{"paper":{"title":"A Wasserstein Inequality and Minimal Green Energy on Compact Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.CA","authors_text":"Stefan Steinerberger","submitted_at":"2019-07-21T20:00:46Z","abstract_excerpt":"Let $M$ be a smooth, compact $d-$dimensional manifold, $d \\geq 3,$ without boundary and let $G: M \\times M \\rightarrow \\mathbb{R} \\cup \\left\\{\\infty\\right\\}$ denote the Green's function of the Laplacian $-\\Delta$ (normalized to have mean value 0). We prove a bound on the cost of transporting Dirac measures in $\\left\\{x_1, \\dots, x_n\\right\\} \\subset M$ to the normalized volume measure $dx$ in terms of the Green's function of the Laplacian $$ W_2\\left( \\frac{1}{n} \\sum_{k=1}^{n}{\\delta_{x_k}}, dx\\right) \\lesssim_M \\frac{1}{n^{1/d}} + \\frac{1}{n} \\left| \\sum_{k, \\ell=1 \\atop k \\neq \\ell}^{n}G(x_k"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.09023","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"}