{"paper":{"title":"$L^{2}$ harmonic forms on complete special holonomy manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Teng Huang","submitted_at":"2018-01-13T14:12:23Z","abstract_excerpt":"In this article, we consider $L^{2}$ harmonic forms on a complete non-compact Riemannian manifold $X$ with a nonzero parallel form $\\omega$. The main result is that if $(X,\\omega)$ is a complete $G_{2}$- ( or $Spin(7)$-) manifold with a $d$(linear) $G_{2}$- (or $Spin(7)$-) structure form $\\omega$, the $L^{2}$ harmonic $2$-forms on $X$ will be vanish. As an application, we prove that the instanton equation with square integrable curvature on $(X,\\omega)$ only has trivial solution. We would also consider the Hodge theory on the principal $G$-bundle $E$ over $(X,\\omega)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.04443","kind":"arxiv","version":3},"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"}