{
  "id": "2010.15663",
  "version": "v1",
  "published": "2020-10-29T15:13:05.000Z",
  "updated": "2020-10-29T15:13:05.000Z",
  "title": "$d_p$ convergence and $ε$-regularity theorems for entropy and scalar curvature lower bounds",
  "authors": [
    "Man-Chun Lee",
    "Aaron Naber",
    "Robin Neumayer"
  ],
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "Consider a sequence of Riemannian manifolds $(M^n_i,g_i)$ with scalar curvatures and entropies bounded below by small constants $R_i,\\mu_i \\geq-\\epsilon_i$. The goal of this paper is to understand notions of convergence and the structure of limits for such spaces. Even in the seemingly rigid case $\\epsilon_i\\to 0$, we construct examples showing that such a sequence may converge wildly in the Gromov-Hausdorff or Intrinsic Flat sense. 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 introduce $d_p$ convergence, a weaker notion of convergence that is valid for a class of rectifiable Riemannian spaces. These rectifiable spaces have well-behaved topology, measure theory, and analysis, though potentially there will be no reasonably associated distance function. Under the $d_p$ notion of closeness, a space with almost nonnegative scalar curvature and small entropy bounds must in fact be close to Euclidean space; this will constitute our $\\epsilon$-regularity theorem. More generally, we have a compactness theorem saying that sequences of Riemannian manifolds $(M^n_i,g_i)$ with small lower scalar curvature and entropy bounds $R_i,\\mu_i \\geq -\\epsilon$ must $d_p$ converge to such a rectifiable Riemannian space $X$. Comparing to the first paragraph, the distance functions of $M_i$ may be degenerating, even though in a well-defined sense the analysis cannot be. Applications for manifolds with small scalar and entropy lower bounds include an $L^\\infty$-Sobolev embedding and apriori $L^p$ scalar curvature bounds for $p<1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-29T15:13:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "scalar curvature lower bounds",
      "regularity theorem",
      "convergence",
      "rectifiable riemannian space",
      "small lower scalar curvature"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}