{"paper":{"title":"Toric aspects of the first eigenvalue","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SG"],"primary_cat":"math.DG","authors_text":"Eveline Legendre, Rosa Sena-Dias","submitted_at":"2015-05-07T12:11:52Z","abstract_excerpt":"In this paper we study the smallest non-zero eigenvalue $\\lambda_1$ of the Laplacian on toric K\\\"ahler manifolds. We find an explicit upper bound for $\\lambda_1$ in terms of moment polytope data. We show that this bound can only be attained for $\\mathbb{CP}^n$ endowed with the Fubini-Study metric and therefore $\\mathbb{CP}^n$ endowed with the Fubini-Study metric is spectrally determined among all toric K\\\"ahler metrics. We also study the equivariant counterpart of $\\lambda_1$ which we denote by $\\lambda_1^T$. It is the the smallest non-zero eigenvalue of the Laplacian restricted to torus-invar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.01678","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"}