论文标题
质量，批量和等级的中心和渐近平面$ 3 $ - manifolds
Mass, center of mass and isoperimetry in asymptotically flat $3$-manifolds
论文作者
论文摘要
我们在渐近平坦的$ 3 $ manifolds（没有和非紧密的边界）的情况下，重新审视质量，质量中心和某些等速器商的大规模行为之间的相互作用。在无边界的情况下，我们首先检查涉及总平均曲率的等速缺陷是否在渐近极限中恢复ADM质量，从而扩大了由于G. huisken而导致的经典结果。接下来，在Schwarzschild的渐近学下，假设质量是阳性的，我们表明了R. Ye并通过L.-H的精制进行隐式函数方法是如何开创的。黄可以适应以确定无穷大街区的叶面的存在，满足相应的曲率条件。在存在非紧凑型边界的情况下，恢复质量作为相应相对等级赤字的渐近极限也是正确的，在schwarzschild渐近学下，我们再次获得了在schwarzschild的渐近剂中，通过自由边界恒定的均值均值曲率的范围，从而足够众多的相对体积，从而占据了典型的相对范围，从而足够众所周知，这是典型的相对范围。 M. Eichmair和J. Metzger结果。同样，在此处处理的每种情况下，我们将叶面的几何中心与由哈密顿方法定义的歧管质量中心联系起来。
We revisit the interplay between the mass, the center of mass and the large scale behavior of certain isoperimetric quotients in the setting of asymptotically flat $3$-manifolds (both without and with a non-compact boundary). In the boundaryless case, we first check that the isoperimetric deficits involving the total mean curvature recover the ADM mass in the asymptotic limit, thus extending a classical result due to G. Huisken. Next, under a Schwarzschild asymptotics and assuming that the mass is positive we indicate how the implicit function method pioneered by R. Ye and refined by L.-H. Huang may be adapted to establish the existence of a foliation of a neighborhood of infinity satisfying the corresponding curvature conditions. Recovering the mass as the asymptotic limit of the corresponding relative isoperimetric deficit also holds true in the presence of a non-compact boundary, where we additionally obtain, again under a Schwarzschild asymptotics, a foliation at infinity by free boundary constant mean curvature hemispheres, which are shown to be the unique relative isoperimetric surfaces for all sufficiently large enclosed volume, thus extending to this setting a celebrated result by M. Eichmair and J. Metzger. Also, in each case treated here we relate the geometric center of the foliation to the center of mass of the manifold as defined by Hamiltonian methods.
