Constant mean curvature foliations of flat space--times

Details Constant mean curvature foliations of flat space--times Constant mean curvature foliations of flat space--times

2001-10-22

arxiv.org/abs/math/0110245v1

Comm. Anal. Geom. vol. 10, pp. 1094-1115, 2002

Mathematics

Differential Geometry

20 pages

Scientific paper

Let $V$ be a maximal globally hyperbolic flat $n+1$--dimensional space--time with compact Cauchy surface of hyperbolic type. We prove that $V$ is globally foliated by constant mean curvature hypersurfaces $M_{\tau}$, with mean curvature $\tau$ taking all values in $(-\infty, 0)$. For $n \geq 3$, define the rescaled volume of $M_{\tau}$ by $\Ham = |\tau|^n \Vol(M,g)$, where $g$ is the induced metric. Then $\Ham \geq n^n \Vol(M,g_0)$ where $g_0$ is the hyperbolic metric on $M$ with sectional curvature -1. Equality holds if and only if $(M,g)$ is isometric to $(M,g_0)$.

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