Perelman's Entropy and Doubling Property on Riemannian Manifolds

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).