{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:KLNWUH7CF2RXPFVPMCSF4NI56L","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":"3a4239ebd73ac58636aa3ab4568a19f11b8a171c0fc4e1df93342f1c89f56bc6","cross_cats_sorted":["math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-12-13T12:13:20Z","title_canon_sha256":"d7484f6935cea75c4e61adf3782913ecbe73d32ec02dab8611d125369346966a"},"schema_version":"1.0","source":{"id":"1612.04119","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.04119","created_at":"2026-07-05T00:56:21Z"},{"alias_kind":"arxiv_version","alias_value":"1612.04119v1","created_at":"2026-07-05T00:56:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.04119","created_at":"2026-07-05T00:56:21Z"},{"alias_kind":"pith_short_12","alias_value":"KLNWUH7CF2RX","created_at":"2026-07-05T00:56:21Z"},{"alias_kind":"pith_short_16","alias_value":"KLNWUH7CF2RXPFVP","created_at":"2026-07-05T00:56:21Z"},{"alias_kind":"pith_short_8","alias_value":"KLNWUH7C","created_at":"2026-07-05T00:56:21Z"}],"graph_snapshots":[{"event_id":"sha256:169dcd4cc1374803f638e06b7d9f1eab53ad18890fca5412405da2906e8d7796","target":"graph","created_at":"2026-07-05T00:56:21Z","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/1612.04119/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Levy-Gromov inequality states that round spheres have the least isoperimetric profile (normalized by total volume) among Riemannian manifolds with a fixed positive lower bound on the Ricci tensor. In this note we study critical metrics corresponding to the Levy-Gromov inequality and prove that, in two-dimensions, this criticality condition is quite rigid, as it characterizes round spheres and projective planes.","authors_text":"Andrea Mondino, Fabio Cavalletti, Francesco Maggi","cross_cats":["math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-12-13T12:13:20Z","title":"Rigidity for critical points in the Levy-Gromov inequality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.04119","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":"sha256:6deddfc1e9b578614b7ffd0af89235df371499b9010aca8e6bf52e2ea2f29c1e","target":"record","created_at":"2026-07-05T00:56:21Z","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":"3a4239ebd73ac58636aa3ab4568a19f11b8a171c0fc4e1df93342f1c89f56bc6","cross_cats_sorted":["math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-12-13T12:13:20Z","title_canon_sha256":"d7484f6935cea75c4e61adf3782913ecbe73d32ec02dab8611d125369346966a"},"schema_version":"1.0","source":{"id":"1612.04119","kind":"arxiv","version":1}},"canonical_sha256":"52db6a1fe22ea37796af60a45e351df2c8ccff026447e0e2e9318552a19ac1b7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"52db6a1fe22ea37796af60a45e351df2c8ccff026447e0e2e9318552a19ac1b7","first_computed_at":"2026-07-05T00:56:21.972525Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:56:21.972525Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KsREQJjK+G+Vp8lJx9Jp+OuO/DSWadpLtVeZCxDLDa00BNIS/Es5fiSZ7a68hxcpGWJMbNEsHFTDtkO+T7k/AQ==","signature_status":"signed_v1","signed_at":"2026-07-05T00:56:21.972990Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.04119","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6deddfc1e9b578614b7ffd0af89235df371499b9010aca8e6bf52e2ea2f29c1e","sha256:169dcd4cc1374803f638e06b7d9f1eab53ad18890fca5412405da2906e8d7796"],"state_sha256":"e21bf049998cbd8b34a61f139e7d42eac202fb4232dcb4ff01d8de673d440827"}