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Based on perturbation theory, we present the exact first-order solution tothe Einstein equations for the exterior static gravitational field of anisolated non-rotating star in a spatially finite universe having the topologyof a flat 3-torus. Since the method of images leads to a divergent Poincare'series, one needs a regularization which we achieve by using the Appellrespectively the Epstein zeta function. The solution depends on a new positiveconstant which is completely fixed by the mass of the star and the spatialvolume of the universe. The physical interpretation is that a stable ormetastable equilibrium requires a topological dark energy which fills the wholeuniverse with positive energy density and negative pressure. The properties ofthe gravitational field are discussed in detail. In particular, its anisotropyis made explicit by deriving an exact multipole expansion which shows that inthis case Birkhoff's theorem does not hold. While the monopole describes theNewtonian potential, there is no dipole but always a non-vanishing quadrupolewhich leads to a repulsive force experienced by a planet at rest. Finally, weput forward the conjecture that black holes exist in a toroidal universe andthat their gravitational field is in the weak-field limit well approximated bythe first-order field.

Do Black Holes Exist in a Finite Universe Having the Topology of a Flat 3-Torus?