Embedding of RCD⁎(K,N) spaces in L2 via eigenfunctions

In this paper we study the family of embeddings Φt of a compact RCD⁎(K,N) space (X,d,m) into L2(X,m) via eigenmaps. Extending part of the classical results 10,11 known for closed Riemannian manifolds, we prove convergence as t↓0 of the rescaled pull-back metrics Φt⁎gL2 in L2(X,m) induced by Φt. Moreover we discuss the behavior of Φt⁎gL2 with respect to measured Gromov-Hausdorff convergence and t. Applications include the quantitative Lp-convergence in the noncollapsed setting for all p<∞, a result new even for closed Riemannian manifolds and Alexandrov spaces.