Compact hyperbolic universe and singularities

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Abstract
Recently many people have discussed the possibility that the universe is hyperbolic and was in an inflationary phase at an early stage. Under these assumptions, it is shown that the universe cannot have compact hyperbolic time slices. Though the universal covering space of the universe has a past Cauchy horizon and can be extended analytically beyond it, the extended region has densely many points which correspond to singularities of the compact universe. The result is essentially attributed to the ergodicity of the geodesic flow on a compact negatively curved manifold. The relationship with the strong cosmic censorship conjecture also is discussed.
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