期刊
JOURNAL OF FUNCTIONAL ANALYSIS
卷 275, 期 2, 页码 478-515出版社
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jfa.2018.02.001
关键词
Heat equation; Li-Yau bound; Ricci L-P norm; Ricci Kato norm; Ricci flow
类别
资金
- Simons' Foundation [282153]
- Siyuan Foundation through Nanjing University [1215]
- National Natural Science Foundation of China [11501206]
- Science and Technology Commission of Shanghai Municipality [13dz2260400]
We prove Li-Yau type gradient bounds for the heat equation either on manifolds with fixed metric or under the Ricci flow. In the former case the curvature condition is vertical bar Ric(-)vertical bar epsilon L-P for some p > n/2, or sup(M) integral(M) vertical bar Ric(-)vertical bar(2)(y)d(2-n) (x, y)dy < infinity, where n is the dimension of the manifold. In the later case, one only needs scalar curvature being bounded. We will explain why the conditions are nearly optimal and give an application. The Li-Yau bound for the heat equation on manifolds with fixed metric seems to be the first one allowing Ricci curvature not bounded from below. (C) 2018 Elsevier Inc. All rights reserved.
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