Reconstruction of Collapsed Manifolds

摘要

In this article, we consider the problem of reconstructing collapsed manifolds in a moduli space by means of geometric or analytic data of the limit spaces. The moduli space of our main interest is that consisting of closed Riemannian manifolds of fixed dimension with a lower sectional curvature and an upper diameter bound. In this moduli space, we can reconstruct the topology of three-dimensional or four-dimensional collapsed manifolds in terms of the singularities of the limit Alexandrov spaces. In the general dimension, we define a new covering invariant and prove the uniform boundedness of it with an application to Gromov's Betti number theorem. Finally we discuss the reconstruction and stability problems of collapsed manifolds by using analytic spectral data, where we assume an additional upper sectional curvature bound.