Title: Some New Gradient Estimates for the Heat Equations on Manifolds
by Qi, Zhang (UCR) Jan. 14;

Abstract: We  derive a localized version of elliptic type gradient estimates
for the heat equation which are akin to the Cheng-Yau estimate for the Laplace 
equation and Hamilton's estimate for the heat equation on compact manifolds. 
As applications, we generalize Yau's celebrated Liouville theorem (with 
necessary modification) for positive harmonic functions to positive eternal 
solutions of the heat equation.  This was unexpected since Liouville theorem 
for positive eternal solutions of the heat equation does not hold even in 
${\bf R}^n$.  We also prove a sharpened gradient estimate for the heat kernel. 
This is a joint work with Philippe Souplet.