Yamabe flow and ADM mass on asymptotically flat manifolds

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−2 2 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−2 2 for n > 4. Mathematics Subject Classification (2000): 53C21,51P05 (defined in Definition 2.1) in the coordinates {x i } induced on M ∞ . And the coordinates {x i } are called asymptotic coordinates.Definition 1.2. We say that u(x, t) is a fine solution of Yamabe flow, 0 ≤ t < t max , on a complete manifold (M n for any T < t max , such that either lim t→tmax sup M |Rm|(·, t) = ∞ for t max < ∞ or t max = ∞, where Rm(g) is the Riemannian curvature of the metric g := g(t) = u 4/(n−2) g 0 .

A probabilistic method for gradient estimates of some geometric flows

“…Note that by [4, Lemma 2.2], for each (t, x) ∈ [0, T ] ×U , we have } and the operators F t := − a t n−1 ,F t := − 3a t n−1 . By applying Theorem 3.2, we can give a local gradient estimate for the Yamabe flow (see [7,Theorem 3.4]). for some positive constant C(n).…”

Infinite-time incompleteness of noncompact Yamabe flow

“…For instance, the Euclidean space is a manifold with one end and a cylinder is a manifold with two ends. In this direction, for instance, Chen and Zhu [4], Ma [21] and Chen and Wang [3] studied Riemannian manifolds with one end which is asymptotically flat or Euclidean. These results are strongly related to ADM mass.…”

Convergence rate of the prescribed curvature flow

“…The existence and convergence of the Yamabe flow has been studied in [5,6,14,34,38]. See also [3,12,16,18,19,20,29,33] and references therein for results related to the Yamabe flow. In [7], Carlotto, Chodosh and Rubinstein studied the rate of convergence of the Yamabe flow (1.1).…”