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:.
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.
Monotone Sequences of Metric Spaces with Compact Limits
by R. Perales, C. Sormani (DMS1612049, PSC-CUNY) (arxiv) 36 pages
Invited research article for Special Volume in Honor of Xiaochun Rong
to appear in Journal of Mathematical Study, Vol. 58 (2025), Iss. 1 : pp. 96–132
Abstract: In this paper, we consider a fixed metric space (possibly an oriented Riemannian manifold with boundary) with an increasing sequence of distance functions and a uniform upper bound on diameter. When the metric space endowed with the pointwise limit of these distances is compact, then there is uniform and Gromov-Hausdorff (GH) convergence to this limit. When the metric space also has an integral current structure of uniformly bounded total mass (as is true for an oriented Riemannian manifold with boundary that has a uniform bound on total volume), we prove volume preserving intrinsic flat convergence to a subset of the GH limit whose closure is the whole GH limit. We provide a review of all notions and have a list of open questions at the end. Dedicated to Xiaochun Rong.
SWIF Convergence of Smocked Metric Spaces (work with student team under revision)
by Dinowitz, Drillick, Farahzad, Sormani, 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:
"Geometric Convergence for an Extreme Limit with Nonnegative Scalar Curvature” and related papers
with Wenchuan Tian and Wai-Ho Yeung (DMS1612049) (Fields)
"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)