We study a class of non-smooth asymptotically flat manifolds on which metrics fail to be C 1 across a hypersurface Σ. We first give an approximation scheme to mollify the metric, then we show that the Positive Mass Theorem [8] still holds on these manifolds if a geometric boundary condition is satisfied by metrics separated by Σ. e-print archive: http://lanl.arXiv.org/abs/math-ph/0212025

Abstract. We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and "small" hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional.

Abstract. Let R be a constant. Let M R γ be the space of smooth metrics g on a given compact manifold Ω n (n ≥ 3) with smooth boundary Σ such that g has constant scalar curvature R and g| Σ is a fixed metric γ on Σ. Let V (g) be the volume of g ∈ M R γ . In this work, we classify all Einstein or conformally flat metrics which are critical points of V (·) in M R γ .

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