{"as_of":"2026-08-19T04:36:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:903f9e0cca5c3240a4fac60f75017ff9a2156c3056921cab365c666ef94f7a17","coverage":[{"denominator":34,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":34,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-02T18:30:23.566623Z","state":"measured"},{"denominator":36,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":36,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-18T06:34:40.430872+00:00","state":"measured"},{"denominator":2,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":2,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-02T14:59:29.436979Z","state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"arxiv_reference","source_observed_at":"2026-05-11T21:36:13.514248Z","state":"measured"}],"external_citation_measurements":[],"inbound":[{"citation":{"cited_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"cited_work":{"arxiv_id":"2603.12025","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":"2603.12025","snapshot_observed_at":"2026-07-21T02:20:36.045063Z","title":"Brendle, Geometric inequalities and the Alexandrov-Bakelman-Pucci technique, Preprint, available at","venue":null,"work_id":"09e27d76-246d-4002-ba7a-0e1f27199e5c","year":null},"citing_paper":{"arxiv_id":"2605.06074","last_updated":"2026-07-16T09:56:56Z","snapshot_observed_at":"2026-08-08T12:58:47.534047Z","submitted_at":"2026-05-07T11:59:15Z","title":"A comparison theorem with applications to sharp geometric inequalities for submanifolds","version":1},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-05-08T05:03:15.133395Z"},"links":{"cited_paper":"/paper/2603.12025","citing_paper":"/paper/2605.06074"},"observation_digest":"sha256:59754e0b7a565c214f3078c6ff1a639dd4a2d41c5e37edb5335ff0db9024c534","observation_id":"64cce054-0b9a-4096-aaea-706160629940","resolution":{"observed_at":"2026-07-21T02:20:36.045063Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2603.12025","snapshot_observed_at":"2026-08-02T14:59:29.436979Z","title":"Brendle, Geometric inequalities and the Alexandrov-Bakelman-Pucci technique, Preprint, available at arXiv:2603.12025","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2605.06074","last_updated":"2026-07-16T09:56:56Z","snapshot_observed_at":"2026-08-08T12:58:47.534047Z","submitted_at":"2026-05-07T11:59:15Z","title":"A comparison theorem with applications to sharp geometric inequalities for submanifolds","version":2},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-02T14:59:29.436979Z"},"links":{"cited_paper":"/paper/2603.12025","citing_paper":"/paper/2605.06074"},"observation_digest":"sha256:c10f4898b2d4e84f781181cb6a95bc5722e2b006d65e7ccad730de4096ceab44","observation_id":"c7cdf6c0-5128-497f-973d-567eb7f05771","resolution":{"observed_at":"2026-08-02T14:59:29.436979Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}}],"links":{"evidence":"/evidence","html":"/paper/2603.12025/citation-record","integrity":"/paper/2603.12025/integrity","json":"/paper/2603.12025/citation-record.json","paper":"/paper/2603.12025"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-02T18:30:20.412148Z","title":"Agostiniani, M","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:20.412148Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:7edfd53a3e7707e22a9ef268da06f77f9c9b875e42bd2b6001156a8b6ce599b2","observation_id":"f2dad893-3c5e-42e5-a201-6108702f21b0","resolution":{"observed_at":"2026-08-02T18:30:20.412148Z","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":"2026-08-02T18:30:20.492703Z","title":"Allard,On the first variation of a varifold,Ann","venue":null,"work_id":null,"year":1972},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:20.492703Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:e64eafe0cd8f0ef49093348b1dde1e06bdeba8709c424ccc8b6383de01032090","observation_id":"8d490aae-5803-4be8-a2de-694174dca718","resolution":{"observed_at":"2026-08-02T18:30:20.492703Z","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":"2026-08-02T18:30:20.585513Z","title":"Almgren, Jr.,Optimal isoperimetric inequalities,Indiana Univ","venue":null,"work_id":null,"year":1986},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:20.585513Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:eed520b301ffd4298228c42f9ea4015a9a4df9c81bf775dcd75ee25dfbb4ab7f","observation_id":"2c96f047-a78b-47f2-90f3-791799cffebf","resolution":{"observed_at":"2026-08-02T18:30:20.585513Z","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":"2026-08-02T18:30:20.690344Z","title":"Balogh and A","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:20.690344Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:6f68aa312e9792043c2c639a5b841e84186c2853aed21ae3ec0c4b3112933d1c","observation_id":"fac358fa-8d67-4aa0-aeaf-d99d8e73c680","resolution":{"observed_at":"2026-08-02T18:30:20.690344Z","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":"2026-08-02T18:30:20.780150Z","title":"Brendle,The isoperimetric inequality for a minimal submanifold in Euclidean