{"paper":{"title":"On critical maps of the horizontal energy functional between Riemannian foliations","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hui Liu, Tian Chong, Xin Huang, Yuxin Dong","submitted_at":"2025-04-02T08:50:59Z","abstract_excerpt":"In this paper, we consider critical points of the\n  horizontal energy $E_{\\HH}(f)$ for a smooth map $f$ between two Riemannian\n  foliations. These critical points are referred to as horizontally harmonic\n  maps. In particular, if the maps are foliated, they become transversally\n  harmonic maps. By utilizing the stress-energy tensor, we establish some\n  monotonicity formulas for horizontally harmonic maps from Euclidean spaces,\n  the quotients $K_{m}$ of Heisenberg groups and also for transversally\n  harmonic maps from Riemannian foliations with appropriate curvature pinching\n  conditions. Fina"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.01498","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/2504.01498/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"}