{"as_of":"2026-08-07T17:39:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:ce909790b2950abfc4e139ee464afe4398832560105ae1a329964b296c1efd2a","coverage":[{"denominator":99,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":99,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-04T15:45:09.023079Z","state":"measured"},{"denominator":99,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":99,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-07T06:34:17.273281+00:00","state":"measured"},{"denominator":0,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":0,"source":"paper_references, paper_reference_links","source_observed_at":null,"state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"cited_works","source_observed_at":null,"state":"measured"}],"external_citation_measurements":[],"inbound":[],"links":{"evidence":"/evidence","html":"/paper/2509.18618/citation-record","integrity":"/paper/2509.18618/integrity","json":"/paper/2509.18618/citation-record.json","paper":"/paper/2509.18618"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.640546Z","title":"Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.640546Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:e4fb2a7ffdf455eb7d95822eb98a30bdc093a7dd09e6a76921cced95a27d33e4","observation_id":"d76709ea-ad59-44d0-8ad4-ff79eb00b97e","resolution":{"observed_at":"2026-08-04T15:45:08.640546Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.645295Z","title":"Uniqueness theorems for surfaces in the large. I","venue":null,"work_id":null,"year":1956},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.645295Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:81200ecfd40656c63faaefa46a6ba9f9686d31e4fe99c55dd298eaeedde37089","observation_id":"8f80419f-96b8-49e9-b077-997f8b712561","resolution":{"observed_at":"2026-08-04T15:45:08.645295Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.649102Z","title":"Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces","venue":null,"work_id":null,"year":2001},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.649102Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:6de2c4219190f340b5a0749de6bf1d2e93b4790144c9ee430ad47ee0ae769673","observation_id":"19f21cd9-db33-436e-815e-c156b4697694","resolution":{"observed_at":"2026-08-04T15:45:08.649102Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.652483Z","title":"Calculus, heat flow and curvature-dimension bounds in metric mea- sure spaces","venue":null,"work_id":null,"year":2018},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.652483Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:e85de30483d47f7584c5786f1439029aabac50984eefb39314b3e649d3c1b9f2","observation_id":"65f3efa0-b5ef-4735-8545-f9c17da6ca08","resolution":{"observed_at":"2026-08-04T15:45:08.652483Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.656068Z","title":"Ambrosio, E","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.656068Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:29f3babe1fd7e0a8352c757fa1a906e05578439ceae4a64d079b5db0b7494da0","observation_id":"2d734195-5482-4cfe-8405-d74b1473cc2b","resolution":{"observed_at":"2026-08-04T15:45:08.656068Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.659806Z","title":"Rigidity of the 1-Bakry- ´Emery inequality and sets of finite perimeter in RCD spaces","venue":null,"work_id":null,"year":2019},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.659806Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:eb7efa4fe1b22f584dc3f3ca5bbd702e3412be68104eb8a7812d2ed9860dc51d","observation_id":"b17ab89b-83ef-434f-a860-2306dc041d9f","resolution":{"observed_at":"2026-08-04T15:45:08.659806Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1016/j.na.2016.03.010","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Perimeter as relaxed Minkowski content in metric measure spaces","venue":"Nonlinear Analysis","work_id":"89bbdf62-d4ad-42db-9097-fdb16042075b","year":2017},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.663426Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:63d8ca0b1b36498b60db0972d61a3c98a849ef019b4e23161cef38910a391bae","observation_id":"caf6263c-ab60-4eff-9daf-acc03b2da3a2","resolution":{"observed_at":"2026-08-04T15:51:50.843990Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.667013Z","title":"Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below","venue":null,"work_id":null,"year":2014},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.667013Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:df973f0e4f3baf5bb5a583e59b89b12c3878d5d23f5ef4c86acb402e936dbeb2","observation_id":"33f8cdf0-c0c4-4fe3-b0a9-659171f7eda4","resolution":{"observed_at":"2026-08-04T15:45:08.667013Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.670328Z","title":"Metric measure spaces with Riemannian Ricci curvature bounded from below","venue":null,"work_id":null,"year":2014},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.670328Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:e42e8861e0443fcdbba59bf2b8d0936d23db1f2fe60881fdb275e94b6c8fb869","observation_id":"26ee8103-0b41-4fbf-8c3b-0e3e43700f9e","resolution":{"observed_at":"2026-08-04T15:45:08.670328Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.678423Z","title":"The rank-one theorem on RCD spaces","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.678423Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:3e55e531ac6622effcb9674816c3b60b35084e1eb281cbeca530d3db84963cb6","observation_id":"3b4c1584-878a-4537-b479-6cd97b654236","resolution":{"observed_at":"2026-08-04T15:45:08.678423Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.682197Z","title":"On the existence of isoperi- metric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.682197Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:0b5d6e203725d6f8c53c67e5dab98ca7c66fbef0f329be1d23c537da9c3dec44","observation_id":"4a69964f-a751-4931-bd85-f76e207c86cc","resolution":{"observed_at":"2026-08-04T15:45:08.682197Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1142/s0219199722500687","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"The isoperimetric problem on Rie- mannian manifolds via Gromov-Hausdorff asymptotic analysis","venue":"Communications in Contemporary Mathematics","work_id":"48905dba-c884-479c-b928-b6b6ba7190cc","year":2024},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.689929Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:a8b2d196384826772c910bab0c0e891028085e8d7474af1daf53e819e65677de","observation_id":"87eaed61-d94a-43b5-a266-50692a6a06c6","resolution":{"observed_at":"2026-08-04T15:51:50.808912Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1515/crelle-2024-0032","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Nonexistence of isoperimetric sets in spaces of pos- itive curvature","venue":"Journal für die reine und angewandte Mathematik (Crelles