Celebratio Mathematica — Choquet-Bruhat

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Y. Choquet-Bruhat, A. Fisher, and J. Marsden: “Équations des contraintes sur une variété non compacte,” C. R. Acad. Sci. Paris Sér. A-B 284 : 16 (1977), pp. A975–A978. MR 433498 Zbl 0364.58010 article

The basic data one needs for the “initial-value problem” in general relativity are: a three-dimensional manifold \( S \) with a Riemannian metric \( g \) and a symmetric rank two tensor field \( k \) which satisfy certain constraint equations. Theorems from p.d.e. state that \( g \) and \( k \) will generate a four-dimensional space-time which satisfies Einstein field equations and contains \( S \) as a spacelike hypersurface with second fundamental form \( k \). The constraint equations are said to be linearization stable at a given solution if every nonzero solution of the linearized constraint equations is tangent to a curve of solutions of the nonlinear constraint equations. This article gives a new simple method for proving linearization stability of the constraint equations in the case where \( S \) is not necessarily compact, but trace \( (k) \) is constant.