Preprints and Works in Progress by Professor Sormani
See also google scholar and my published papers page. Survey articles, chapters, conference proceedings, and undergraduate research are on my surveys page.
Preprints on the ArXiv:.
Gromov's Compactness Theorem for the Intrinsic Timed-Hausdorff Distance
by M Che, R Perales, C Sormani (arxiv) 37 pages (SCGP, psc-cuny)
Abstract: We prove Gromov's Compactness Theorem for the Intrinsic timed Hausdorff convergence of timed-metric-spaces using timed-Fréchet maps. Our proof introduces the notion of "addresses" and provides a new way of stating Gromov's original compactness theorem for Gromov-Hausdorff (GH) convergence of metric spaces. We also obtain a new Arzela-Ascoli Theorem for real valued uniformly bounded Lipschitz functions on GH converging compact metric spaces. The intrinsic timed-Hausdorff distance between timed-metric-spaces was first defined by Sakovich-Sormani to define a weak notion of convergence for space-times, and our compactness theorem will soon be applied to advance their work in this direction.
Existence of uniform Temple charts and applications to null distance
Authors: B Meco, A Sakovich, C Sormani (arxiv) 37 pages (DMS1612049, psc-cuny)
Abstract: In this paper, we prove that Temple's cylindrical future null coordinate charts can be constructed uniformly and we estimate the gradients of their optical functions. We then apply these charts to study a spacetime (N,g) that has been converted into a definite metric space (N, d^T), where d^T is the null distance of Sormani and Vega defined using a locally anti-Lipschitz (in the sense of Chrusciel, Grant, and Minguzzi) generalized time function τ. In particular, in the case when τ is Lipschitz we prove that (N, d^T) is a rectifiable metric space, where the causal structure is locally encoded by T and d^T. As a consequence, applying a classical theorem of Hawking and following a technique developed by Sakovich and Sormani, we can prove a Lorentzian isometry theorem, generalizing our earlier result.
Introducing Various Notions of Distances between Space-Times
by A. Sakovich, C. Sormani (DMS1612049, psc-cuny, MSRI, Fields) (arxiv) 135 pages
Abstract: We introduce the notion of causally-null-compactifiable space-times which can be canonically converted into a compact timed-metric-spaces using the cosmological time of Andersson-Howard-Galloway and the null distance of Sormani-Vega. We produce a large class of such space-times including future developments of compact initial data sets and regions which exhaust asymptotically flat space-times. We then present various notions of intrinsic distances between these space-times (introducing the timed-Hausdorff distance) and prove some of these notions of distance are definite in the sense that they equal zero iff there is a time-oriented Lorentzian isometry between the space-times. These definite distances enable us to define various notions of convergence of space-times to limit space-times which are not necessarily smooth. Many open questions and conjectures are included throughout.
Geometric Convergence to an Extreme Limit Space with nonnegative scalar curvature
by C Sormani, Wenchuan Tian and Wai-Ho Yeung (DMS1006059) (arxiv) 33 pages
Abstract: In 2014, Gromov conjectured that sequences of manifolds with nonnegative scalar curvature should have subsequences which converge in some geometric sense to limit spaces with some notion of generalized nonnegative scalar curvature. In recent joint work with Changliang Wang, the authors found a sequence of warped product Riemannian metrics on $\Sph^2\times \Sph^1$ with nonnegative scalar curvature whose metric tensors converge in the $W^{1,p}$ sense for $p<2$ to an extreme warped product limit space where the warping function hits infinity at two points. Here we study this extreme limit space as a metric space and as an integral current space and prove the sequence converges in the volume preserving intrinsic flat and measured Gromov-Hausdorff sense to this space. This limit space may now be used to test any proposed definitions for generalized nonnegative scalar curvature. One does not need expertise in Geometric Measure Theory or in Intrinsic Flat Convergence to read this paper.
SWIF Convergence of Smocked Metric Spaces (work with student team under revision)
by Sormani with Dinowitz, Drillick, Farahzad, and Yamin (DMS1612049) (SCGP and GC) (arxiv)
In this paper we explore a special class of metric spaces called smocked metric spaces and study their tangent cones at infinity. We prove that under the right hypotheses, the rescaled limits of balls converge in both the Gromov-Hausdorff and Intrinsic Flat sense to normed spaces. This paper will be applied in upcoming work by Kazaras and Sormani concerning Gromov's conjectures on the properties of GH and SWIF limits of Riemannian manifolds with positive scalar curvature.
Works in Progress:
"Spacetime Intrinsic Flat Convergence" and related papers
by A Sakovich and C Sormani (DMS 1309360 and DMS1612049)(SCGP)(Fields)
"GH and SWIF Convergence of Metric Spaces"
by S. Lakzian and C Sormani (DMS1612049)
Other Projects:
Compactness Theorem for SIF Convergence Project Title TBD
by A Sakovich, M. Graf, and C Sormani (Fields)
Geometry and Physics of Cosmic Strings
with Geshnizjani, Graf, Gregory, Gunasekaran, and Zavala (Fields, WIG3)
Smocking and Scalar Curvature Project Title TBD
by D Kazaras and C Sormani (DMS1612049)(SCGP)
Big Bang Spacetimes Project (this project has been divided between the two authors)
by C Sormani and C Vega (Vega NSF DMS1309360 and Sormani DMS1612049)