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.

Continuity of nonlinear eigenvalues in $CD(K,infty)$ spaces with respect to measured Gromov-Hausdorff convergence

In this note we prove in the nonlinear setting of $CD(K,infty)$ spaces the stability of the Krasnoselskii spectrum of the Laplace operator $-Delta$ under measured Gromov-Hausdorff convergence, under an additional compactness assumption satisfied, for instance, by sequences of $CD^*(K,N)$ metric measure spaces with uniformly bounded diameter. Additionally, we show that every element $lambda$ in the Krasnoselskii spectrum is indeed an eigenvalue, namely there exists a nontrivial $u$ satisfying the eigenvalue equation $- Delta u = lambda u$.