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.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Type
Standard Grant (Standard)
Application #
1105045
Program Officer
Joanna Kania-Bartoszynska
Project Start
Project End
Budget Start
2011-08-15
Budget End
2014-07-31
Support Year
Fiscal Year
2011
Total Cost
$141,863
Indirect Cost
Name
Georgia Tech Research Corporation
Department
Type
DUNS #
City
Atlanta
State
GA
Country
United States
Zip Code
30332