Convergence of Riemannian Manifolds

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Convergence of Riemannian ManifoldsOver the past three decades mathematicians have gained deep new insight into Riemannian manifolds by applying the methods of Gromov-Hausdorff, Lipschitz and metric measure convergence. Such techniques have been particularly useful for studying manifolds with bounds on sectional or Ricci curvature, but a new weaker notion of convergence is needed to understand manifolds without such strong conditions. Recently the PI and Dr. Wenger have applied work of Drs. Ambrosio and Kirchheim to introduce a new distance between manifolds: the intrinsic flat distance. While the convergence is weaker than previous forms of convergence, the limit spaces, called Integral Current Spaces, are countably H^mrectifiable. Applying work of Cheeger-Colding, Gromov and Perelman, the PI and Dr. Wenger have shown that the Gromov-Hausdorff and intrinsic flat limits of manifolds with nonnegative Ricci curvature agree. However, in general the limit spaces are different and sequences which do not converge in the Gromov-Hausdorff sense may still converge in the intrinsic flat sense. The PI will study the properties of these limit spaces under a variety of conditions on the sequence of manifolds and prove stability theorems under these weaker conditions. In particular the PI proposes to improve her results on the stability of the Friedmann model.The spacelike universe is described in Friedmann cosmology as an isotropic three dimensional Riemannian manifold that expands in time starting from the initial Big Bang. In reality the universe is not isotropic because it is bent by gravity in a nonuniform way. Weak gravitational lensing (due to dust) and strong gravitational lensing (due to massive objects) has been observed by the Hubble to distort regions of space. The universe is thus, at best, close to the Friedmann model in some sense. In prior work, under strong assumptions, the PI has shown that a Riemannian manifold which is almost isotropic (in a way which allows for weak gravitational lensing and localized strong gravitational lensing) is close to the Friedmann model in the Gromov-Hausdorff sense. This is proven by studying the Gromov-Hausdorff limits of increasingly isotropic manifolds. Now the PI proposes to prove that under weaker assumptions, the universe is close to the Friedmann model in the intrinsic flat sense by studying the intrinsic flat limits of Riemannian manifolds. Using the intrinsic flat distance will not only allow for weak and strong gravitational lensing but also allow for the possible existence of wormholes.---------------------------------------------------------------------------