  In this paper, we investigate the behavior of ADM mass and Einstein-Hilbert
functional under the Yamabe flow. Through studying the Yamabe flow by weighted
spaces, we show that ADM mass and Einstein-Hilbert functional are well-defined
and monotone non-increasing under the Yamabe flow on $n$-dimensional, $n\geq
3$, asymptotically flat manifolds. In the case of dimension $n=3$ or 4, we
obtain that the ADM mass is invariant under the Yamabe flow and the Yamabe flow
is the gradient flow of Einstein-Hilbert functional on asymptotically flat
manifolds
