Embedding of $RCD^*(K,N)$ spaces in $L^2$ via eigenfunctions

In this paper we study the family of embeddings $Phi_t$ of a compact $RCD^*(K,N)$ space $(X,d,m)$ into $L^2(X,m)$ via eigenmaps. Extending part of the classical results by Berard, Berard-Besson-Gallot, known for closed Riemannian manifolds, we prove convergence as $tdownarrow 0$ of the rescaled pull-back metrics $Phi_t^*g_{L^2}$ in $L^2(X,m)$ induced by $Phi_t$. Moreover we discuss the behavior of $Phi_t^*g_{L^2}$ 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.