dp-convergence and ε-regularity theorems for entropy and scalar curvature lower bounds

Consider a sequence of Riemannian manifolds (Min, gi) whose scalar curvatures and entropies are bounded from below by small constants Ri, mu i >= -ei. The goal of this paper is to understand notions of convergence and the structure of limits for such spaces. As a first issue, even in the seemingly rigid case ei-* 0, we will construct examples showing that from the Gromov-Hausdorff or intrinsic flat points of view, such a sequence may converge wildly, in particular to metric spaces with varying dimensions and topologies and at best a Finsler-type structure. On the other hand, we will see that these classical notions of convergence are the incorrect ones to consider. Indeed, even a metric space is the wrong underlying category to be working on.Instead, we will introduce a weaker notion of convergence called dp-convergence, which is valid for a class of rectifiable Riemannian spaces. These rectifiable spaces will have a well-behaved topology, measure theory and analysis. This includes the existence of gradients of functions and absolutely continuous curves, though potentially there will be no reasonably associated distance function. Under this dp notion of closeness, a space with almost nonnegative scalar curvature and small entropy bounds must in fact always be close to Euclidean space, and this will constitute our e-regularity theorem. In particular, any sequence (Min, gi) with lower scalar curvature and entropies tending to zero must dp-converge to Euclidean space.More generally, we have a compactness theorem saying that sequences of Riemannian manifolds (Min, gi) with small lower scalar curvature and entropy bounds Ri, mu i >= -e must dp-converge to such a rectifiable Riemannian space X. In the context of the examples from the first paragraph, it may be that the distance functions of Mi are degenerating, even though in a well-defined sense the analysis cannot be. Applications for manifolds with small scalar and entropy lower bounds include an L1-Sobolev embedding and a priori Lp scalar curvature bounds for p < 1.