space,J","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:20.780150Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:565073053799b1a8b6639a60da45dcf78929d08966ad80dce4daaea4d4a0fc0c","observation_id":"1b182608-7e7e-4958-8733-aa64c09acc93","resolution":{"observed_at":"2026-08-02T18:30:20.780150Z","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":"2026-08-02T18:30:20.888811Z","title":"Brendle,The logarithmic Sobolev inequality for a submanifold in Euclidean space, Comm","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:20.888811Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:b2546922ac46b74c8daf75d3d1483f805850f82dabebf828acbb42a1b3b4ed28","observation_id":"406f2e31-de4e-43af-9612-006563d875f8","resolution":{"observed_at":"2026-08-02T18:30:20.888811Z","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":"2026-08-02T18:30:20.971580Z","title":"Brendle,Sobolev inequalities in manifolds with nonnegative curvature,Comm","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:20.971580Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:5d8582860a76f42560e495ce62c31e1e47473f2e259a8dbff4f5e7e04bc2c6ee","observation_id":"e0b19a91-4cce-4f73-b12a-29f4731f250c","resolution":{"observed_at":"2026-08-02T18:30:20.971580Z","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":"2026-08-02T18:30:21.036370Z","title":"Brendle,Minimal hypersurfaces and geometric inequalities,Ann","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:21.036370Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:be9b5b61793222846cc0bf13f10aee1f8e443d4fae827e4e2c88f0b2970fa596","observation_id":"c02b0953-7b5c-4ab7-948b-88f4627ba425","resolution":{"observed_at":"2026-08-02T18:30:21.036370Z","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":"2026-08-02T18:30:21.112842Z","title":"Brendle and M","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:21.112842Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:77dbf92a8eb2dfb03c99893bc8cf8de0958e3a31f22d6bf7b643825962a6290d","observation_id":"08c0ce13-4051-4817-807d-58b4090c806b","resolution":{"observed_at":"2026-08-02T18:30:21.112842Z","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":"2026-08-02T18:30:21.195024Z","title":"Cabr´ e,Nondivergent elliptic equations on manifolds with nonnegative curvature, Comm","venue":null,"work_id":null,"year":1997},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:21.195024Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:2bdb72f6b2caf01927482c3a63a1441c5ca637a0943e0930dbd68e4f3a8464ba","observation_id":"e14bde9d-901d-42c3-88ac-75efaad37b31","resolution":{"observed_at":"2026-08-02T18:30:21.195024Z","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":"2026-08-02T18:30:21.317717Z","title":"Cabr´ e,Equacions en derivades parcials, geometria i control estoc` astic,Butl","venue":null,"work_id":null,"year":2000},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:21.317717Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:3761ed75d11cf8b2c6fdf7442d845eb468ed3e3b63382f5b57b02c216ac98ed6","observation_id":"c19e5935-80d9-4245-9f0f-4e1282f46532","resolution":{"observed_at":"2026-08-02T18:30:21.317717Z","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":"2026-08-02T18:30:21.374397Z","title":"Cabr´ e,Elliptic PDEs in probability and geometry","venue":null,"work_id":null,"year":2008},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:21.374397Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:91d2fae9a6fec233c11edabe56adb29fa92d2580ed99faa1554ae0117cbd768c","observation_id":"67b6db4f-7a6c-44ed-b256-2266827b89c5","resolution":{"observed_at":"2026-08-02T18:30:21.374397Z","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":"2026-08-02T18:30:21.473975Z","title":"Carleman,Zur Theorie der Minimalfl¨ achen,Math","venue":null,"work_id":null,"year":1921},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:21.473975Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:c3f4eb1d7aa2d60ee0de068eb697ef612803eac57bcc8f395e667e584f0b1bfd","observation_id":"daac13a8-9350-474a-82fc-d1f914dca680","resolution":{"observed_at":"2026-08-02T18:30:21.473975Z","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":"2026-08-02T18:30:21.541431Z","title":"Castillon,Submanifolds, isoperimetric inequalities and optimal transportation,J","venue":null,"work_id":null,"year":2010},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:21.541431Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:92da1dc95dd097669755bb487fe5044557d3f32d4a889876819518a47c5f68ee","observation_id":"0a02efe4-d7de-46a5-9afd-46ef65ee4e93","resolution":{"observed_at":"2026-08-02T18:30:21.541431Z","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":"2026-08-02T18:30:21.582819Z","title":"Chen,On the total curvature of immersed manifolds, I: an inequality of Fenchel- Borsuk-Willmore,Amer","venue":null,"work_id":null,"year":1971},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:21.582819Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:df48f1657361e0b962161dd1b418e3dc7f2463eb27cf838df5b3e7eefb3b8b76","observation_id":"ac7404ac-3123-4d5e-915d-45c11fbf3713","resolution":{"observed_at":"2026-08-02T18:30:21.582819Z","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":"2026-08-02T18:30:21.665960Z","title":"Choe,The