On the structure of spaces with Ricci curvature bounded below. III (Q1609763)

The reviewed paper is the third in a series (see Zbl 0902.53034 and Zbl 1004.94517) devoted to the study of the structure of complete connected Riemannian manifolds, whose Ricci curvature has a definite lower bound and of the Gromow-Hausdorff limits, \(Y\), of sequences of such manifolds. This paper mainly concerns the spectral information of \(Y\). First the authors show that the renormalized limit measures determine a unique measure class. Next they introduce \(\mu\)-rectifiable spaces, which are the abstract version of finite unions of rectifiable varifolds, and study analysis on such spaces. The authors show that there is a unique self-adjoint operator, \(\Delta\), the Laplacian on Lipschitz functions. Finally, they prove the upper semicontinuity of the spectrum of the Laplacian under measured Gromov-Hausdorff convergence for rectifiable spaces, and the continuity of the eigenfunctions and eigenvalues for limit spaces.