In this paper we study the family of embeddings ${\mathrm{\Phi }}_t$ of a compact ${\mathrm{RCD}}^{⁎}(K,N)$ space $(X,\mathsf{d},\mathfrak{m})$ into $L^2(X,\mathfrak{m})$ via eigenmaps. Extending part of the classical results [10], [11] known for closed Riemannian manifolds, we prove convergence as $t\downarrow 0$ of the rescaled pull-back metrics ${\mathrm{\Phi }}_t^{⁎}{g}_{L^2}$ in $L^2(X,\mathfrak{m})$ induced by ${\mathrm{\Phi }}_t$. Moreover we discuss the behavior of ${\mathrm{\Phi }}_t^{⁎}{g}_{L^2}$ with respect to measured Gromov-Hausdorff convergence and t. Applications include the quantitative $L^p$-convergence in the noncollapsed setting for all $p<\infty$, a result new even for closed Riemannian manifolds and Alexandrov spaces.

中文翻译：

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RCD ^ding（K，N）空间通过特征函数嵌入L ^2中。

在本文中，我们研究了嵌入族 ${\mathrm{\Phi }}_{Ť}$ 紧凑的 ${\mathrm{刚果民盟}}^{⁎}（ķ，ñ）$ 空间 $（X，\mathsf{d}，\mathfrak{米}）$ 进入 ${\mathrm{大号}}^2（X，\mathfrak{米}）$通过特征图 扩展已知的经典黎曼流形[10]，[11]的部分结果，证明收敛为$Ť\downarrow 0$ 重新调整后的拉回指标 ${\mathrm{\Phi }}_{Ť}^{⁎}{G}_{{\mathrm{大号}}^2}$ 在 ${\mathrm{大号}}^2（X，\mathfrak{米}）$ 由...介绍 ${\mathrm{\Phi }}_{Ť}$。此外，我们讨论了${\mathrm{\Phi }}_{Ť}^{⁎}{G}_{{\mathrm{大号}}^2}$关于测得的Gromov-Hausdorff收敛和t。应用包括定量${\mathrm{大号}}^p$-在所有的非折叠设置中收敛 $p<\infty$，即使对于封闭的黎曼流形和Alexandrov空间，也是一个新结果。