We derive necessary and sufficient conditions for the existence of marginally stable circular orbits (MSCOs) of test particles in a stationary axisymmetric (SAS) spacetime which possesses a reflection symmetry with respect to the equatorial plane; photon orbits and marginally bound orbits (MBOs) are also addressed. Energy and angular momentum are shown to decouple from metric quantities, rendering a purely geometric characterization of circular orbits for this general class of metrics. The subsequent system is analyzed using resultants, providing an algorithmic approach for finding MSCO conditions. MSCOs, photon orbits and MBOs are explicitly calculated for concrete examples of physical interest.

We present a conjecture, based on computational results, on the area minimizing way to enclose and separate two arbitrary volumes in the flat cubic three-torus T 3 . For comparable small volumes, we prove that an area minimizing double bubble in T 3 is the standard double bubble from R 3 .

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We consider a system representing self-gravitating balls of dust in an expanding Universe.It is demonstrated that one can prescribe data for such a system at infinity and evolve it backward in time without the development of shocks or singularities. The resulting solution to the Einstein-λ-dust equations exists for an infinite amount of time in the asymptotic region of the spacetime. Furthermore, we find that if the density is small compared to the Cosmological constant, then it is possible to construct Cosmological solutions to the Einstein constraint equations on a standard Cauchy hypersurface representing self-gravitating balls of dust. If, in addition, the density is assumed to be sufficiently small, then this initial data gives rise to a future geodesically complete solution to the Einstein-λ-dust equations admitting a smooth conformal extension at infinity which can be regarded as a perturbation of de Sitter spacetime. The main technical tool in this analysis are Friedrich's conformal Einstein field equations for the Einstein-λ-dust written in terms of a gauge in which the flow lines of the dust are recast as conformal geodesics.

After giving the most general formulation to date of the notion of integrability for axially symmetric harmonic maps from R 3 into symmetric spaces, we give a complete and rigorous proof that, subject to some mild restrictions on the target, all such maps are integrable. Furthermore, we prove that a variant of the inverse scattering method, called vesture (dressing) can always be used to generate new solutions for the harmonic map equations starting from any given solution. In particular we show that the problem of finding N -solitonic harmonic maps into a noncompact Grassmann manifold SU (p, q)/S(U (p) × U (q)) is completely reducible via the vesture (dressing) method to a problem in linear algebra which we prove is solvable in general. We illustrate this method by explicitly computing a 1-solitonic harmonic map for the two cases (p = 1, q = 1) and (p = 2, q = 1); and we show that the family of solutions obtained in each case contains respectively the Kerr family of solutions to the Einstein vacuum equations, and the Kerr-Newman family of solutions to the Einstein-Maxwell equations. * beheshti@math.rutgers.edu †