Journal)","work_id":"5496c5a7-9182-49d2-b449-d20651ad3bf1","year":2024},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.693497Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:21538c33daedd3298c49fd0f71d28f3e961f53547574a8e9ace86009c1bc6beb","observation_id":"b2a09197-66fb-4d23-b8b2-3e9bf2394e64","resolution":{"observed_at":"2026-08-04T15:51:50.794513Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.696730Z","title":"The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.696730Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:84c4bcc32e5af537c03f59123cae22ee90f5f67039c13f503903209deb3b4296","observation_id":"7132acc1-eb25-4391-9111-1cfa34afd164","resolution":{"observed_at":"2026-08-04T15:45:08.696730Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.700053Z","title":"Isoperimetric sets in spaces with lower bounds on the Ricci curvature","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.700053Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:7e91403cffeb019c3e9459771d0b4f5c553135add36551a436991cc02085cff6","observation_id":"196517c3-022d-468c-a29f-3f6d8249a188","resolution":{"observed_at":"2026-08-04T15:45:08.700053Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1007/s00208-023-02674-y","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds","venue":"Mathematische Annalen","work_id":"90a7500f-16da-4f14-b2ca-6edb12c0d15c","year":2024},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.703321Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:82ac0953d9e2f3a3f8d86d9e21ae44a235e744ea69e7dd19d04d7413316cf4e6","observation_id":"d6c621ec-501b-4aa0-92d9-ba526fa0795d","resolution":{"observed_at":"2026-08-04T15:51:50.780456Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.706909Z","title":"Sharp isoperimetric comparison on non-collapsed spaces with lower Ricci bounds","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.706909Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:1f1e87e96b5f822f18d4d505907d003bf143c61ae11fdd8b1fe41057eb4ee67f","observation_id":"d0de2644-640e-472b-87a8-b873dd32b880","resolution":{"observed_at":"2026-08-04T15:45:08.706909Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1017/prm.2023.105","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Topological regularity of isoperimetric sets in PI spaces having a deformation property","venue":"Proceedings of the Royal Society of Edinburgh Section A Mathematics","work_id":"f1eefbb2-d34d-4fd9-b34e-5f3d910b5f54","year":2025},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.709876Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:0c6c3269ed92f082cd3a9a73fbedab896b268c5f3c3e70e8e4aec493588995d6","observation_id":"ca482517-5b55-4688-9461-a8b0d2c1c035","resolution":{"observed_at":"2026-08-04T15:51:50.767067Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.713561Z","title":"Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.713561Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:fd2598f8e252f1d960a2a11c23714a643ba4b742c79ca875c8ad02a2ed3bff20","observation_id":"2f708819-bf2a-4927-99c5-e94816b8eeb5","resolution":{"observed_at":"2026-08-04T15:45:08.713561Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.717426Z","title":"A note on the stable Bernstein theorem","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.717426Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:c1b20cc2386d31596e85e10dd0240f2e74cdc27dfa95074d55c6748bb778b40d","observation_id":"b312fe5b-34a3-49b1-a38a-896eade0c118","resolution":{"observed_at":"2026-08-04T15:45:08.717426Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2405.08918","last_updated":"2026-07-11T09:25:50Z","snapshot_observed_at":"2026-08-07T13:50:24.758187Z","submitted_at":"2024-05-14T19:15:17Z","title":"New spectral Bishop-Gromov and Bonnet-Myers theorems and applications to isoperimetry","version":4},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2405.08918","snapshot_observed_at":"2026-08-04T15:45:08.722495Z","title":"New spectral Bishop–Gromov and Bonnet–Myers theorems and applications to isoperimetry","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.722495Z"},"links":{"cited_paper":"/paper/2405.08918","citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:ca28b50e500d45ac045dc5ca34594d8f7877c31c252779cc9b106fba30c59f84","observation_id":"cd921834-53e6-41a3-a458-374a949df393","resolution":{"observed_at":"2026-08-04T15:45:08.722495Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.726875Z","title":"Ballmann, M","venue":null,"work_id":null,"year":1985},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.726875Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:2d188c64dc3e753c7aff94c8db7d610e7af2231418cba05660091236253ea943","observation_id":"17458a5c-6a0b-4176-8e29-666fe5882ae8","resolution":{"observed_at":"2026-08-04T15:45:08.726875Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1007/s00208-022-02380-1","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature","venue":"Mathematische Annalen","work_id":"5494ae69-75bc-484c-92f6-b69cbd50e27e","year":2023},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.730371Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:d98437dd6fd9f280ed7428d73fac0b37116f3d9e22ce2fa1a6bf54bd4b46a8e2","observation_id":"249c4de0-5c68-4af8-aaad-816d58b44120","resolution":{"observed_at":"2026-08-04T15:51:50.747162Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.733995Z","title":"Stability of hypersurfaces with constant mean curvature","venue":null,"work_id":null,"year":1984},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.733995Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:70499e620bcc17a5375fac8c92bd69c63e165fa35716188ddd9e17668c6b8359","observation_id":"d5ba28e0-7a9b-43fe-8086-04a03e4044fd","resolution":{"observed_at":"2026-08-04T15:45:08.733995Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.737697Z","title":"Stability of hypersurfaces of con- stant mean curvature in Riemannian manifolds","venue":null,"work_id":null,"year":1988},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.737697Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:ab1d574b6cffbb32de66ca56769df5d6be461704de1a6c4b644f5c268059c73c","observation_id":"d6d449cd-27ae-440b-a635-4314d20d0b8a","resolution":{"observed_at":"2026-08-04T15:45:08.737697Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.747342Z","title":"Sur le volume minimal de R2","venue":null,"work_id":null,"year":1986},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.747342Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:3c2e190f2e2ff44207d45688a18cdd4e39df4b5c45aa3bff53ff6fbdfb4b1e99","observation_id":"5aceb8c6-813e-47fe-bfde-68fdcc8b3d2e","resolution":{"observed_at":"2026-08-04T15:45:08.747342Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.751335Z","title":"A differential inequality for the isoperimetric profile","venue":null,"work_id":null,"year":2004},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.751335Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:f5efb06c4f12be545ec9cce1f29b0a99f0bd35b47547ee7b333faec561d19d0c","observation_id":"a79fdd4f-7c0b-4949-b6d8-ccad1daf14f4","resolution":{"observed_at":"2026-08-04T15:45:08.751335Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.755087Z","title":"Extensions