612.08512v2","created_at":"2026-05-18T00:49:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.08512","created_at":"2026-05-18T00:49:36Z"},{"alias_kind":"pith_short_12","alias_value":"Z4OD27ZO7WW3","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_16","alias_value":"Z4OD27ZO7WW3FWV5","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_8","alias_value":"Z4OD27ZO","created_at":"2026-05-18T12:30:53Z"}],"graph_snapshots":[{"event_id":"sha256:cd1eab0d96226065480edcfd294a648a148e5e01c8baea926aeeedf9f99c7c5d","target":"graph","created_at":"2026-05-18T00:49:36Z","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":"The gradient shrinking $\\rho$-Einstein soliton is a triple $(M^n,g,f)$ such that $$R_{ij}+f_{ij}=(\\rho R+\\lambda) g_{ij},$$ where $(M^n,g)$ is a Riemannian manifold, $\\lambda>0, \\rho\\in\\mathbb{R}\\setminus\\{0\\}$ and $f$ is the potential function on $M^n$. In this paper, using algebraic curvature estimates and the Yamabe-Sobolev inequality, we prove some integral pinching rigidity results for compact gradient shrinking $\\rho$-Einstein solitons.","authors_text":"Guangyue Huang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-12-27T06:47:05Z","title":"Integral pinched gradient shrinking $\\rho$-Einstein solitons"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.08512","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":"sha256:3fd7a3d3232236f0da743b64fa795865b7bce7e7df72d3a5c681a6b38573ce2b","target":"record","created_at":"2026-05-18T00:49:36Z","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":"426ee3d14efafc8743b98b25f26608baf112b06a6adcd152b528e789e79c0522","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-12-27T06:47:05Z","title_canon_sha256":"3d12335e42184bd047d4a53f04955930bba9328c75c6ee95e11ee8e4f67a2543"},"schema_version":"1.0","source":{"id":"1612.08512","kind":"arxiv","version":2}},"canonical_sha256":"cf1c3d7f2efdadb2dabdf514db69792b042320c59401a161afe5a45e6b45c1d4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cf1c3d7f2efdadb2dabdf514db69792b042320c59401a161afe5a45e6b45c1d4","first_computed_at":"2026-05-18T00:49:36.322475Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:49:36.322475Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LkE3O6RAG6SJT0zB/t+NJ//XKcUEM8pMjuK7rkHzF91+X1d6VNwjM2EWk751U0CYcKMoDKA32prvu1yAH6mFDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:49:36.323146Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.08512","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3fd7a3d3232236f0da743b64fa795865b7bce7e7df72d3a5c681a6b38573ce2b","sha256:cd1eab0d96226065480edcfd294a648a148e5e01c8baea926aeeedf9f99c7c5d"],"state_sha256":"de730578797dfda6ef708c0b82b9ac14b8923dc90951feaa45d1ab8e41ad259e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"weTMUNR7B+BRuhUObSrG2YEz/u2lHDerov9pWva4v5b2lR8r2rhWEE5bZkffuc2ILB3WicrBIXgyu0hsbyd2Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T20:29:45.266837Z","bundle_sha256":"5dd9994400325a6328ac3d6e6035fbc71d6baf773fa85b0cdc41b19d4aeb7a2c"}}