Principal Investigator: Igor Belegradek
This project is concerned with the topology of curved Riemannian manifolds. One objective is to study collapsed manifolds, i.e. manifolds that appear lower dimensional at a certain small but definite scale. Specifically, it is planned to investigate almost nonnegatively curved manifolds, collapsed manifolds of bounded nonnegative curvature, and cusp cross-sections of finite volume nonpositively curved manifolds. In a somewhat different direction it is proposed to study negatively pinched manifolds, including pinching estimates, rigidity in case of optimal pinching, finiteness theorems forced by assumptions on pinching, asymptotic geometry of horospheres, and negatively pinched groups. The project continues earlier work of Belegradek and collaborators, and especially the recent results with Kapovitch on diffeomorphism classification of negatively pinched manifolds with amenable fundamental groups (that settled a problem of Bowditch), and on optimal pinching estimates for such manifolds (that answered a question of Gromov).
While the project mostly deals with intrinsic development of global Riemannian geometry, it is also to some extent motivated by Sciences and Engineering, where curved spaces are plentiful and occur naturally. The theoretical insights obtained in the project may be applicable to concrete problems in those fields.