isoperimetric inequality for a minimal surface with radially connected boundary,Ann","venue":null,"work_id":null,"year":1990},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:21.665960Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:dc4ac3ca01a2d0d27a017f64f083b90b85c35f4c9e27a16fcec5d5ae48a9caf8","observation_id":"a0ad1997-c459-4de4-b233-a0bd7004b6b7","resolution":{"observed_at":"2026-08-02T18:30:21.665960Z","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":"2026-08-02T18:30:21.742111Z","title":"Cordero-Erausquin, R.J","venue":null,"work_id":null,"year":2001},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:21.742111Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:f7fb3879d588f96f7669e4f3d6674c36a1bd50292dd4e3ef4fad7d3cfb8c7ac9","observation_id":"b13d63b5-5a00-4b75-aa9f-a8823f273b5e","resolution":{"observed_at":"2026-08-02T18:30:21.742111Z","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":"2026-08-02T18:30:21.811913Z","title":"Ecker,Logarithmic Sobolev inequalities on submanifolds of Euclidean sapce,J","venue":null,"work_id":null,"year":2000},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:21.811913Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:9cf28f77f0b1ae6921682603185dae0c49d6d878879e50c7c5a6bd334ab90d50","observation_id":"68a435fd-6a5c-4cb2-9abf-2b07ddaec0ce","resolution":{"observed_at":"2026-08-02T18:30:21.811913Z","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":"2026-08-02T18:30:21.891257Z","title":"Feinberg,The isoperimetric inequality for doubly-connected minimal surfaces in Rn,J","venue":null,"work_id":null,"year":1977},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:21.891257Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:bc454318bd993c897cc1b3758169bf772275972827ec8338c9e5bd782e20a2e5","observation_id":"1ee9c495-5f43-4276-a3f6-798d3a53d2d8","resolution":{"observed_at":"2026-08-02T18:30:21.891257Z","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":"2026-08-02T18:30:22.021299Z","title":"Fenchel, ¨Uber die Kr¨ ummung und Windung geschlossener Randkurven,Math","venue":null,"work_id":null,"year":1929},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:22.021299Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:d0bd2865dac8249cd2253a985da0af3f6b899e0832472082f6a2edd0f891aa81","observation_id":"e946b5df-f1ac-416a-8d78-550c87761fd3","resolution":{"observed_at":"2026-08-02T18:30:22.021299Z","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":"2026-08-02T18:30:22.148406Z","title":"Gromov,Metric structures for Riemannian and non-Riemannian spaces,Progress in Mathematics vol","venue":null,"work_id":null,"year":1999},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:22.148406Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:fc4acda16b05b931f94d032eb8ecdb720f95c38320facc2382273e110b38399d","observation_id":"f7a341d2-f547-45f5-b36f-2488935d55b0","resolution":{"observed_at":"2026-08-02T18:30:22.148406Z","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":"2026-08-02T18:30:22.249718Z","title":"Heintze and H","venue":null,"work_id":null,"year":1978},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:22.249718Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:26e6c76f2e9f46ae06cb234b70552e459cd23c127eac29667c8afaf05291784d","observation_id":"515f4114-4b4b-42f8-a79b-de17663059c0","resolution":{"observed_at":"2026-08-02T18:30:22.249718Z","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":"2026-08-02T18:30:22.349504Z","title":"Hsiung,Isoperimetric inequalities for two-dimensional Riemannian manifolds with boundary,Ann","venue":null,"work_id":null,"year":1961},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:22.349504Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:d094a958aa1e33a1e7105bbb5457cbdcefe5a3230303b371afa704c24ffbc10a","observation_id":"9623bfc8-a17d-436e-a7cd-723bba24c2ee","resolution":{"observed_at":"2026-08-02T18:30:22.349504Z","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":"2026-08-02T18:30:22.468100Z","title":null,"venue":null,"work_id":null,"year":1984},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:22.468100Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:abf6c46d3b7e113bb1d29f2efc9adccd6df7fb8f6015c41670a17e77ae034d57","observation_id":"333e0abc-6804-48fa-a5e0-dddcf5cc9172","resolution":{"observed_at":"2026-08-02T18:30:22.468100Z","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":"2026-08-02T18:30:22.594428Z","title":"Lusternik,Die Brunn-Minkowskische Ungleichung f¨ ur beliebige meßbare Mengen, Dokl","venue":null,"work_id":null,"year":1935},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:22.594428Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:b8af24bf92aa5bf6c2d1ad287ac83954db2d667ed6c4081260e30d7f346a7b00","observation_id":"dec33ba8-41c5-4a49-baf0-80bd2010543e","resolution":{"observed_at":"2026-08-02T18:30:22.594428Z","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":"2026-08-02T18:30:22.688229Z","title":"McCann,A Convexity Theory for Interacting Gases and Equilibrium