{"paper":{"title":"A Sharp Bound for the Ratio of the First Two Dirichlet Eigenvalues of a Domain in a Hemisphere of S^n","license":"","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.SP","authors_text":"Mark S. Ashbaugh, Rafael D. Benguria","submitted_at":"2000-08-11T20:45:10Z","abstract_excerpt":"For a domain $\\Omega$ contained in a hemisphere of the $n$-dimensional sphere $\\SS^n$ we prove the optimal result $\\lambda_2/\\lambda_1(\\Omega) \\le \\lambda_2/\\lambda_1(\\Omega^{\\star})$ for the ratio of its first two Dirichlet eigenvalues where $\\Omega^{\\star}$, the symmetric rearrangement of $\\Omega$ in $\\SS^n$, is a geodesic ball in $\\SS^n$ having the same $n$-volume as $\\Omega$. We also show that $\\lambda_2/\\lambda_1$ for geodesic balls of geodesic radius $\\theta_1$ less than or equal to $\\pi/2$ is an increasing function of $\\theta_1$ which runs between the value $(j_{n/2,1}/j_{n/2-1,1})^2$ f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0008088","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"}