Collapsing, pinching, and curvature bounds

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AbstractAward: DMS-0503864Principal Investigator: Igor BelegradekThis project is concerned with the topology of curved Riemannianmanifolds. One objective is to study collapsed manifolds, i.e.manifolds that appear lower dimensional at a certain small butdefinite scale. Specifically, it is planned to investigate almostnonnegatively curved manifolds, collapsed manifolds of boundednonnegative curvature, and cusp cross-sections of finite volumenonpositively curved manifolds. In a somewhat different directionit is proposed to study negatively pinched manifolds, includingpinching estimates, rigidity in case of optimal pinching,finiteness theorems forced by assumptions on pinching, asymptoticgeometry of horospheres, and negatively pinched groups. Theproject continues earlier work of Belegradek and collaborators,and especially the recent results with Kapovitch ondiffeomorphism classification of negatively pinched manifoldswith amenable fundamental groups (that settled a problem ofBowditch), and on optimal pinching estimates for such manifolds(that answered a question of Gromov).While the project mostly deals with intrinsic development ofglobal Riemannian geometry, it is also to some extent motivatedby Sciences and Engineering, where curved spaces are plentifuland occur naturally. The theoretical insights obtained in theproject may be applicable to concrete problems in those fields.