{"paper":{"title":"First variation of the fractional $k$-dimensional measure: extending the concept of nonlocal curvature to submanifolds","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Brian Seguin, Cornelia Mihaila","submitted_at":"2026-06-23T01:04:20Z","abstract_excerpt":"The fractional $k$-dimensional measure of a submanifold of $\\mathbb{R}^n$ is a generalization of the fractional perimeter and fractional length appearing in the literature and depends on a parameter $\\sigma$ between $0$ and $1$. Here its first variation is computed. The resulting formula is used to define a nonlocal version of the mean-curvature vector for embedded submanifolds. It is shown that in the case where $k=n-1$, this agrees with the nonlocal mean-curvature that has been widely studied."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.24043","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/2606.24043/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"}