Jonathan Glöckle: Initial data rigidity and spacetimes

Initial data sets \((g,k)\)on a manifold \(M\) consist of a metric \(g\) and a symmetric \(2\)-tensor \(k\) on \(M\). They are used to describe the gravitational data found on a spacelike hypersurface \(M\) within a time-oriented Lorentzian manifold \((\overline{M},\overline{g})\) - \(g\) being the induced metric and \(k\) the induced second fundamental form. We suppose that \((\overline{M},\overline{g})\) satisfies the dominant energy condition, a curvature condition that is assumed to hold for all physically relevant spacetimes. Then \((g,k)\) is subject to \(\rho \geq |j|_g\), where \(\rho\) and \(j\) are quantities defined in terms of \((g,k)\). In this talk, we will visit situations, where this inequality only holds in the marginal sense \(\rho = |j|_g\) and where it is not possible to locally deform to \(\rho > |j|_g\) without destroying \(\rho \geq |j|_g\) somewhere else. The goal is to show that in this case the spacetime \((\overline{M},\overline{g})\) is locally essentially uniquely determined.