of Schoen-Simon-Yau and Schoen-Simon theorems via iteration ` a la De Giorgi","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.755087Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:558389c145282f1458c37b7e5702b8809e2c4b597ec4e9961bc15a60f0473173","observation_id":"9782e985-62c4-4e0d-b780-f494f74ccb78","resolution":{"observed_at":"2026-08-04T15:45:08.755087Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.759146Z","title":"Minimal cones and the Bernstein prob- lem","venue":null,"work_id":null,"year":1969},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.759146Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:fbbe80c739355ba3853410abe92150a2b1faed6b6414ed647a3e26e039948339","observation_id":"d62f53d0-0c86-4f1a-b132-156413f6922b","resolution":{"observed_at":"2026-08-04T15:45:08.759146Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.762982Z","title":null,"venue":null,"work_id":null,"year":1997},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.762982Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:6389dda276cd4db45a394d4f3eb3235beeaa1893c297e00854c55c9c46481be5","observation_id":"c0fd1f24-cffa-43b9-8d8f-670ef10b0b71","resolution":{"observed_at":"2026-08-04T15:45:08.762982Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.766264Z","title":"The isoperimetric inequality for a minimal submanifold in Euclidean space","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.766264Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:ee7c3b4556f70710e57efd9deb3e02fe5136fdc741d9e815d7ac1c23d213299d","observation_id":"51533a03-c4ff-46c9-b408-350a712ddefe","resolution":{"observed_at":"2026-08-04T15:45:08.766264Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.769701Z","title":"Sobolev inequalities in manifolds with nonnegative curvature","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.769701Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:282fcec84fe4b9d9daf92b0b149bd6a43df330a4263740ddfeaea98914fa92e2","observation_id":"9846f508-f946-4da5-b3e9-a0f8ed0d65ab","resolution":{"observed_at":"2026-08-04T15:45:08.769701Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.773258Z","title":"Constancy of the dimension in codimension one and locality of the unit normal on RCD( K, N) spaces","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":33,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.773258Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:66d303a3ac23a38babcc29741c5db9539c62322e52ff393505eb6e59568d19a4","observation_id":"db248afc-7188-4de2-a711-972c518a5f49","resolution":{"observed_at":"2026-08-04T15:45:08.773258Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.4171/jems/1217","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Rectifiability of the reduced boundary for sets of finite perimeter over RCD( K, N) spaces","venue":"Journal of the European Mathematical Society","work_id":"d3e270cd-f866-4b0e-9508-caf85d8d0e29","year":2023},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":34,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.776828Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:f31f11f745d504ba9e08c17b79ba7839c8f7655eb8a3b8a0a5aa320a06b8f3d6","observation_id":"0f472d70-0208-4b11-bd88-e146c3590aa0","resolution":{"observed_at":"2026-08-04T15:51:50.687949Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.780431Z","title":"Burago, Y","venue":null,"work_id":null,"year":2001},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":35,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.780431Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:fc0318291c6c72b1e7a9924f0c9b6be51a29498c60ea2a1e7f8f34c6ccc8f8af","observation_id":"2802f280-a5d6-4426-a5bb-2ab7668768d4","resolution":{"observed_at":"2026-08-04T15:45:08.780431Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.783978Z","title":"An elementary regularity theory of minimal sur- faces","venue":null,"work_id":null,"year":1993},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":36,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.783978Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:f2d8715bfcf5e3a6a447be2013c0c94e3f3010d988e430a7addc3a83e144494e","observation_id":"5c0e32c5-62ee-430e-8b5e-a297771e477c","resolution":{"observed_at":"2026-08-04T15:45:08.783978Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.787319Z","title":"Stable complete minimal surfaces in R3 are planes","venue":null,"work_id":null,"year":1979},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":37,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.787319Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:d95817598159999f5c082b1303c4498cf47794547f895bd57b0875f693821066","observation_id":"bb1447a1-ac4f-416a-9cd7-751e8f24a1c9","resolution":{"observed_at":"2026-08-04T15:45:08.787319Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.794121Z","title":"Two rigidity results for stable minimal hypersurfaces","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":38,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.794121Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:b44110b410b2145715146ef3cdcbb464300956259c940aa4dbe926f44c7fe69a","observation_id":"5e772987-f684-4afb-af4a-e86bfbe1f53f","resolution":{"observed_at":"2026-08-04T15:45:08.794121Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1090/proc/15945","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Isoperimetric inequality in noncompact MCP spaces","venue":"Proceedings of the American Mathematical Society","work_id":"d0c36578-4cf6-4a77-b808-4be6ab7dec52","year":2022},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":39,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.800825Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:e1132f6874e64304067820783cc55404c1de8f7698e1991c335be5e26c0bcc0a","observation_id":"2543eb98-add6-44e7-b286-90fc324032d4","resolution":{"observed_at":"2026-08-04T15:51:50.636281Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1007/s00526-022-02193-9","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"doi: 10.1007/s00526-022-02193-9","venue":"Calculus of Variations and Partial Differential Equations","work_id":"9e7cc89a-8a3c-42fd-b29c-4607664770cb","year":null},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":40,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.686340Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:894614a9aa4a7df4f5a87437f9be27d2667758065ca3acd90ab008e855eda32f","observation_id":"cbfb330e-cc4a-4a15-a8a5-58df741e78fc","resolution":{"observed_at":"2026-08-04T15:51:50.822535Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.807652Z","title":"The globalization theorem for the curvature-dimension