具有光滑反平均曲率叶理的黎曼三空间的构造
On the construction of Riemannian three-spaces with smooth inverse mean curvature foliation
摘要：
考虑双球面上的单参数光滑黎曼度量族$\mathscr{S}$。通过选择一个单参数的光滑平移和平移族，这些黎曼双球面总是可以组装成光滑的黎曼三空间，度量$h_{ij}$在一个由单参数双球面族$\mathscr{S}_ρ$划分的三流形$Σ$上。它表明，首先，我们总是可以选择的移位，使$\mathscr{S}_ρ$表面形成一个光滑的逆平均曲率叶理的$Σ$。一个integodifferential表达式，只指的水平集和失效函数的面积，也可以用来量化的Geroch质量。如果所构造的黎曼三维空间恰好是渐近平坦的，并且积分微分表达式的$ρ$ -积分是非负的，那么不仅正质量定理成立，而且如果其中一个$\mathscr{S}_ρ$水平集是极小曲面，彭罗斯不等式也成立。值得注意的是，上述结果都不要求构造的三度量的标量曲率是非负的。
英文摘要：
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 $Σ$ foliated by a one-parameter family of two-spheres $\mathscr{S}_ρ$. It is shown first that we can always choose the shift such that the $\mathscr{S}_ρ$ surfaces form a smooth inverse mean curvature foliation of $Σ$. 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 $ρ$-integral of the integrodifferential expression is non-negative, then not only the positive mass theorem but, if one of the $\mathscr{S}_ρ$ 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.
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