Journal
JOURNAL OF GEOMETRIC ANALYSIS
Volume 21, Issue 4, Pages 1119-1131Publisher
SPRINGER
DOI: 10.1007/s12220-010-9180-x
Keywords
Doubling property; Volume comparison theorem; Perelman's entropy
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Funding
- NSF [DMS 0907326, DMS-0701001]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1001317] Funding Source: National Science Foundation
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The purpose of this work is to study some monotone functionals of the heat kernel on a complete Riemannian manifold with nonnegative Ricci curvature. In particular, we show that on these manifolds, the gradient estimate of Li and Yau (Acta Math. 156, 153-201, 1986), the gradient estimate of Ni (J. Geom. Anal. 14(1), 87-100, 2004), the monotonicity of the Perelman's entropy and the volume doubling property are all consequences of an entropy inequality recently discovered by Baudoin and Garofalo, arXiv:0904.1623, 2009. The latter is a linearized version of a logarithmic Sobolev inequality that is due to D. Bakry and M. Ledoux (Rev. Mat. Iberoam. 22, 683-702, 2006).
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