condition","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":41,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.807652Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:f2995b302a2bc024553b253446929bc4a20cc83d448b6b76ce320c2f7abe1ab6","observation_id":"7d9fd539-590d-4824-9008-3e6edc6326e0","resolution":{"observed_at":"2026-08-04T15:45:08.807652Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.810913Z","title":"Sharp and rigid isoperimetric inequalities in metric- measure spaces with lower Ricci curvature bounds","venue":null,"work_id":null,"year":2017},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":42,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.810913Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:83410acf609018fcd805483d8dc42db2fe3b4ad32e35dd54a231c1466dc292b8","observation_id":"4597b9f5-6f6d-41ac-a882-b59bcf338da8","resolution":{"observed_at":"2026-08-04T15:45:08.810913Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.2140/apde.2020.13.2091","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"New formulas for the Laplacian of distance functions and applications","venue":"Analysis & PDE","work_id":"8e955662-dc39-47cb-8f65-75f089b1cb90","year":2020},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":43,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.814061Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:793e1a7a0c87fae0609edda2fa27b9c220e9b4bb8b67833a301244263f6225b1","observation_id":"31b9f1b7-508d-4a2d-a76d-ac9fc9e7938b","resolution":{"observed_at":"2026-08-04T15:51:50.609918Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.817186Z","title":"Differentiability of Lipschitz Functions on Metric Measure Spaces","venue":null,"work_id":null,"year":1999},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":44,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.817186Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:b5a70867004953c7b2b874ab50c25207409821e2e15213d6f0a70bdf919a38e8","observation_id":"d30f2345-fa76-468e-ae17-966922ecfbc5","resolution":{"observed_at":"2026-08-04T15:45:08.817186Z","resolver_source":null,"status":"malformed_identifier"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2207.03423","last_updated":"2024-08-07T15:56:17Z","snapshot_observed_at":"2026-07-06T13:28:54.165063Z","submitted_at":"2022-07-07T16:48:55Z","title":"Rigidities of Isoperimetric inequality under nonnegative Ricci curvature","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2207.03423","snapshot_observed_at":"2026-08-04T15:45:08.804002Z","title":"Rigidities of Isoperimetric inequality under nonnega- tive Ricci curvature","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":45,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.804002Z"},"links":{"cited_paper":"/paper/2207.03423","citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:8699bf4202f4913172717d220e0880718973545fa989b9a93bc7a13854566af7","observation_id":"f7625b24-730e-4dee-8f3e-23552be6537b","resolution":{"observed_at":"2026-08-04T15:45:08.804002Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.824697Z","title":"On the structure of spaces with Ricci curvature bounded below. I","venue":null,"work_id":null,"year":1997},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":46,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.824697Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:5d6db4a969bb76d020118d9d1533037cd78a6a7a1a9efc5ba572b87ca584f25b","observation_id":"9af9c3f8-d82f-4b48-89f8-c9d41a0b758a","resolution":{"observed_at":"2026-08-04T15:45:08.824697Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.828223Z","title":"Stable anisotropic minimal hypersurfaces in R4","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":47,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.828223Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:b6065573f785f02b9a324cb698bd84771bd3e40c853eacea0e7ef90165293b09","observation_id":"e175a8c6-933f-4c71-a700-6c315ad7f006","resolution":{"observed_at":"2026-08-04T15:45:08.828223Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.831742Z","title":"Stable minimal hypersurfaces in R4","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":48,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.831742Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:a2b27e449cab533ba44053251c71e12099944f9660443b6edb15682f3e5772fa","observation_id":"9e6822b2-8f8d-4919-98e0-053299716df9","resolution":{"observed_at":"2026-08-04T15:45:08.831742Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.835110Z","title":"Stable minimal hypersurfaces in R5","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":49,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.835110Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:fff76bb24f40b96475f4be708eb4b6094cb681e8b42717f3491013d835cad13a","observation_id":"78b405a9-5fa2-473c-8f5d-ec129debf7fb","resolution":{"observed_at":"2026-08-04T15:45:08.835110Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.820681Z","title":"Lower bounds on Ricci curvature and the almost rigidity of warped products","venue":null,"work_id":null,"year":1996},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":50,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.820681Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:e3dcc5b9fb4df388634926b9d6e05786f2cb8ec1a86174744701bcdf90ba180f","observation_id":"4d9c5ca7-789e-4c60-a258-a682b9eb86fb","resolution":{"observed_at":"2026-08-04T15:45:08.820681Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1007/s00039-012-0202-7","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Characterization of tangent cones of noncollapsed limits with lower Ricci bounds and applications","venue":"Geometric and Functional Analysis","work_id":"f1338158-4765-479d-a250-f1251738b124","year":2013},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":51,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.848716Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:92e27a1a4ed94bc26c547e4a1d1b48ef183e8c8dec183927ba9dcb2739125795","observation_id":"204a3ffd-eb35-4165-9a9d-8331d3b1c9ce","resolution":{"observed_at":"2026-08-04T15:51:50.557161Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.2307/2000936","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Volumes of small balls on open manifolds: lower bounds and examples","venue":"Transactions of the American Mathematical Society","work_id":"bf2090ae-fb5d-4e20-8a1c-427497e9aac4","year":1988},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":52,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.852226Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:4a1fc99675c6a11129a0775ec0785cddc89cfff372f2a2f3cef7d52d8b8b2c96","observation_id":"032f6242-cdb4-44aa-8c11-f4d017846c0e","resolution":{"observed_at":"2026-08-04T15:51:50.539981Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.855888Z","title":null,"venue":null,"work_id":null,"year":1969},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":53,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.855888Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:590f3cb99dc8003e37c4a8b8057606dc8fb8fb9297bb5dd077159064cb51e6c9","observation_id":"b1e07dcf-ae54-4112-a103-b50a9a543cc3","resolution":{"observed_at":"2026-08-04T15:45:08.855888Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.860414Z","title":"The