{"paper":{"title":"A comparison theorem with applications to sharp geometric inequalities for submanifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"An explicit formula for the Jacobian determinant of the normal exponential map on a submanifold leads to a new comparison theorem and sharp geometric inequalities.","cross_cats":[],"primary_cat":"math.DG","authors_text":"Chengyang Yi, Shengliang Pan","submitted_at":"2026-05-07T11:59:15Z","abstract_excerpt":"In this paper, we derive an explicit expression for the Jacobian determinant of the normal exponential map on a submanifold, establishing a relationship with its ambient counterpart. This formula leads to a new comparison theorem which is closely related to the comparison theorem of Heintze-Karcher. As applications, we obtain a Fenchel-Borsuk-Chern-Lashof-type inequality and a Willmore-Chen-type inequality on closed submanifolds in complete noncompact manifolds with nonnegative curvature and Euclidean volume growth."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We derive an explicit expression for the Jacobian determinant of the normal exponential map on a submanifold, establishing a relationship with its ambient counterpart. This formula leads to a new comparison theorem which is closely related to the comparison theorem of Heintze-Karcher and the estimate of Brendle. As applications, we obtain a Fenchel-Borsuk-Chern-Lashof-type inequality and a Willmore-Chen-type inequality on closed submanifolds in complete noncompact manifolds with nonnegative curvature and Euclidean volume growth.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The ambient manifold is complete and noncompact with nonnegative sectional curvature and Euclidean volume growth; the submanifolds are closed (and the normal exponential map is well-defined up to the cut locus).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"An explicit Jacobian formula for the normal exponential map produces a comparison theorem that implies Fenchel-Borsuk-Chern-Lashof-type and Willmore-Chen-type inequalities for closed submanifolds in complete noncompact manifolds with nonnegative curvature and Euclidean volume growth.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"An explicit formula for the Jacobian determinant of the normal exponential map on a submanifold leads to a new comparison theorem and sharp geometric inequalities.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"2a5ece32ac3084f96317d18da1390aee0f0684e718121a58eb194df8576a0afe"},"source":{"id":"2605.06074","kind":"arxiv","version":2},"verdict":{"id":"9a9e348e-d8fd-466a-a1f1-8259ec0760fc","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T04:59:16.173102Z","strongest_claim":"We derive an explicit expression for the Jacobian determinant of the normal exponential map on a submanifold, establishing a relationship with its ambient counterpart. This formula leads to a new comparison theorem which is closely related to the comparison theorem of Heintze-Karcher and the estimate of Brendle. As applications, we obtain a Fenchel-Borsuk-Chern-Lashof-type inequality and a Willmore-Chen-type inequality on closed submanifolds in complete noncompact manifolds with nonnegative curvature and Euclidean volume growth.","one_line_summary":"An explicit Jacobian formula for the normal exponential map produces a comparison theorem that implies Fenchel-Borsuk-Chern-Lashof-type and Willmore-Chen-type inequalities for closed submanifolds in complete noncompact manifolds with nonnegative curvature and Euclidean volume growth.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The ambient manifold is complete and noncompact with nonnegative sectional curvature and Euclidean volume growth; the submanifolds are closed (and the normal exponential map is well-defined up to the cut locus).","pith_extraction_headline":"An explicit formula for the Jacobian determinant of the normal exponential map on a submanifold leads to a new comparison theorem and sharp geometric inequalities."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.06074/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T13:02:04.354218Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-20T08:38:00.674043Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T19:01:19.546451Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T12:57:21.064466Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"d6161c586c43f78362ea1d67e2c9ce5db0672f1cb35b3e66dc1f2f9ec8964bd3"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}