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

Rácz, István (2022) On the construction of Riemannian three-spaces with smooth inverse mean curvature foliation. GENERAL RELATIVITY AND GRAVITATION, 54 (6). ISSN 0001-7701

Preview

Text 1912.02576.pdf Download (731kB) | Preview

Abstract

Consider a one-parameter family of smooth Riemannian metrics on a two-sphere, S. By choosing a one-parameter family of smooth lapse and shift, these Riemannian twospheres can always be assembled into smooth Riemannian three-space, with metric hi j on a three-manifold Sigma foliated by a one-parameter family of two-spheres S.. It is shown first that we can always choose the shift such that the S. 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 threespace 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 S. 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.

Actions (login required)

Edit Item