structure of complete stable minimal sur- faces in 3-manifolds of nonnegative scalar curvature","venue":null,"work_id":null,"year":1980},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":54,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.860414Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:6e766da147bde0b4eec4203af6a6090f7f59a96358340ba64c921236d19001c0","observation_id":"095e5beb-83ad-41f1-9ac9-df33f03f616d","resolution":{"observed_at":"2026-08-04T15:45:08.860414Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.844742Z","title":"Ricci curvature and volume convergence","venue":null,"work_id":null,"year":1997},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":55,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.844742Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:5a3abd56f4ea2c40443be5b579c8e8b20bead9d14fa6ffa82e8f394eb3b59ca0","observation_id":"7001cccd-1f5c-401a-8660-3ee34eb7414f","resolution":{"observed_at":"2026-08-04T15:45:08.844742Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.867588Z","title":"In´ egalit´ es isop´ erim´ etriques et analytiques sur les vari´ et´ es riemanniennes","venue":null,"work_id":null,"year":1986},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":56,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.867588Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:c9e7b764c4ca5a0c830468e1f5e850ec5d6b91732cd40346e9d187182d1c0f4d","observation_id":"b2e00ae9-f380-4750-9459-8229af8feab6","resolution":{"observed_at":"2026-08-04T15:45:08.867588Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.870990Z","title":"On the differential structure of metric measure spaces and applications","venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":57,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.870990Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:586c383533016246792e61cdbb8599c20541aee81230310ce4762f5c3d484f1a","observation_id":"fb4d9222-2d65-4d8f-8fc9-3914cdf8f0a5","resolution":{"observed_at":"2026-08-04T15:45:08.870990Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2306.14604","last_updated":"2023-06-26T11:27:55Z","snapshot_observed_at":"2026-08-03T04:22:15.997530Z","submitted_at":"2023-06-26T11:27:55Z","title":"De Giorgi and Gromov working together","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2306.14604","snapshot_observed_at":"2026-08-04T15:45:08.876380Z","title":null,"venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":58,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.876380Z"},"links":{"cited_paper":"/paper/2306.14604","citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:d1a3f8e9d270b38c43841fecf0bb62f7d304758a6fe5ac00fd0d7a8160651260","observation_id":"257b79bc-4afc-49f3-87f7-8069c6d899c4","resolution":{"observed_at":"2026-08-04T15:45:08.876380Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.880066Z","title":"Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows","venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":59,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.880066Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:a06066686a395de78a0bfec2e76bd290e218890506bd8d3aa17bab7c1c0a54d7","observation_id":"d82bf457-469d-42e8-aec0-3cba34fa7b8b","resolution":{"observed_at":"2026-08-04T15:45:08.880066Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.864163Z","title":"Minimising hulls, p-capacity and isoperimetric in- equality on complete Riemannian manifolds","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":60,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.864163Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:c873970d8c4e57cb862046af8fb109cbdba14e3f5cd733a6c8a0c58e009f5c12","observation_id":"96bb237c-3529-4dce-879c-d97d6b8194dc","resolution":{"observed_at":"2026-08-04T15:45:08.864163Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.4153/cmb-2018-022-9","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"A note about the strong maximum principle on RCD spaces","venue":"Canadian Mathematical Bulletin","work_id":"a733128f-0ff2-44f5-8a29-836d8a5c163c","year":2019},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":61,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.894075Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:bbe2ccf18bd533ec07813e4338f68de5957c99f9440a6eddb6bbd86ab09d5e9b","observation_id":"250421f6-affc-4ccf-bc22-f7f9cc4afbc3","resolution":{"observed_at":"2026-08-04T15:51:50.476483Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.898702Z","title":"Groups of polynomial growth and expanding maps","venue":null,"work_id":null,"year":1981},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":62,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.898702Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:4890ecfc4bf6172bda330b5978f455cd089e2e01b35ca2ed687bc6969114eb5c","observation_id":"e641ed70-bde3-404e-be8e-a4a14a240f6e","resolution":{"observed_at":"2026-08-04T15:45:08.898702Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.912920Z","title":"Needle decompositions in Riemannian geometry","venue":null,"work_id":null,"year":2017},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":63,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.912920Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:3904915ee72e48d2bed576a279165ab6e64631c8d06e9ba2fe7ff7583c73dfc4","observation_id":"e62f9f01-1d3c-4124-9ff6-acd2e8ffa6c7","resolution":{"observed_at":"2026-08-04T15:45:08.912920Z","resolver_source":null,"status":"malformed_identifier"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1090/memo/1354","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Isoperimetric inequalities in un- bounded convex bodies","venue":"Memoirs of the American Mathematical Society","work_id":"fca5dffa-8e35-4cab-9783-0e38cd48cc47","year":2022},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":64,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.922534Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:fdf8f516f277dcbb2b362cb4915bdf88e5b9719c48c5e2f39f6e145275f93640","observation_id":"2e1cdf08-014a-4f24-a4ff-56fae14bad7a","resolution":{"observed_at":"2026-08-04T15:51:50.467392Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1090/proc/16550","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"On the notion of Laplacian bounds on RCD spaces and applications","venue":"Proceedings of the American Mathematical