{"paper":{"title":"Reciprocal lower bound on modulus of curve families in metric surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Kai Rajala, Matthew Romney","submitted_at":"2018-01-24T22:30:26Z","abstract_excerpt":"We prove that any metric space $X$ homeomorphic to $\\mathbb{R}^2$ with locally finite Hausdorff 2-measure satisfies a reciprocal lower bound on modulus of curve families associated to a quadrilateral. More precisely, let $Q \\subset X$ be a topological quadrilateral with boundary edges (in cyclic order) denoted by $\\zeta_1, \\zeta_2, \\zeta_3, \\zeta_4$ and let $\\Gamma(\\zeta_i, \\zeta_j; Q)$ denote the family of curves in $Q$ connecting $\\zeta_i$ and $\\zeta_j$; then $\\text{mod} \\Gamma(\\zeta_1, \\zeta_3; Q) \\text{mod} \\Gamma(\\zeta_2, \\zeta_4; Q) \\geq 1/\\kappa$ for $\\kappa = 2000^2\\cdot (4/\\pi)^2$. Th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.08223","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"}