{"paper":{"title":"Horizontal $\\alpha$-Harmonic Maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Francesca Da Lio, Tristan Rivi\\`ere","submitted_at":"2016-04-19T07:55:47Z","abstract_excerpt":"Given a $C^1$ planes distribution $P_T$ on all ${\\mathbb R}^m$ we consider {\\em horizontal $\\alpha$-harmonic maps}, $\\alpha\\ge 1/2$, with respect to such a distribution. These are maps $u\\in H^{\\alpha}({{\\mathbb R}}^k,{{\\mathbb R}}^m)$ satisfying $P_T\\nabla u=\\nabla u$ and $P_T(u)(-\\Delta)^{\\alpha}u=0$ in ${\\mathcal D}'({{\\mathbb R}}^k).$\n  If the distribution of planes is integrable then we recover the classical case of $\\alpha$-harmonic maps with values into a manifold. In this paper we shall focus our attention to the case $\\alpha=1/2$ in dimension $1$ and $\\alpha=2$ in dimension $2$ and we"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.05461","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":""},"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"}