Society","work_id":"81f2e204-4543-4e29-884f-a73a7677d11e","year":2024},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":65,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.883386Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:47ffe46acaefd3378ae450aafd0b9e88920a9f8543ce9e08bfeffac3dac9403d","observation_id":"99d3f972-8f27-45b0-a732-c7487df6953a","resolution":{"observed_at":"2026-08-04T15:51:50.490496Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.929909Z","title":null,"venue":null,"work_id":null,"year":2012},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":66,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.929909Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:ef0db3ba70fa96015b43a9e74e7893a3a29ae095dd771e993c548b7323cbd664","observation_id":"84e4f2ba-bf7d-420b-b45e-3c328b897fed","resolution":{"observed_at":"2026-08-04T15:45:08.929909Z","resolver_source":null,"status":"malformed_identifier"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2405.14676","last_updated":"2024-05-23T15:11:53Z","snapshot_observed_at":"2026-07-06T18:18:43.328203Z","submitted_at":"2024-05-23T15:11:53Z","title":"Stable minimal hypersurfaces in $\\mathbb R^6$","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2405.14676","snapshot_observed_at":"2026-08-04T15:45:08.933082Z","title":"Stable minimal hypersurfaces in R6","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":67,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.933082Z"},"links":{"cited_paper":"/paper/2405.14676","citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:cbc408c051a2f7bd6fcf75ac17a7571524076ce3e8802bdf6e2d2906f88ca9a9","observation_id":"8e669b4c-dc85-457f-8e2e-f83314274d3e","resolution":{"observed_at":"2026-08-04T15:45:08.933082Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.937096Z","title":"A convexity principle for interacting gases","venue":null,"work_id":null,"year":1997},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":68,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.937096Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:4506185a3c4b65541dbb9fa6cfbed787bebfd5baaa13439efa1f25596857a43a","observation_id":"7f2d55bd-6ace-4cb6-a430-7f7a138eeb73","resolution":{"observed_at":"2026-08-04T15:45:08.937096Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.940420Z","title":"Sobolev and mean-value inequalities on generalized submanifolds of Rn","venue":null,"work_id":null,"year":1973},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":69,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.940420Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:36f9299cdc34ef90a988497f434ddf9c7baa077f4522c02931ee7d68c6298708","observation_id":"b6b1795c-768c-4c32-83e3-5eb597aad685","resolution":{"observed_at":"2026-08-04T15:45:08.940420Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.4007/annals.2009","metadata_source":"doi_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:51:50.448644Z","title":"Ricci curvature for metric-measure spaces via optimal trans- port","venue":null,"work_id":"6beefd76-d89d-4488-8122-0d8fe08e9393","year":2009},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":70,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.926133Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:2fa65991d746daaabf2207e9053903fff36378f7fb2a8b02826faf4f12927c70","observation_id":"e12e305a-a103-4da2-9496-5a2b6a03d28e","resolution":{"observed_at":"2026-08-04T15:51:50.453631Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.951172Z","title":null,"venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":71,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.951172Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:a3ed296157bb8f94f8611be864bb001232e85599be1383a3db52b30759e91dc5","observation_id":"93e002ab-2892-4110-a6c2-f95f5271ad59","resolution":{"observed_at":"2026-08-04T15:45:08.951172Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1016/s0021-7824(03","metadata_source":"doi_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:51:50.412441Z","title":"Functions of bounded variation on “good","venue":null,"work_id":"26423c1a-b1c9-4a7f-99b4-082c5c4ad505","year":2003},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":72,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.954289Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:efe6787237c84756af7e441f8e33eb353814c4592d6abf42a2347faa42888980","observation_id":"5d3bf967-b253-40b8-af0b-90330d154125","resolution":{"observed_at":"2026-08-04T15:51:50.417593Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.957924Z","title":"Existence of isoperimetric regions in non-compact Riemannian manifolds under Ricci or scalar curvature conditions","venue":null,"work_id":null,"year":2016},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":73,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.957924Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:16384b6ed99b7260fb10ad82a455a53bf600033860b0ec064425dea60729cdd8","observation_id":"acf4c4e4-ce20-4a7e-a740-ad797eae2511","resolution":{"observed_at":"2026-08-04T15:45:08.957924Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2107.12344","last_updated":"2023-02-16T12:51:29Z","snapshot_observed_at":"2026-07-06T11:32:37.657047Z","submitted_at":"2021-07-26T17:34:39Z","title":"Weak Laplacian bounds and minimal boundaries in non-smooth spaces with Ricci curvature lower bounds","version":3},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2107.12344","snapshot_observed_at":"2026-08-04T15:45:08.960970Z","title":"Weak Laplacian bounds and minimal boundaries on non smooth spaces with Ricci curvature lower bounds","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":74,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.960970Z"},"links":{"cited_paper":"/paper/2107.12344","citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:80d9a4c08299e14043a31e235059095c283389d08ab6909067eb7477b8a4ad61","observation_id":"b4989e50-4ca8-41f9-98b2-5c838577a292","resolution":{"observed_at":"2026-08-04T15:45:08.960970Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.944230Z","title":"Sharp isoperimetric inequalities and model spaces for the curvature- dimension-diameter condition","venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":75,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.944230Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:d1a7e4aef27303139ae83f3bcd629c9078ec1274852278ad8eefb21a8f13faad","observation_id":"d295e686-d482-4c09-8aef-7724f09da340","resolution":{"observed_at":"2026-08-04T15:45:08.944230Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.968035Z","title":"Generalized existence of isoperimetric regions in non-compact Rie- mannian manifolds and applications to the isoperimetric profile","venue":null,"work_id":null,"year":2014},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":76,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.968035Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:840eaaf62c5bfc7d50c69af27e7b9f218f7860cd37c6b63b81a9a31eb54e70ce","observation_id":"ffc8af26-d100-4653-af43-f8c172f41606","resolution":{"observed_at":"2026-08-04T15:45:08.968035Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.971742Z","title":"An