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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 $$(M,g)$$.

MSC:
 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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