{"paper":{"title":"Dimension of polynomial growth harmonic functions on locally conformally flat manifolds with nonnegative Ricci curvature","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Mijia Lai, Xiaohan Cai","submitted_at":"2026-08-05T07:46:41Z","abstract_excerpt":"Let $\\mathcal{H}_d(M)$ denote the space of harmonic functions of polynomial growth at most degree $d$ on a complete Riemannian manifold. Yau raised two fundamental questions regarding $\\mathcal{H}_d(M)$ on complete manifolds with nonnegative Ricci curvature. The first question is the finite dimensionality of $\\mathcal{H}_d(M)$, which is confirmed by Colding and Minicozzi. The second question asks whether a sharp upper bound by its Euclidean analog $\\operatorname{dim}\\mathcal{H}_{d}(\\mathbb{R}^n)$ is true. We verify that if $(M,g)$ is a locally conformally flat manifold with nonnegative Ricci c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.04553","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.04553/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}