Embedding of RCD* (K, N) spaces in L2 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 eigen-maps. Extending part of the classical results [10], [11] known for closed Riemannian manifolds, we prove convergence as t down arrow 0 of the rescaled pull-back metrics Phi(t)*g(L2) in L-2(X, m) induced by Phi(t). Moreover we discuss the behavior of Phi(t)*g(L2) with respect to measured Gromov-Hausdorff convergence and t. Applications include the quantitative L-p-convergence in the noncollapsed setting for all p < infinity, a result new even for closed Riemannian manifolds and Alexandrov spaces. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper studies the family of embeddings Phi(t) of a compact RCD*(K, N) space into L-2(X, m) using eigen-maps. It proves convergence of the rescaled pull-back metrics Phi(t)*g(L2) as t approaches 0 in L-2(X, m), and discusses its behavior with respect to measured Gromov-Hausdorff convergence and t. Applications include quantitative L-p convergence in the noncollapsed setting.