optimal Poincar´ e inequality for convex domains","venue":null,"work_id":null,"year":1960},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":77,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.971742Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:23994c2f534367bb2cda30ea72670bb4e8019edc3f80efccc9e624feca2cbf61","observation_id":"c7f6663e-932b-4b0a-ab4e-2c2e72c4bb91","resolution":{"observed_at":"2026-08-04T15:45:08.971742Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.974946Z","title":"A complete Riemannian manifold of positive Ricci curvature with Euclidean volume growth and nonunique asymptotic cone","venue":null,"work_id":null,"year":1993},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":78,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.974946Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:98e40edee17502703d669deea6674b72fb33d93a8d11b4374305f0170d16b3ee","observation_id":"8e120a5f-e54e-4c82-9429-a5577b8b1b5a","resolution":{"observed_at":"2026-08-04T15:45:08.974946Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.978024Z","title":"Harmonic functions on Alexandrov spaces and their applications","venue":null,"work_id":null,"year":2003},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":79,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.978024Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:86a9db7f5d84de7d178ae3abe3d4eb41e79303081680af08538c2fe3cdf4d123","observation_id":"d3d10ba9-bb3f-43aa-afa3-b520cdcd1d13","resolution":{"observed_at":"2026-08-04T15:45:08.978024Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.980966Z","title":"On the stability of minimal surfaces","venue":null,"work_id":null,"year":1981},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":80,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.980966Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:2f49549f7a7b61159ef5046e46693378aeb5ea226ee65268602134b8d96cb601","observation_id":"b19c8a2d-8494-40e7-b940-b3efe88ed0e0","resolution":{"observed_at":"2026-08-04T15:45:08.980966Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1007/s10711-018-0416-4","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Local H¨ older continuity of the isoperimetric profile in complete noncompact Riemannian manifolds with bounded geometry","venue":"Geometriae Dedicata","work_id":"97bf29df-3987-4144-8f17-0b077dd03cf6","year":2019},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":81,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.964431Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:c18d16d2c939e6f8eb2156233d2b5b1052cf88cb51d456bdc68bd2a7522f6e9f","observation_id":"e29ef60d-dd50-4bef-a56c-1ce940f009f7","resolution":{"observed_at":"2026-08-04T15:51:50.402113Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1007/pl00004326","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Examples of constant mean curvature surfaces obtained from harmonic maps to the two sphere","venue":"Mathematische Zeitschrift","work_id":"e2a419cd-e283-4200-af65-82244b1e2aa9","year":1997},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":82,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.987066Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:cfb93dd6476602bd6d3278ddbf2deb98e6c6638644199ad21c5ece07287169e2","observation_id":"78838601-1587-47fe-a4e2-e7f65fd91c7f","resolution":{"observed_at":"2026-08-04T15:51:50.365031Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1007/bf02922017","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"The isoperimetric problem in complete surfaces of nonnegative curva- ture","venue":"Journal of Geometric Analysis","work_id":"d4329fc1-fe5a-40c5-85b0-33b422d8ae76","year":2001},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":83,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.990142Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:feff38c778f53a09f74b466b097f6ca26c86d4a3494c7edd8fe79867130677b9","observation_id":"b4061a06-96ac-4b21-bb7c-9e05ab19caaa","resolution":{"observed_at":"2026-08-04T15:51:50.350279Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1007/978-3-031-37901-7","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Ritor´ e","venue":"Progress in mathematics","work_id":"3f7735b3-3174-473f-9486-0be073c74eb7","year":2023},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":84,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.993113Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:145fce9507c46a9c1d84acd10ef93d62cbd4662e0071c605a873ce69d98c66f3","observation_id":"6d62643f-a93e-4bf1-a530-9124a522ebe8","resolution":{"observed_at":"2026-08-04T15:51:50.332535Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.996561Z","title":"Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones","venue":null,"work_id":null,"year":2004},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":85,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.996561Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:77b865e0c2d70e7bc182ee0f6fd3e54d3b8e04d360e249c2db4e8fc95da88266","observation_id":"79e804eb-49a5-4d55-a8c7-9057f7ee1712","resolution":{"observed_at":"2026-08-04T15:45:08.996561Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.999874Z","title":"The isoperimetric problem","venue":null,"work_id":null,"year":2005},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":86,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.999874Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:c488c8c98241044f51d8a314124ac205dcb93853f95733a49ceb5f6bd06b484f","observation_id":"d6bea40a-2411-442e-a6b6-b8880ef90be1","resolution":{"observed_at":"2026-08-04T15:45:08.999874Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1007/978-981-97-6984-1","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Isoperimetry on manifolds with Ricci bounded below: overview of recent results and methods","venue":"Springer INdAM series","work_id":"5bc17fd4-8ac1-4a61-9df8-84070e68383b","year":2024},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":87,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.983917Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:d5f76b239b13641ab72bd5efcef5fa6a57762718cb1ea0631c11876a5d4f6c53","observation_id":"c7ec4a66-d104-4779-a783-59563de6e4ce","resolution":{"observed_at":"2026-08-04T15:51:50.374952Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:09.006417Z","title":"On the geometry of metric measure spaces. I","venue":null,"work_id":null,"year":2006},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":88,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:09.006417Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:ff43caaae87296c77cd78c88ca6193db8d241946e5569fc69cae1271271e406b","observation_id":"a30d3862-84e5-46e0-8968-7d24692a99ee","resolution":{"observed_at":"2026-08-04T15:45:09.006417Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:09.009754Z","title":"On the geometry of metric measure