  In this paper, we study the existence of complete Yamabe metric with zero
scalar curvature on an n-dimensional complete Riemannian manifold $(M,g_0)$,
$n\geq 3$. Under suitable conditions about the initial metric, we show that
there is a global fine solution to the Yamabe flow. The interesting point here
is that we have no curvature assumption about the initial metric. We show that
on an n-dimensional complete Riemannian manifold $(M,g_0)$ with non-negative
Ricci curvature, $n\geq 3$, the Yamabe flow enjoys the local $L^1$-stability
property from the view-point of the porous media equation. Complete Yamabe
metrics with zero scalar curvature on an n-dimensional Riemannian model space
are also discussed.
