{"paper":{"title":"Area of H\\\"older curves and coarea formula on the Heisenberg group","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Lipschitz maps from the subriemannian Heisenberg group to R^{2n} obey the coarea formula.","cross_cats":["math.CA","math.DG"],"primary_cat":"math.MG","authors_text":"Gioacchino Antonelli, Robert Young","submitted_at":"2026-05-15T14:20:09Z","abstract_excerpt":"We prove the coarea formula for Lipschitz maps from the subriemannian $n$th Heisenberg group $\\mathbb H_n$ to $\\mathbb R^{2n}$. Our result is new even when $n=1$ and provides the simplest vector-valued instance of the coarea formula in subriemannian geometry. This answers a question left open in the works of Magnani, Kozhevnikov, Magnani--Stepanov--Trevisan, and Julia--Nicolussi Golo--Vittone.\n  The main difficulty of the proof is that a fiber of a $C^1_{\\mathrm{H}}$ map $f: \\mathbb H_n\\to \\mathbb R^{2n}$ is typically an unrectifiable curve. Its measure depends on the symplectic area of its pr"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove the coarea formula for Lipschitz maps from the subriemannian nth Heisenberg group H_n to R^{2n}.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The geometric condition ensuring convergence of the new integral holds for almost every fiber of the map, established via beta-number estimates from the Fassler-Orponen Dorronsoro Theorem (abstract, final paragraph).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves the coarea formula for Lipschitz maps from H_n to R^{2n} via a new integral defining symplectic area for 1/2-Holder curves and beta-number estimates showing convergence for almost every fiber.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Lipschitz maps from the subriemannian Heisenberg group to R^{2n} obey the coarea formula.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"64f3fc97d7853022a2b57b71fed62a46344672eae5c9ec046faa29d9d08d8138"},"source":{"id":"2605.15987","kind":"arxiv","version":1},"verdict":{"id":"9fd4902c-65b1-447f-978d-5b5f1952972c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T17:14:29.922400Z","strongest_claim":"We prove the coarea formula for Lipschitz maps from the subriemannian nth Heisenberg group H_n to R^{2n}.","one_line_summary":"Proves the coarea formula for Lipschitz maps from H_n to R^{2n} via a new integral defining symplectic area for 1/2-Holder curves and beta-number estimates showing convergence for almost every fiber.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The geometric condition ensuring convergence of the new integral holds for almost every fiber of the map, established via beta-number estimates from the Fassler-Orponen Dorronsoro Theorem (abstract, final paragraph).","pith_extraction_headline":"Lipschitz maps from the subriemannian Heisenberg group to R^{2n} obey the coarea formula."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15987/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:44.048648Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T17:31:18.437702Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T17:26:19.338102Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T17:01:55.671721Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"0a8dd5c590ed9a91660620586470a3d29e78676a4b9dd852f7837a49d7efcdd8"},"references":{"count":23,"sample":[{"doi":"","year":2000,"title":"Rectifiable sets in metric and Ba- nach spaces","work_id":"ab61ab9f-1ccb-4841-a363-e02f5d60022c","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"Antonelli.Rectifiability in Carnot groups","work_id":"ea7de3b6-7138-4bcf-a3ae-f5d7ba9f5987","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Vertical curves and vertical fibers in the Heisenberg group","work_id":"01acddc3-14da-43dc-a279-5245b70533ec","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1990,"title":"A T (b) theorem with remarks on analytic capacity and the Cauchy integral","work_id":"a99f945d-a387-41cb-a645-5e4623632a12","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"A reverse coarea-type inequality in Carnot groups","work_id":"3511a456-844e-49a3-b98b-0771c666aa3e","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":23,"snapshot_sha256":"c73853ab15b1250bb1024734005d0ea229155a0cfa248a9b6cc96acd014bc1ae","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"d9eae46862ac4be9a8cf6a58ce3191f8da0f38a2554cbf0a1894195fd340e3dd"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}