  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, the gradient estimate of Ni, the monotonicity of the Perelman's entropy
and the volume doubling property are all consequences of an entropy inequality
recently discovered by Baudoin-Garofalo. The latter is a linearized version of
a logarithmic Sobolev inequality that is due to D. Bakry and M. Ledoux.
