Embedding of RCD<sup>⁎</sup>(K,N) spaces in L<sup>2</sup> via eigenfunctions
Luigi Ambrosio, Shouhei Honda, Jacobus W. Portegies, David Tewodrose
Abstract
In this paper we study the family of embeddings Φ<sub>t</sub> of a compact RCD<sup>⁎</sup>(K,N) space (X,d,m) into L<sup>2</sup>(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 Φ<sub>t</sub><sup>⁎</sup>g<sub>L<sup>2</sup></sub> in L<sup>2</sup>(X,m) induced by Φ<sub>t</sub>. Moreover we discuss the behavior of Φ<sub>t</sub><sup>⁎</sup>g<sub>L<sup>2</sup></sub> with respect to measured Gromov-Hausdorff convergence and t. Applications include the quantitative L<sup>p</sup>-convergence in the noncollapsed setting for all p<∞, a result new even for closed Riemannian manifolds and Alexandrov spaces.