{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:VN6MM3FWUWFLQPXHGCV34IWL7C","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":"b6a149dc9ab5be122daf7b7c6f6fa4d139c5c8047e4938d9cda21e90a7f14cb8","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-10-26T08:18:28Z","title_canon_sha256":"2a19f5012134322ab7c54e41080f8d50c6edeef82ca0c8fd8362bfeea9fdd108"},"schema_version":"1.0","source":{"id":"1310.7082","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.7082","created_at":"2026-05-17T23:46:56Z"},{"alias_kind":"arxiv_version","alias_value":"1310.7082v1","created_at":"2026-05-17T23:46:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.7082","created_at":"2026-05-17T23:46:56Z"},{"alias_kind":"pith_short_12","alias_value":"VN6MM3FWUWFL","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_16","alias_value":"VN6MM3FWUWFLQPXH","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_8","alias_value":"VN6MM3FW","created_at":"2026-05-18T12:28:04Z"}],"graph_snapshots":[{"event_id":"sha256:90d387ee95e516a17f7f37cc9d26c6d940d322d31284e208f7fb4d732cbe2524","target":"graph","created_at":"2026-05-17T23:46:56Z","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":"Given a 3-dimensional Riemannian manifold $(M,g)$, we prove that if $(\\Phi_k)$ is a sequence of Willmore spheres (or more generally area-constrained Willmore spheres), having Willmore energy bounded above uniformly strictly by $8 \\pi$, and Hausdorff converging to a point $\\bar{p}\\in M$, then $Scal(\\bar{p})=0$ and $\\nabla Scal(\\bar{p})=0$ (resp. $\\nabla Scal(\\bar{p})=0$). Moreover, a suitably rescaled sequence smoothly converges, up to subsequences and reparametrizations, to a round sphere in the euclidean 3-dimensional space. This generalizes previous results of Lamm and Metzger contained in \\","authors_text":"Andrea Mondino, Paul Laurain","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-10-26T08:18:28Z","title":"Concentration of small Willmore spheres in Riemannian 3-manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.7082","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":"sha256:39ebd24b6256de51bc353baa743972547091e605e5bcf810e41b6e6e7c0bd39a","target":"record","created_at":"2026-05-17T23:46:56Z","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":"b6a149dc9ab5be122daf7b7c6f6fa4d139c5c8047e4938d9cda21e90a7f14cb8","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-10-26T08:18:28Z","title_canon_sha256":"2a19f5012134322ab7c54e41080f8d50c6edeef82ca0c8fd8362bfeea9fdd108"},"schema_version":"1.0","source":{"id":"1310.7082","kind":"arxiv","version":1}},"canonical_sha256":"ab7cc66cb6a58ab83ee730abbe22cbf8b386c489712ff136f8ce0d66967644e3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ab7cc66cb6a58ab83ee730abbe22cbf8b386c489712ff136f8ce0d66967644e3","first_computed_at":"2026-05-17T23:46:56.047081Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:46:56.047081Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dngTcpPDK7P4rQGXetiZf0XBlOENNzin868QRjjw3DQJp6QumUbU79qZSVUboSDcQ/fwKsnrwGxx12VWD3gMBg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:46:56.047681Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.7082","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:39ebd24b6256de51bc353baa743972547091e605e5bcf810e41b6e6e7c0bd39a","sha256:90d387ee95e516a17f7f37cc9d26c6d940d322d31284e208f7fb4d732cbe2524"],"state_sha256":"d0868444eb38c3067c4cfe895b28c309ac2ed98cc1d8c29477054ada2821b5fe"}