d_p –convergence and –regularity theorems for entropy and scalar curvature lower bounds

Consider a sequence of Riemannian manifolds.Mi^{n; g}i/whose scalar curvatures and entropies are bounded from below by small constants R_i; 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 ! 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 d_p –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 d_p 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 –regularity theorem. In particular, any sequence.Mi^{n; g}i/with lower scalar curvature and entropies tending to zero must d_p –converge to Euclidean space. More generally, we have a compactness theorem saying that sequences of Riemannian manifolds.Mi^{n; g}i/with small lower scalar curvature and entropy bounds R_i; must d_p –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 M_i 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 L^p scalar curvature bounds for p < 1.