Stability of Euclidean 3-space for the positive mass theorem (Q6659473)

The paper proves that Euclidean \(3\)-space is stable for the positive mass theorem, more precisely the authors prove the following main theorem: \N\NTheorem. Let \((M_i ,g_i )\) be a sequence of complete asymptotically flat \(3\)-manifolds with nonnegative scalar curvature. Suppose that the ADM mass \(m(g_i)\) of one end \(\mathcal{E}_{i}\) of \(M_i\) converges to \(0\). Then for all \(i\) there is an open domain \(Z_i\) in \(M_i\) with smooth compact boundary \(\partial Z_i\), such that\N\begin{itemize}\N\item[1.] \(M_i \setminus Z_i\) contains the given end \(\mathcal{E}_i\) ,\N\item[2.] the area of \(\partial Z_i\) converges to \(0\),\N\item[3.] for any choice of basepoint \(p_i \in M_i \setminus Z_i\) , the sequence \((M_i \setminus Z_i , \hat{d}_{g_{i}} , p i )\) converges to \((\mathbb{R}^{3}, d_{Eucl}, 0)\) in the pointed measured Gromov-Hausdorff topology, where \(\hat{d}_{g_{i}}\) is the length metric on \(M_{i}\setminus Z_{i}\) induced by \(g_i\). Moreover, if \(m(g_i) > 0\) for all \(i\), the region \(Z_i\) can be chosen so that the area of \(\partial Z_i\) is almost bounded quadratically by the mass in the following sense: for any positive continuous function \(\xi : (0, +\infty) \rightarrow (0, +\infty)\) with\N\[\N\lim_{x\to 0^{+}} \xi(x)=0,\N\]\Nfor all large \(i\), \(Z_{i}\) can be chosen depending on \(\xi\) such that\N\[\N\mathrm{Area}(\partial Z_{i}) \leq \frac{m(g_{i})^{2}}{\xi(m(g_{i}))}.\N\]\N\end{itemize}\N\NAs a consequence of the main theorem, the authors establish the rigidity of the Bartnik capacity \(c_{B}(\Omega)\) of an admissible open Riemannian 3-manifold \(\Omega\): they show that \(c_{B}(\Omega)>0\), unless there is a Riemannian isometric embedding of \(\Omega\) into the Euclidean 3-space.