Date of Degree
Dan A. Lee
Geometry and Topology
Convergence of Riemannian Manifolds, Scalar Curvature
In this thesis, we develop a new method of performing surgery on 3-dimensional manifolds called "sewing" and use this technique to construct sequences of Riemannian manifolds with positive or nonnegative scalar curvature. The foundation of our method is a strengthening of the Gromov-Lawson tunnel construction which guarantees the existence of “tiny” and arbitrarily “short” tunnels. We study the limits of sequences of sewn spaces under the Gromov-Hausdorff (GH) and Sormani-Wenger Instrinsic-Flat (SWIF) distances and discuss to what extent the notion of scalar curvature extends to these spaces. We give three applications of the sewing technique to demonstrate that stability theorems for sequences of manifolds with positive or nonnegative scalar curvature fail to hold under these weak notions of convergence. The first application provides a counter-example to a conjecture of Gromov about sequences of manifolds with scalar curvature bounded from below. In fact, Gromov conjectured that SWIF-limits of sequences of Riemannian manifolds with nonnegative scalar curvature should have nonnegative scalar curvature in some generalized sense. We provide examples that shows this is false when the generalized notion on the limit-spaces is taken to be the classical volume-limit formula for scalar curvature. The second application demonstrates how the Scalar Torus Rigidity Theorem is not stable under GH- or SWIF-limits. The final application gives a sequence of sewn asymptotically flat three manifolds with nonnegative scalar curvature with ADM mass decreasing to zero that converges in the (pointed) GH and SWIF sense to a limit space that is homeomorphic to euclidean space but that is not isometric. Thus, the positive mass theorem on limits of manifolds with positive scalar curvature is false.
Basilio, Jorge E., "Manifold Convergence: Sewing Sequences of Riemannian Manifolds with Positive or Nonnegative Scalar Curvature" (2017). CUNY Academic Works.