On an asymptotically flat manifold M n with nonnegative scalar curvature, with outer minimizing boundary Σ, we prove a Penrose-like inequality in dimensions n < 8, under suitable assumptions on the mean curvature and the scalar curvature of Σ.exists and is known as the ADM mass ([1]) of M.Here ω n−1 is the area of the standard unit (n − 1)-sphere in R n , S r = {x | |x| = r}, ν is the Euclidean outward unit normal to S r , dσ is the Euclidean area element on S r , and summation is implied over repeated indices. Under suitable conditions, it was proved by Bartnik [2] and

Abstract. Let g be a metric on the 2-sphere S 2 with positive Gaussian curvature and H be a positive constant. Under suitable conditions on (g, H), we construct smooth, asymptotically flat 3-manifolds M with non-negative scalar curvature, with outer-minimizing boundary isometric to (S 2 , g) and having mean curvature H, such that near infinity M is isometric to a spatial Schwarzschild manifold whose mass m can be made arbitrarily close to a constant multiple of the Hawking mass of (S 2 , g, H). Moreover, this constant multiplicative factor depends only on (g, H) and tends to 1 as H tends to 0. The result provides a new upper bound of the Bartnik mass associated to such boundary data.

Abstract. The Minkowski inequality is a classical inequality in differential geometry, giving a bound from below, on the total mean curvature of a convex surface in Euclidean space, in terms of its area. Recently there has been interest in proving versions of this inequality for manifolds other than R n ; for example, such an inequality holds for surfaces in spatial Schwarzschild and AdSSchwarzschild manifolds. In this note, we adapt a recent analysis of Y. Wei to prove a Minkowski-like inequality for general static asymptotically flat manifolds.