Yamabe metrics, Fine solutions to the Yamabe flow, and local L1-stability

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.