{"paper":{"title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Wen-Qi Li, Zhikai Zhang","submitted_at":"2025-05-25T12:13:52Z","abstract_excerpt":"In this paper, we establish a parabolic Harnack inequality for positive solutions of the $\\phi$-heat equation and prove Gaussian upper and lower bounds for the $\\phi$-heat kernel on weighted Riemannian manifolds under lower $N$-Ricci curvature bound with $\\varepsilon$-range. Building on these results, we demonstrate: The $L^1_\\phi$-Liouville theorem for $\\phi$-subharmonic functions, $L^1_\\phi$-uniqueness property for solutions of the $\\phi$-heat equation and lower bounds for eigenvalues of the weighted Laplacian $\\Delta_\\phi$.\n  Furthermore, leveraging the Gaussian upper bound of the weighted "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.19113","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/2505.19113/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"}