{"paper":{"title":"Spectral Estimates, Contractions and Hypercontractivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.FA"],"primary_cat":"math.SP","authors_text":"Emanuel Milman","submitted_at":"2015-08-03T21:37:08Z","abstract_excerpt":"Sharp comparison theorems are derived for all eigenvalues of the (weighted) Laplacian, for various classes of weighted-manifolds (i.e. Riemannian manifolds endowed with a smooth positive density). Examples include Euclidean space endowed with strongly log-concave and log-convex densities, extensions to $p$-exponential measures, unit-balls of $\\ell_p^n$, one-dimensional spaces and Riemannian submersions. Our main tool is a general Contraction Principle for \"eigenvalues\" on arbitrary metric-measure spaces. Motivated by Caffarelli's Contraction Theorem, we put forth several conjectures pertaining"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.00606","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"}