NSF Grant DMS - 1309360 (2013-2016)
Applications of the Convergence of Riemannian Manifolds to General Relativity
Christina Sormani
Abstract:
The PI will apply Intrinsic Flat convergence between Riemannian manifolds to better understand how close space-like manifolds studied in Mathematical General Relativity approximate the standard well known models. The Intrinsic Flat distance, first introduced by the PI with Stefan Wenger using methods of Ambrosio-Kirchheim, is particularly well-suited to some questions arising in General Relativity because increasingly thin gravity wells disappear under this convergence. In joint work with Dan Lee, the PI has shown that spherically symmetric Riemannian manifolds with increasingly small ADM mass converge to Euclidean space in the pointed intrinsic flat sense, and here proposes to generalize this result. In addition, the PI proposes to develop two new notions of convergence: the first will allow mathematicians to study Lorentzian manifolds directly, and the second will prevent regions from disappearing due to orientation and cancellation. Both notions are specifically adapted to questions arising in General Relativity.
Einstein's Theory of General Relativity describes how space is curved by gravity. Even within our own solar system, when computing the trajectories of spacecraft heading to Mars, engineers must take into account the curvature caused by the mass of the planets and the sun. Each planet forms a gravity well. If the mass of a planet is small, one would like to know in what sense the space around it is almost flat. In fact, the space around a planet of arbitrarily small mass could be very highly curved (and have a very deep but thin gravity well). In joint work with Dr. Stefan Wenger, the PI has developed a new means of measuring the closeness between curved spaces and, in joint work with Dr. Dan Lee, she has estimated how close the space around a single perfectly spherical planet is to Euclidean space. In this project, she will develop tools allowing one to better understand the space around groups of planets which are not perfect spheres: like the ones in our own solar system.
NSF Grant DMS - 10060059 (2010-2014)
Convergence of Riemannian Manifolds
Christina Sormani
Abstract:
Over 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^m rectifiable. 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.