Geometry & Topology最新文献_第3页

dp–convergence and 𝜖–regularity theorems forentropy and scalar curvature lower bounds p -收敛和𝜖-regularity定理，熵定理和标量曲率下界

Geometry & Topology Pub Date : 2020-10-29 DOI: 10.2140/gt.2023.27.227

Man-Chun Lee, A. Naber, Robin Neumayer

{"title":"dp–convergence and 𝜖–regularity theorems for\u0000entropy and scalar curvature lower bounds","authors":"Man-Chun Lee, A. Naber, Robin Neumayer","doi":"10.2140/gt.2023.27.227","DOIUrl":"https://doi.org/10.2140/gt.2023.27.227","url":null,"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 $epsilon_ito 0$, 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 $p<1$.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"117 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115276321","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}