论文标题
$ d_p $收敛和$ε$ - 熵定理和标态曲率下限
$d_p$ convergence and $ε$-regularity theorems for entropy and scalar curvature lower bounds
论文作者
论文摘要
考虑一系列riemannian歧管$（m^n_i，g_i）$，带有标态曲率和熵，下面由小常数$ r_i，μ_i\ geq-ε_i$。本文的目的是了解融合的概念和此类空间的限制结构。即使在看似刚性的情况下，$ε_i\ to 0 $，我们构建了示例，表明这种序列可能会在gromov-hausdorff或固有的平坦感中疯狂收敛。另一方面，我们将看到这些经典的融合概念是要考虑的不正确的观念。确实，即使是度量空间也是错误的基本类别。 取而代之的是，我们引入了$ D_P $ Convergence，这是一个较弱的收敛概念，适用于一类可纠正的Riemannian空间。这些可调的空间具有良好的拓扑，测量理论和分析，尽管可能没有合理相关的距离函数。在$ d_p $的亲密概念下，一个几乎非负标态曲率和小熵边界的空间实际上必须接近欧几里得空间。这将构成我们的$ε$ -REGULANITY定理。 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 $R_i,μ_i \geq -ε$ must $d_p$ converge to such a rectifiable Riemannian space $X$.与第一段相比，$ m_i $的距离函数可能是退化的，即使在明确的意义上，分析不能进行。具有小标量和熵下限的流形的应用包括$ l^\ infty $ -sobolev嵌入和apriori $ l^p $标量曲率稳定性范围，$ p <1 $。
Consider a sequence of Riemannian manifolds $(M^n_i,g_i)$ with scalar curvatures and entropies bounded below by small constants $R_i,μ_i \geq-ε_i$. 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 $ε_i\to 0$, 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 $ε$-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 $R_i,μ_i \geq -ε$ 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 $p<1$.
