$d_p$ convergence and $\epsilon$-regularity theorems for entropy and scalar curvature lower bounds.
$d_p$ convergence and $\epsilon$-regularity theorems for entropy and scalar curvature lower bounds.
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发表时间：
2020-10
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通讯作者：
Man-Chun Lee;A. Naber;Robin Neumayer
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作者：
Man-Chun Lee;A. Naber;Robin Neumayer
考虑一个黎曼流形序列$(M^n_i,g_i)$，其标量曲率和熵在下面由小常数$R_i,\mu_i \geq-\epsilon_i$限定。本文的目的是了解收敛的概念和这种空间的极限结构。即使在看似刚性的情况下$\epsilon_i\to 0$，我们构造的例子表明，这样的序列可能在Gromov-Hausdorff或内在平坦意义下疯狂收敛。另一方面，我们会发现这些经典的收敛概念是不正确的。事实上，即使是度量空间也是一个错误的潜在范畴。相反，我们引入$d_p$收敛性，一个较弱的收敛性概念，它对一类可整流黎曼空间有效。这些可整流空间具有良好的拓扑、测量理论和分析，尽管可能没有合理的相关距离函数。在$d_p$接近性的概念下，具有几乎非负标量曲率和小熵界的空间实际上必须接近欧几里德空间；这就构成了$\epsilon$ -正则性定理。更一般地说，我们有一个紧性定理，表明具有较小的低标量曲率和熵界$R_i,\mu_i \geq -\epsilon$的黎曼流形序列$(M^n_i,g_i)$必须$d_p$收敛于这样一个可整流黎曼空间$X$。与第一段相比，$M_i$的距离函数可能是退化的，即使在一个明确定义的意义上，分析不能退化。具有小标量和熵下界的流形的应用包括$L^\infty$ -Sobolev嵌入和$p<1$的先验$L^p$标量曲率边界。
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 $p<1$.