spaces. II","venue":null,"work_id":null,"year":2006},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":89,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:09.009754Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:c9c44fc930536127b01dd6811f2565648136d3f6d34d389dfa17bff450189d0e","observation_id":"54e5e928-1c7e-404f-af99-78efb63e0179","resolution":{"observed_at":"2026-08-04T15:45:09.009754Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2409.04947","last_updated":"2024-09-21T05:59:01Z","snapshot_observed_at":"2026-07-06T19:12:02.473662Z","submitted_at":"2024-09-08T01:56:32Z","title":"Some estimates on stable minimal hypersurfaces in Euclidean space","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2409.04947","snapshot_observed_at":"2026-08-04T15:45:09.012798Z","title":null,"venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":90,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:09.012798Z"},"links":{"cited_paper":"/paper/2409.04947","citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:540a15f3c3cd99e39111649ee75b39034168e9b1cd40b82bdc5705aadbfda35e","observation_id":"6951b6e8-27f1-4602-9b66-442ba19bfdcc","resolution":{"observed_at":"2026-08-04T15:45:09.012798Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:09.016457Z","title":"Controlling area blow-up in minimal or bounded mean curvature vari- eties","venue":null,"work_id":null,"year":2016},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":91,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:09.016457Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:1bde916412886268d9cb96cc2eb896683eea92a7aef55577286661d1316432b5","observation_id":"186b9783-02e4-41ed-9ef0-5746e05bfdf5","resolution":{"observed_at":"2026-08-04T15:45:09.016457Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:09.019470Z","title":"On the geometry at infinity of manifolds with linear volume growth and nonnegative Ricci curvature","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":92,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:09.019470Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:60470905b73004a46c9819045ad264a2f19084b3959798affe425a74dc8ef5bd","observation_id":"e77a87d6-9cb3-45ae-ba24-92be8d0c8f28","resolution":{"observed_at":"2026-08-04T15:45:09.019470Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:09.003057Z","title":"Curvature estimates for minimal hypersur- faces","venue":null,"work_id":null,"year":1975},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":93,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:09.003057Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:f98b3a8a19d79700982f0b92aca8855193ae199ffbcf95e016d1a5fea80fc26a","observation_id":"d87072c8-2857-4ece-ba48-314ab5a5a7c0","resolution":{"observed_at":"2026-08-04T15:45:09.003057Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.741241Z","title":"doi: 10.1007/BF01161634","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":138,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.741241Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:36822858248eb8545157d51f6e6d1f825a1d106984d22ec11883dbc9c24ba390","observation_id":"67a4ca33-096d-4876-89a6-c3e94e2af91b","resolution":{"observed_at":"2026-08-04T15:45:08.741241Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1090/tran/9261","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"doi: 10.1090/tran/9261","venue":"Transactions of the American Mathematical Society","work_id":"2559b973-5ae8-4b6e-a522-2356367aae72","year":null},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":526,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:09.023079Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:2e83534571b0a15106ac84ef5673cfbed5d46fbbca6b0eae2c7c750bf9f192e6","observation_id":"3bffd301-4de7-4fc4-a324-9f832a2659df","resolution":{"observed_at":"2026-08-04T15:51:50.293850Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.4171/jems/526","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"doi: 10.4171/JEMS/526","venue":"Journal of the European Mathematical Society","work_id":"ddd9c716-a5be-4e50-ab31-10eabd7ea346","year":null},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":1078,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.947696Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:c945c2cda9b19c2f2b78b05d52f8d5fdaa67172916457339d887bdfa59be570c","observation_id":"07107103-a70b-4125-a5b2-ddc11e4535c6","resolution":{"observed_at":"2026-08-04T15:51:50.432162Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.674425Z","title":"doi: 10.1215/00127094-2681605","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":1490,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.674425Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:1814bee4e5e399f417eb02128cb8d25c649bf57339ca842e70252f91dc8d5926","observation_id":"62fdca5c-90b7-45c3-a151-7dc60e35dd26","resolution":{"observed_at":"2026-08-04T15:45:08.674425Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T15:45:08.797280Z","title":null,"venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":8970,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.797280Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:2e524df1f6189590f0c300712c7fa901a4e8a3578ce07287b4417d3f42ff69a2","observation_id":"0e43be74-1012-4f9b-8f04-5dfe420b4b1d","resolution":{"observed_at":"2026-08-04T15:45:08.797280Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1090/s0273-0979-1979-14689-5","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":null,"venue":"Bulletin of the American Mathematical Society","work_id":"a81f8e15-03cb-49fc-ae9c-21f994eb4a6f","year":1979},"citing_paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature","version":2},"reference_index":9485,"source":"pdf_text","source_observed_at":"2026-08-04T15:45:08.790542Z"},"links":{"citing_paper":"/paper/2509.18618"},"observation_digest":"sha256:146e6ab2abaecffa682e29dfbe10e1afeb887cab992430cd5442afc2b03c3ade","observation_id":"0f924e16-9c4e-45e9-a425-0fcd30c374c7","resolution":{"observed_at":"2026-08-04T15:51:50.662218Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}}],"paper":{"arxiv_id":"2509.18618","last_updated":"2026-07-18T15:48:31Z","latest_version":2,"primary_category":"math.DG","snapshot_observed_at":"2026-08-06T17:10:02.128544Z","submitted_at":"2025-09-23T04:08:51Z","title":"Isoperimetric problems and lower bounds on curvature"},"reference_resolution":{"displayed":99,"state_counts":{"malformed_identifier":3,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":71,"verified_exact":25,"verified_fuzzy":0},"total_outbound_references":99},"refusal":"A citation records a reference. It does not transfer a finding from one paper to another.","schema":"pith.paper-citation-record.v1","standing_sources":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"thesis":"As of 7 August 2026, this Paper Citation Record lists 99 of 99 outbound references and 0 inbound Pith citation observations for arXiv:2509.18618."}