Abstract
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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Communicated by Jiaping Wang.
First author supported in part by NSF Grant DMS 0907326.
Second author supported in part by NSF Grant DMS-0701001.
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Baudoin, F., Garofalo, N. Perelman’s Entropy and Doubling Property on Riemannian Manifolds. J Geom Anal 21, 1119–1131 (2011). https://doi.org/10.1007/s12220-010-9180-x
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DOI: https://doi.org/10.1007/s12220-010-9180-x