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

## TL;DR

This paper investigates eigenfunction-based embeddings of compact $RCD^*(K,N)$ spaces into $L^2$, demonstrating convergence of induced metrics and exploring their behavior under Gromov-Hausdorff convergence, with applications to $L^p$-convergence.

## Contribution

It extends classical eigenmap results to $RCD^*(K,N)$ spaces, showing metric convergence and analyzing stability under measured Gromov-Hausdorff convergence.

## Key findings

- Rescaled pull-back metrics converge in $L^2$ as $t o 0$.
- Embeddings are stable under measured Gromov-Hausdorff convergence.
- Quantitative $L^p$-convergence results are established for all $p<
$.

## Abstract

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 B\'erard, B\'erard-Besson-Gallot, known for closed Riemannian manifolds, we prove convergence as $t\downarrow 0$ 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.