Convergence and Regularity of Manifolds with Scalar Curvature and Entropy Lower Bounds

We discuss in this short survey the notion of dp convergence and its application to studying limits of sequence of Riemannian manifolds (Mⁿ_i, gi) whose scalar curvatures and entropies are bounded from below by small constants. We will build examples showing how the more classical notions of Gromov–Hausdorff and Intrinsic Flat convergence must fail in the context of lower scalar bounds. More fundamentally, we will see how a notion of a metric space itself is the wrong notion for such limits, as distance functions may degenerate. We will see how to fix these problems by weakening the convergence criteria with respect to the dp-structure. The d∞-structure corresponds to the classical metric structure, but for any p < ∞ a dp-structure still comes equipped with a well behaved analysis. Our main result is to show that if a manifold with lower scalar and entropy bounds R, μ ≥ −_(n), then a ball must be dp-close to a Euclidean ball for p = p(n, ϵ) < ∞. We also study more general limits, and have applications which include a priori estimates on such spaces.