On the construction of Riemannian three-spaces with smooth inverse mean curvature foliation

TL;DR

This paper demonstrates how to construct smooth Riemannian three-spaces with inverse mean curvature foliations on two-spheres, deriving expressions to quantify Geroch mass and establishing conditions for positive mass and Penrose inequalities without scalar curvature constraints.

Contribution

It introduces a method to assemble Riemannian three-spaces with inverse mean curvature foliations using smooth lapse and shift functions, without requiring non-negative scalar curvature.

Findings

Constructed Riemannian three-spaces with smooth inverse mean curvature foliation.

Derived an integrodifferential expression for Geroch mass based on area and lapse.

Established conditions for positive mass and Penrose inequality without scalar curvature assumptions.

Abstract

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.