论文标题
一些稳定性结果的阳性定理均匀渐近平面$ 3 $ -Manifolds
Some stability results of positive mass theorem for uniformly asymptotically flat $3$-manifolds
论文作者
论文摘要
在本文中，我们表明，对于一系列可定向的完全均匀的均匀扁平$ 3 $ -manifolds $（m_i，g_i）$，带有非负量表曲率和adm $ m（g_i）$倾向于零，通过减去一些开放式subsets $ z_i $，他们的边界$ \ kyq y mathers n of tobles $ \ s req eq。 cm（g_i）^{1/2- \ \ varepsilon} $，对于任何基本点$ p_i \ in m_i \ setminus z_i $，$（m_i \ setMinus z_i，g_i，p_i，p_i，p_i）$可以忽略的音量感。此外，如果我们假设RICCI曲率是从下方统一的，则$（m_i，g_i，p_i）$收敛到$（\ Mathbb {r}^3，g_e，0）$中的Gromov-Hausdorff拓扑。
In this paper, we show that for a sequence of orientable complete uniformly asymptotically flat $3$-manifolds $(M_i , g_i)$ with nonnegative scalar curvature and ADM mass $m(g_i)$ tending to zero, by subtracting some open subsets $Z_i$, whose boundary area satisfies $\mathrm{Area}(\partial Z_i) \leq Cm(g_i)^{1/2 - \varepsilon}$, for any base point $p_i \in M_i\setminus Z_i$, $(M_i\setminus Z_i,g_i,p_i)$ converges to the Euclidean space $(\mathbb{R}^3,g_E,0)$ in the $C^0$ modulo negligible volume sense. Moreover, if we assume that the Ricci curvature is uniformly bounded from below, then $(M_i, g_i, p_i)$ converges to $(\mathbb{R}^3,g_E,0)$ in the pointed Gromov-Hausdorff topology.
