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

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