Homologically maximizing geodesics in conformally flat tori

Details Homologically maximizing geodesics in conformally flat tori Homologically maximizing geodesics in conformally flat tori

2010-03-11

arxiv.org/abs/1003.2322v1

Mathematics

Differential Geometry

16 pages, submitted to Adv. in Lor. geometry

Scientific paper

We study homologically maximizing timelike geodesics in conformally flat tori. A causal geodesic $\gamma$ in such a torus is said to be homologically maximizing if one (hence every) lift of $\gamma$ to the universal cover is arclength maximizing. First we prove a compactness result for homologically maximizing timelike geodesics. This yields the Lipschitz continuity of the time separation of the universal cover on strict sub-cones of the cone of future pointing vectors. Then we introduce the stable time separation $\mathfrak{l}$. As an application we prove relations between the concavity properties of $\mathfrak{l}$ and the qualitative behavior of homologically maximizing geodesics.

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