{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:RGQMHNQWJQVIYAAOWILFSHLMSN","short_pith_number":"pith:RGQMHNQW","schema_version":"1.0","canonical_sha256":"89a0c3b6164c2a8c000eb216591d6c9353ab84aa5e2fb6ff9e314043313ea6d9","source":{"kind":"arxiv","id":"1607.05951","version":1},"attestation_state":"computed","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 $00$, 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