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Large isoperimetric surfaces in initial data sets. (English) Zbl 1269.53071
For \(m>0\), if \(M_m=\mathbb R^3\setminus\{0\}\) and \(g_m=\varphi^4_m\sum_{i=1}^3d\,x_i^2\), where \(\varphi_m=1+\frac{m}{2\,r}\) with \(r=\sqrt{x_1^2+x_2^2+x_3^2}\), \((M_m,g_m)\) is a totally geodesic space-like slice of the Schwarzschild spacetime of mass \(m\) and it is called the “Schwarzschild metric of mass \(m>0\)”. An initial data set \((M,g)\) is a connected complete Riemannian \(3\)-manifold such that there exists a bounded open set \(U\subset M\) with \(M\setminus U\cong_x\mathbb R^3\setminus B(0,\frac12)\) and such that \(r|g_{ij}-\delta_{ij}|+r^2|\partial_kg_{ij}|+r^3|\partial^2_kg_{ij}|\leq C\). For any initial data set \((M,g)\) there is a complete Riemannian manifold \((\widetilde M,\widetilde g)\) diffeomorphic to \(\mathbb R^3\) that contains \((M,g)\) isometrically. A Borel set \(U\subset\widetilde M\) is said to contain the horizon if \(\widetilde M\setminus M\subset U\). If \(U\) has locally finite perimeter, then its reduced boundary \(\partial^*U\) of \((\widetilde M,\widetilde g)\) is supported in \(M\) and \({\mathcal H}^2_g(\partial^*U)={\mathcal H}^2_{\widetilde g}(\partial^*U)\). The isoperimetric area function \(A_g\) is defined as \(A_g(V)=\inf\{{\mathcal H}^2_{g}(\partial^*U)\}\), where \(U\subset\widetilde M\) is a Borel set containing the horizon and of finite perimeter with \({\mathcal L}_g^3(U)=V\). A Borel set \(\Omega\subset\widetilde M\) containing the horizon and of finite perimeter with \({\mathcal L}_g^3(U)=V\) and \(A_g(V)=\inf\{{\mathcal H}^2_{g}(\partial^*U)\}\) is called an isoperimetric region of \((M,g)\) of volume \(V\).
In this paper, the authors study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds \((M, g)\) that are \(\mathcal{C}^0\)-asymptotic to the Schwarzschild metric of mass \(m>0\). They show that if \((M,g)\) is an initial data set that is \(C^0\)-asymptotic to the Schwarzschild metric of mass \(m>0\), then there exists \(V_0>0\) such that for every \(V\geq V_0\), the infimum in \(A_g(V)\) is achieved and every minimizer has a smooth bounded representative whose boundary consists of the horizon and a connected surface that is close to a centered coordinate sphere. Also, it is shown that if the initial data set \((M,g)\) is \(C^4\)-asymptotic to the Schwarzschild metric of mass \(m>0\), then the boundaries of the large isoperimetric regions coincide with the volume-preserving stable constant mean curvature surfaces. In particular, for every sufficiently large volume there exists a unique isoperimetric region in \((M,g)\) of that volume. The boundaries of these regions foliate the complement of a bounded subset of \((M,g)\).

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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