A convergence notion combining volume-preserving SWIF and W^{1,p} control

Develop a notion of convergence, and preferably a metric on the space of Riemannian manifolds, for sequences of pairwise non-diffeomorphic Riemannian manifolds (M_j, g_j) that implies volume-preserving Sormani–Wenger intrinsic flat convergence and simultaneously captures the information of a W^{1,p} limit on diffeomorphic regions while allowing the complement to have vanishing volume.

Background

The authors seek a convergence framework that unifies SWIF’s geometric-measure behavior (including volume preservation) with analytic control afforded by W^{1,p} convergence on large diffeomorphic regions, accommodating small-volume “bad” sets.

Such a framework would deepen understanding of limit spaces with singular sets and potentially yield a robust distance on the moduli space of Riemannian manifolds reflecting both geometric and analytic convergence.