  In this note, we obtain the rigidity of the sharp Cheng-Yau gradient estimate
for positive harmonic functions on surfaces with nonegative Gaussian curvature,
the rigidity of the sharp Li-Yau gradient estimate for positive solutions to
heat equations and the related estimates for Dirichlet Green's functions on
Riemannian manifolds with nonnegative Ricci curvature. Moreover, we also obtain
the corresponding stability results.
