Relation between SWIF limits and metric completions of W^{1,p} limits
Establish whether, when (M^3, g_j) converges to (M^3, g_\infty) in the W^{1,p} sense (for p<2), the Sormani–Wenger intrinsic flat limit (M^3, d_\infty) is isometric to the metric completion of (M^3 \ S, g_\infty), where S is the set on which g_\infty is infinite-valued or degenerate; if not, characterize the relationship between these spaces and ascertain whether a distributional nonnegative scalar curvature condition on g_\infty yields constraints on the SWIF limit.
References
Open Question 3: Recall that we already know there is a SWIF limit space, (M3, d_\infty), which is a rectifiable metric space (possibly the zero space) in the setting of the IAS MinA Compactness Conjecture. If (M3, g_j) \to (M3, g_\infty) have g_j \to g_\infty in the W{1,p} sense can one prove the SWIF limit (M3, d_{\infty}) is isometric to the metric completion of (M3\setminus S, g_\infty) where S is the singular set where g_\infty is infinite valued or degenerate? Are there additional hypothesis that can guarantee they are isometric? If these spaces are not isometric, how are they related? Can one use distributional Scal\ge 0 on the W{1,p} limit to say something about the SWIF limit?