Smooth convergence away from singular sets (Q357205)

The paper under review investigates the situation when a sequence \(g_j\) of Riemannian metrics on a manifold \(M\) converges smoothly (as tensors) outside some set \(S\) to \(g_\infty\) and it asks under which assumptions \((M,g_i)\) Gromov-Hausdorff converges to the metric completion of \((M-S,g_\infty)\). The authors prove a number of theorems and present a number of examples with and without Ricci curvature bounds. One fairly general result is the following:NEWLINENEWLINEIf a sequence of oriented compact Riemannian manifolds with uniform lower curvature bounds and uniformly bounded diameter converges smoothly away from a codimension 2 submanifold \(S\) and if there exists a connected precompact exhaustion \(W_j\) of \(M-S\) with \(\mathrm{Vol}_{g_i}(\partial W_j)\) uniformly bounded and \(\mathrm{Vol}_{g_i}(M-W_j)\leq V_j\) for some sequence \(V_j\rightarrow 0\), then \((M,g_i)\) Gromov-Hausdorff converges to the metric completion of \((M-S,g_\infty)\).NEWLINENEWLINEThe paper is well written with many examples and many pictures.