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^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.
. With Sajjad Lakzian, the PI explored the smooth convergence of manifolds away from singularities considering settings where cones, cusps or even strands may form and disappear in the limit. Lakzian applied this research in his CUNY doctoral dissertation on the Ricci flow through Neck Pinch Singularities in the sense of Angenant-Caputo-Knopf. With Raquel Perales, the PI explored the Gromov-Hausdorff convergence of manifolds with boundary and then, in current work for her Stony Brook doctoral dissertation, Perales completed a paper on the intrinsic flat convergence of such manifolds assuming Ricci curvature bounds and mean curvature bounds on the boundary. With Jorge Basilio, the PI explored the intrinsic flat convergence of manifolds with nonnegative scalar curvature. These upcoming results will form part of Basilio’s upcoming CUNY doctoral dissertation. These three doctoral students and other doctoral students, including Jacobus Portegies of Courant Institute, met regularly in the PIs reading seminar at CUNY. During Fall 2013, when the PI served as a Visiting Research Professor at MSRI, additional doctoral students and postdocs participated in the PI’s reading seminar including: Alessandro Carlotto, Matthais Erbar, Davi Maximo, Anna Sakovich, Zahra Sinaei, Ling Xiao, Shuanjian Zhang, and Xin Zhou. The PI directly supervised MSRI postdoc Dr. Zahra Sinaei and together they completed a paper analyzing the behavior of the covering spectrum under intrinsic flat convergence. Earlier the PI and her senior collaborator, Prof. Guofang Wei, completed a paper on various covering spectra of complete Riemannian manifolds also funded in part by this grant. One of the primary goals in the proposal was to apply intrinsic flat convergence to study stability problems arising in General Relativity. In this direction, the PI and junior colleague, Dr. Dan Lee, have published two works funded by this grant. In these papers, they consider an isolated spherically symmetric gravitational system, like a black hole, star, planet or dust cloud. It was proven by Schoen-Yau in their Positive Mass Theorem that such a system, endowed with the positive energy condition, has positive ADM mass. The mass in the system curves the spacetime much as a marble lying on a sheet of rubber curves the rubber sheet. Schoen-Yau proved that if the ADM mass is 0, then the space is flat Euclidean space with no curvature at all. The PI and Lee proved that if one has a sequence of spherically symmetric spaces satisfying these conditions whose ADM mass converges to 0, then the sequence of spaces converges in the intrinsic flat sense to Euclidean space. The PI has had the opportunity to describe these results in a short video for Frank Morgan’s Huffington Post blog. The PI and Dr. Dan Lee have also considered the Penrose Inequality. Recall that Huisken-Ilmanen and Bray proved Penrose’s Inequality which states that the ADM mass at infinity is greater than the Hawking mass of an apparent horizon of a black hole within the space. If the ADM mass and Hawking mass agree then the space is exactly a Schwarzschild space, with a Schwarzschild black hole at the center. The PI and Dr. Lee considered spherically symmetric spaces where the ADM mass decreases towards the Hawking mass and proved that such a sequence of spaces converges in the intrinsic flat sense to Schwarzschild space. In the Spring of 2014, funding from this grant was used by the PI to hire a one semester postdoc, Dr. Carlos Vega, who specializes in Lorentzian manifolds. They are working together to develop a notion of intrinsic flat convergence for space times instead of just spacelike slices of the universe. Towards this goal they have been studying a new notion called the null distance to better understand big bang spacetimes. While serving as a postdoc at CUNY, Dr. Vega was actively engaged with the minority undergraduates at Lehman College as well as inner city high school students attending the PIs College Now precalculus course. Further research applying intrinsic flat convergence to questions arising in general relativity is being funded by the PIs current grant: NSF-DMS-1309360 (2013-2016).
