NON SINGULAR VACUUM STATIONARY SPACE TIMES WITH A NEGATIVE COSMOLOGICAL CONSTANT

NON-SINGULAR, VACUUM, STATIONARY SPACE-TIMES WITH A NEGATIVE COSMOLOGICAL CONSTANT PIOTR T. CHRUSCIEL AND ERWANN DELAY Abstract. We construct infinite dimensional families of non-singular stationary space times, solutions of the vacuum Einstein equations with a negative cosmological constant. Contents 1. Introduction 1 2. Definitions, notations and conventions 3 3. Isomorphism theorems 4 3.1. An isomorphism on two-tensors 4 3.2. Two isomorphisms on one-forms 5 3.3. An isomorphism on functions in dimension n 6 3.4. An isomorphism on functions in dimension 3 6 4. The equations 8 4.1. The linearised equation 9 4.2. The modified equation 9 5. The construction 12 5.1. The n-dimensional case 12 5.2. The three-dimensional case 14 6. Uniqueness 15 7. Polyhomogeneity 16 Appendix A. “Dimensional reduction” of some operators 17 A.1. Lichnerowicz Laplacian on two-tensor for a warped product metric 17 A.2. The Laplacian on one-forms for a warped product metric 18 References 19 1. Introduction A class of space-times of interest is that of vacuum metrics with a negative cosmological constant admitting a smooth conformal completion at infinity. It is natural to seek for stationary solutions with this property. In this paper we show that a large class of such solutions can be constructed by prescribing the conformal class of a stationary Lorentzian metric on the conformal boundary ∂M , provided that the boundary data are sufficiently close to, e.

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