Crystals, Ph.D","venue":null,"work_id":null,"year":1994},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:22.688229Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:fb4636b3fd5f4d150ded9183fb1cda375aff42b18c57cc7ca740bde7015ed3ae","observation_id":"7a3f9015-5cce-4590-861c-99487843302a","resolution":{"observed_at":"2026-08-02T18:30:22.688229Z","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":"2026-08-02T18:30:22.848588Z","title":"McCann,A convexity principle for interacting gases,Adv","venue":null,"work_id":null,"year":1997},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:22.848588Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:ec6336fc63e3255a32ad865f3fe0538c88fa8368ec3344413fd3244b3e42b13c","observation_id":"cdc7162b-a3bf-40e6-8052-5d29d07e7289","resolution":{"observed_at":"2026-08-02T18:30:22.848588Z","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":"2026-08-02T18:30:22.979066Z","title":"McCann and N","venue":null,"work_id":null,"year":2013},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:22.979066Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:68532bd8f6ca0693ca7d31fa4b8f4f3e96f485a2ee62bcaf95165483462c17eb","observation_id":"3967f8a3-a2a3-4fb8-b8b3-73fb2e548dba","resolution":{"observed_at":"2026-08-02T18:30:22.979066Z","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":"2026-08-02T18:30:23.139783Z","title":"Michael and L.M","venue":null,"work_id":null,"year":1973},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:23.139783Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:815cd34113ed3d6f0b2a0f2219ca7da0d0e600bf7854bb27db5f2445718b2c3e","observation_id":"6a48565e-e1e9-44f9-8ae8-197c3592eefa","resolution":{"observed_at":"2026-08-02T18:30:23.139783Z","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":"2026-08-02T18:30:23.273936Z","title":"Osserman and M","venue":null,"work_id":null,"year":1975},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:23.273936Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:8a7a349359220ac112e03c00f763f718ff9028eae20016d9bf16cc9ac8d78a4d","observation_id":"4dcebd4e-81bd-433a-ae82-b16fd0cfd14a","resolution":{"observed_at":"2026-08-02T18:30:23.273936Z","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":"2026-08-02T18:30:23.343316Z","title":"Reid,The isoperimetric inequality and associated boundary problems,J","venue":null,"work_id":null,"year":1959},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:23.343316Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:462f0badc6e11d23f02a665903ab8c0bdf7dc856fbaddaedc6f4e4750f3b7f17","observation_id":"771b145d-c3e3-4dae-b323-f908f57efef1","resolution":{"observed_at":"2026-08-02T18:30:23.343316Z","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":"2026-08-02T18:30:23.427289Z","title":"Stone,On the isoperimetric inequality on a minimal surface,Calc","venue":null,"work_id":null,"year":2003},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:23.427289Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:fd6577277be81d03857f4df3dc56a9a84604fea708bfd6afa22ee2abd57b6d09","observation_id":"b40f4317-eb48-4ceb-bb58-109e29920ae6","resolution":{"observed_at":"2026-08-02T18:30:23.427289Z","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":"2026-08-02T18:30:23.483703Z","title":"Trudinger,Isoperimetric inequalities for quermassintegrals,Ann","venue":null,"work_id":null,"year":1994},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":33,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:23.483703Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:cd140204b1d7db67f86be9334fe493191e9fe1909bac3592655c1cba94172683","observation_id":"9a56e559-8443-4b1e-9af5-996930aa4762","resolution":{"observed_at":"2026-08-02T18:30:23.483703Z","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":"2026-08-02T18:30:23.566623Z","title":"Willmore,Note on embedded surfaces,A","venue":null,"work_id":null,"year":1965},"citing_paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique","version":3},"reference_index":34,"source":"pdf_text","source_observed_at":"2026-08-02T18:30:23.566623Z"},"links":{"citing_paper":"/paper/2603.12025"},"observation_digest":"sha256:00f369faf6f3521cdb4d8b68b5a70425390fd593410ed846aed6b19f1fc22399","observation_id":"503c3439-a0ed-4a46-9cf9-b5dc48968508","resolution":{"observed_at":"2026-08-02T18:30:23.566623Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}}],"paper":{"arxiv_id":"2603.12025","last_updated":"2026-07-19T19:15:48Z","latest_version":3,"primary_category":"math.DG","snapshot_observed_at":"2026-08-19T04:33:45.632501Z","submitted_at":"2026-03-12T15:04:53Z","title":"Geometric inequalities and the Alexandrov-Bakelman-Pucci technique"},"reference_resolution":{"displayed":34,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":34,"verified_exact":0,"verified_fuzzy":0},"total_outbound_references":34},"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":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"thesis":"As of 19 August 2026, this Paper Citation Record lists 34 of 34 outbound references and 2 inbound Pith citation observations for arXiv:2603.12025."}