Consider a one-parameter family of smooth Riemannian metrics on a two-sphere, \({\mathscr {S}}\). By choosing a one-parameter family of smooth lapse and shift, these Riemannian two-spheres can always be assembled into smooth Riemannian three-space, with metric \(h_{ij}\) on a three-manifold \(\Sigma \) foliated by a one-parameter family of two-spheres \({\mathscr {S}}_\rho \). It is shown first that we can always choose the shift such that the \({\mathscr {S}}_\rho \) surfaces form a smooth inverse mean curvature foliation of \(\Sigma \). An integrodifferential expression, referring only to the area of the level sets and the lapse function, is also derived that can be used to quantify the Geroch mass. If the constructed Riemannian three-space happens to be asymptotically flat and the \(\rho \)-integral of the integrodifferential expression is non-negative, then not only the positive mass theorem but, if one of the \({\mathscr {S}}_{\rho }\) level sets is a minimal surface, the Penrose inequality also holds. Notably, neither of the above results requires the scalar curvature of the constructed three-metric to be non-negative.

考虑在一个二球体\({\mathscr {S}}\)上的平滑黎曼度量的单参数族。通过选择单参数的平滑失效和移位族，这些黎曼二球总是可以组装成平滑黎曼三空间，度量\(h_{ij}\)在三流形\(\Sigma \)由两个球体的单参数族\({\mathscr {S}}_\rho \) 组成。首先表明，我们总是可以选择平移，使得\({\mathscr {S}}_\rho \)曲面形成\(\Sigma \)的平滑逆平均曲率叶状结构. 还导出了一个积分微分表达式，仅涉及水平集的面积和失效函数，可用于量化 Geroch 质量。如果构造的黎曼三空间恰好是渐近平坦的，并且积分微分表达式的\(\rho \) -积分是非负的，那么不仅是正质量定理，而且如果\({\mathscr { S}}_{\rho }\)水平集是一个极小曲面，彭罗斯不等式也成立。值得注意的是，上述结果都不需要构造的三度量的标量曲率是非负的。