Inverse mean curvature flow and the stability of the positive mass theorem (Q6628924)

The positive mass theorem (PMT) says that a complete asymptotically flat \(3\)-manifold \(M\) with nonnegative scalar curvature must have positive ADM mass, and if \(m_{ADM}(M)=0\) then \(M\) is isometric to Euclidean space. Recently, the stability of the PMT has been studied by many authors. In particular, \textit{D. A. Lee} and \textit{C. Sormani} [J. Reine Angew. Math. 686, 187--220 (2014; Zbl 1291.53048)] studied the stability of the positive mass theorem under the assumption of rotational symmetry using the intrinsic flat distance. The author [Ann. Henri Poincaré 19, No. 4, 1283--1306 (2018; Zbl 1409.53034)] studied the stability of the PMT under \(L^2\) convergence on manifolds which can be foliated by a smooth solution of the inverse mean curvature flow (IMCF) which is uniformly controlled.\N\NThe main goal of this paper is to study the stability of the PMT in the case where a sequence of regions of manifolds with positive scalar curvature \(U_T^i\subset M_i^3\) can be foliated by a smooth solution to IMCF which may not be uniformly controlled near the boundary. Firstly, assuming uniform control on various curvature quantities the author obtains the stability result under Gromov-Hausdorff convergence and Sormani-Wenger intrinsic flat (SWIF) convergence. Then the author proves new stability results under SWIF convergence which allow the regions foliated by IMCF to approach the jump regions of the weak IMCF. If \(\partial U_T^i=\Sigma_0^i\cup \Sigma_T^i\), the Hawking mass \(m_H(\Sigma_T^i)\to 0\) and some extra assumptions are satisfied, the author shows that \(U_T^i\) converges to a flat annulus with respect to SWIF convergence. In particular, the lower bound on the mean curvature is allowed to degenerate.