{"paper":{"title":"The PPW conjecture in curved spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Nick Edelen","submitted_at":"2015-03-24T03:47:49Z","abstract_excerpt":"In Euclidean and Hyperbolic space, and the hemisphere in $S^n$, geodesic balls maximize the gap $\\lambda_2 - \\lambda_1$ of Dirichlet eigenvalues, amoung domains with fixed $\\lambda_1$. We prove an upper bound on $\\lambda_2 - \\lambda_1$ for domains in manifolds with certain curvature bounds. The inequality is sharp on geodesic balls in spaceforms."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.06904","kind":"arxiv","version":4},"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"}