{"paper":{"title":"Bounding Curvature Measure on Manifolds with Singularities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Nan Li","submitted_at":"2026-06-08T00:11:16Z","abstract_excerpt":"Let $X$ be an $n$-dimensional Alexandrov space with curvature $\\ge -1$, and let $\\eta > 0$. Define $\\mathcal{S}^{k}_\\eta(X)$ as the set of $(k,\\eta)$-singular points in $X$ whose tangent cones are $\\eta$-away from splitting off $\\mathbb{R}^{k+1}$ isometrically. For a point $p \\in X$, assume that $M = B_2(p) \\setminus (\\mathcal{S}^{n-2}_\\eta(X) \\cup \\partial X)$ is a smooth manifold equipped with the Riemannian metric induced by $X$. We prove that the integral of the scalar curvature of $M$ over $B_1(p)$ is bounded from above by a constant depending only on $n$ and $\\eta$. As a special case, th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08887","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.08887/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}