Some stability results of positive mass theorem for uniformly asymptotically flat $3$-manifolds
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2022
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916139467014144 |
|---|---|
| author | Dong, Conghan |
| author_facet | Dong, Conghan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_06730 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Some stability results of positive mass theorem for uniformly asymptotically flat $3$-manifolds Dong, Conghan Differential Geometry General Relativity and Quantum Cosmology 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. |
| title | Some stability results of positive mass theorem for uniformly asymptotically flat $3$-manifolds |
| topic | Differential Geometry General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2211.06730 |