{"paper":{"title":"On a Liu--Yau type inequality for surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Oussama Hijazi, Sebastian Montiel, Simon Raulot (LMRS)","submitted_at":"2015-02-13T18:53:18Z","abstract_excerpt":"Let $\\Omega$ be a compact and mean-convex domain with smooth boundary $\\Sigma:=\\partial\\Omega$, in an initial data set $(M^3,g,K)$, which has no apparent horizon in its interior. If $\\Sigma$ is spacelike in a spacetime $(\\E^4,g\\_\\E)$ with spacelike mean curvature vector $\\mathcal{H}$ such that $\\Sigma$ admits an isometric and isospin immersion into $\\mathbb{R}^3$ with mean curvature $H\\_0$, then: \n\\begin{eqnarray*}\n\\int\\_{\\Sigma}|\\mathcal{H}|d\\Sigma\\leq\\int\\_{\\Sigma}\\frac{H\\_0^2}{|\\mathcal{H}|}d\\Sigma.\n\\end{eqnarray*}\nIf equality occurs, we prove that there exists a local isometric immersion o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.04087","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"}