{"paper":{"title":"Li-Yau gradient bound for collapsing manifolds under integral curvature condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Meng Zhu, Qi S Zhang","submitted_at":"2016-07-20T13:42:40Z","abstract_excerpt":"Let $(\\M^n, g_{ij})$ be a complete Riemammnian manifold. For some constants $p,\\ r>0$, define $\\displaystyle k(p,r)=\\sup_{x\\in M}r^2\\left(\\oint_{B(x,r)}|Ric^-|^p dV\\right)^{1/p}$, where $Ric^-$ denotes the negative part of the Ricci curvature tensor. We prove that for any $p>\\frac{n}{2}$, when $k(p,1)$ is small enough, certain Li-Yau type gradient bound holds for the positive solutions of the heat equation on geodesic balls $B(O,r)$ in $\\M$ with $0<r\\leq 1$. Here the assumption that $k(p,1)$ being small allows the situation where the manifolds is collapsing. Recall that in \\cite{ZZ}, certain L"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.05951","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"}