{"paper":{"title":"Hypersymplectic 4-manifolds, the $G_2$-Laplacian flow and extension assuming bounded scalar curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SG"],"primary_cat":"math.DG","authors_text":"Chengjian Yao, Joel Fine","submitted_at":"2017-04-25T10:36:08Z","abstract_excerpt":"A hypersymplectic structure on a 4-manifold $X$ is a triple $\\underline{\\omega}$ of symplectic forms which at every point span a maximal positive-definite subspace of $\\Lambda^2$ for the wedge product. This article is motivated by a conjecture of Donaldson: when $X$ is compact $\\underline{\\omega}$ can be deformed through cohomologous hypersymplectic structures to a hyperk\\\"ahler triple. We approach this via a link with $G_2$-geometry. A hypersymplectic structure $\\underline\\omega$ on a compact manifold $X$ defines a natural $G_2$-structure $\\phi$ on $X \\times \\mathbb{T}^3$ which has vanishing "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.07620","kind":"arxiv","version":2},"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"}