Celebratio Mathematica

Y. Choquet-Bruhat and V. Moncrief: “An existence theorem for the reduced Einstein equation,” C. R. Acad. Sci. Paris Sér. I Math. 319 : 2 (1994), pp. 153–159. MR 1288395 Zbl 0813.35060 article

Y. Choquet-Bruhat and V. Moncrief: “Existence theorem for solutions of Einstein’s equations with 1 parameter spacelike isometry groups,” pp. 67–80 in Quantization, nonlinear partial differential equations, and operator algebra (Cambridge, MA, 1994). Edited by W. Arveson, T. Branson, and I. Segal. Proc. Sympos. Pure Math. 59. Amer. Math. Soc. (Providence, RI), 1996. MR 1392984 Zbl 0861.53063 incollection

We shall first show in this paper how one can obtain by direct methods, without making use of a Hamiltonian formalism, the splitting of part of the vacuum Einstein equations with 1 parameter isometry group into a system elliptic on each spacelike slice on the one hand, and on the other hand a hyperbolic system which is essentially a harmonic map equation from a pseudo-Riemannian manifold, whose metric depends on the solution of the elliptic system, into the 2-dimensional symmetric Riemannian space \( \mathcal{H} \).

We then prove global in space, local in time existence and uniqueness of solutions of the Cauchy problem for the hyperbolic-elliptic system so obtained, when \( \Sigma = S^2 \), in appropriate functional spaces. We show that in that case, the obtained solution satisfies the original Einstein equations. We finally discuss the general solution of the system, in the case where \( \Sigma \) is not simply connected. We show that when the genus of \( \Sigma \) is greater than 1, we can determine gt so that it is conformal to a metric at which remains in some chosen cross section of a bundle over the Teiehmuller space of \( \Sigma \) (cf. an analogous method in [Moncrief 1989] for the 3-dimensional Einstein vacuum equations). Here the evolution of at is determined by requiring the mixed elliptic-hyperbolic system obtained previously to be a solution of the original, 4-dimensional, Einstein equations

Y. Choquet-Bruhat and V. Moncrief: “Future complete Einsteinian space times with \( U(1) \) isometry group,” C. R. Acad. Sci. Paris Sér. I Math. 332 : 2 (2001), pp. 137–144. A version of this paper was published as “Future global in time Einsteinian spacetimes with \( U(1) \) isometry group,” in Ann. Henri Poincaré 6, 1007–1064 (2001). MR 1813771 Zbl 0976.83004 article

Y. Choquet-Bruhat and V. Moncrief: “Future global in time Einsteinian spacetimes with \( U(1) \) isometry group,” Ann. Henri Poincaré 2 : 6 (2001), pp. 1007–1064. MR 1877233 Zbl 0998.83007 article

Y. Choquet-Bruhat and V. Moncrief: “Nonlinear stability of an expanding universe with the \( S^1 \) isometry group,” pp. 57–71 in Partial differential equations and mathematical physics (Tokyo, 2001). Edited by K. Kajitani and J. Vaillant. Progr. Nonlinear Differential Equations Appl. 52. Birkhäuser (Boston), 2003. MR 1957625 Zbl 1062.35149 incollection

Y. Choquet-Bruhat, J. Isenberg, and V. Moncrief: “Topologically general U(1) symmetric vacuum space-times with AVTD behavior,” Nuovo Cimento Soc. Ital. Fis. B 119 : 7–9 (2004), pp. 625–638. MR 2136898 article

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