Abstract
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 for parabolic operators, we show that the asymptotically flat property is preserved under the Yamabe flow. We also obtain that ADM mass is invariant under the Yamabe flow and Yamabe flow is the gradient flow of Einstein-Hilbert functional on n-dimensional, n ≥ 3, asymptotically flat manifolds with order τ>n−22 for n = 3, 4 or τ > n − 3 for n > 4. Moreover, we show that ADM mass and Einstein-Hilbert functional are non-increasing under the Yamabe flow on n-dimensional asymptotically flat manifolds if we only assume the order τ>n−22 for n > 4.
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