Some rigidity results for compact initial data sets (Q6567126)

An initial data set \((M,g,K)\) consists of a smooth orientable \(n\)-dimensional manifold \(M\) equipped with a Riemannian metric \(g\) and a symmetric \((0,2)\)-tensor \(K\). The main physical example is when \((M,g,K)\) is an initial data set in a spacetime, i.e., \(M\) is a space-like hypersurface in a Lorentzian manifold \(\overline{M}\), with induced metric \(g\) and second fundamental form \(K\). The local energy density \(\mu\) and the local current density \(J\) of an initial data set \((M,g,K)\) are given by\N\[\N\mu =12(S-|K|^2+(\mathrm{tr}K)^2) \ \ \ \text{and} \ \ \ J=\mathrm{div}(K-\mathrm{tr}(K)g) \ ,\N\]\Nwhere \(S\) is the scalar curvature of \((M,g)\). When \((M,g,K)\) is a spacetime initial data set, these quantities are given by \(\mu=G(u,u)\), \(J=G(u,\cdot)\), where \(G\) is the Einstein tensor.\N\NLet \(\Sigma\) be a closed two-sided surface in \(M\). We say that \(\Sigma\) is an outer trapped surface if \(\theta_+=\mathrm{tr}_{\Sigma}(K)+H < 0\). If \(\theta_+\) vanishes identically we say that \(\Sigma\) is a marginally outer trapped surface (or MOTS). \N\NThe paper under review presents some initial data rigidity results for compact initial data sets. For example, in the main theorem the authors prove that if \((M, g, K)\) satisfies the energy condition \(\mu - |J|\geq c\) for some constant \(c > 0\) and the boundary of \(M\) can be expressed as a disjoint union \(\partial M = \Sigma_0 \cup S\) of nonempty unions of components such that the following conditions hold:\N\begin{itemize}\N\item[1.] \(\theta_+\leq 0\) on \(\Sigma_0\) with respect to the normal vector that points into \(M\);\N\item[2.] \(\theta_+\geq 0\) on \(S\) with respect to the normal vector that points out of \(M\);\N\item[3.] There exists a continuous map \(\rho : M \to\Sigma\) such that \(\rho \circ i: \Sigma\to \Sigma\) is homotopic to \(\mathrm{id}_{\Sigma}\), where \(i :\Sigma \hookrightarrow M\) is the inclusion map;\N\item[4.] The relative homology group \(H^2(M, \Sigma_0)\) vanishes;\N\item[5.] \(\Sigma_0\) minimizes area.\N\end{itemize}\NThen \(\Sigma_0\) is topologically a sphere and its area satisfies \(A(\Sigma_0) \leq 4\pi/c\). Moreover, if \(A(\Sigma_0) = 4\pi/c\), then:\N\begin{itemize}\N\item \((M, g)\) is isometric to \(([0,\ell] \times \Sigma_0, dt^2 + g_0)\) for some \(\ell > 0\), where \(g_0\) is the induced metric on \(\Sigma_0\) and has constant Gaussian curvature \(c\);\N\item \(K = a dt^2\) on \(M\), where \(a \in C^{\infty}(M)\) that depends only on \(t\in [0, \ell]\);\N\item \(\mu = c\) and \(J = 0\) on \(M\).\N\end{itemize}\NThe proof uses in a crucial way the theory of MOTS in initial data sets.