Deformations and Collapsing of Curved Spaces

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This proposal involves studying deformations and collapsing of Riemannian metrics with various curvature bounds. A basic object of interest is the moduli space of complete nonnegatively curved metrics on a given open manifold. Nearby points in the moduli space often correspond to metrics that may appear unrealated and are pulled next to each other by non-obvious diffeomorphisms produced e.g. by surgery theory. Thus to detect nontrivial topology in the moduli space one needs to relate comparison geometry input with machinery of algebraic topology. To this end it is planned to investigate continuity of souls under deformation, especially for manifolds with codimension two souls where the geometry is rather rigid. It is proposed to obtain detailed structure of the moduli space for low-dimensional manifolds. Another topic is studying collapse of souls inside nonnegatively curved open manifolds by establishing relative versions of the fibration theorems. Yet another project is to analyze collapse of cusp cross-sections in finite volume manifolds of bounded negative curvature that admit no negatively pinched metrics.In broad terms the proposal seeks to understand how curvature of the space controls its deformations and degenerations. The spaces considered range from boundaries of three-dimensional convex bodies to their higher dimensional analogs. It is proposed to investigate the manifold of all possible geometric shapes that a given space can assume; the topological features of this manifold of shapes are poorly understood.