{"paper":{"title":"Monotonicity of the first Dirichlet eigenvalue of the Laplacian on manifolds of nonpositive curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jesse Ratzkin, Tom Carroll","submitted_at":"2014-07-03T11:01:35Z","abstract_excerpt":"Let $(M,g)$ be a complete manifold of nonpositive scalar curvature, let $\\Omega\\subset M$ be a suitable domain, and let $\\lambda(\\Omega)$ be the first Dirichlet eigenvalue of the Laplace-Beltrami operator on $\\Omega$. We prove several bounds for the rate of decrease of $\\lambda(\\Omega)$ and $\\Omega$ increases, and a result comparing the rate of decrease of $\\lambda$ before and after a conformal diffeomorphism. Along the way, we prove a reverse-Holder inequality for the first eigenfunction, which generalizes results of Chiti to the monifold setting and may be of independent interest"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.0864","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"}