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Christina Sormani's Preprints

Preprints on the ArXiv:

"Various Covering Spectra on Complete Metric Spaces"
by Christina Sormani and Guofang Wei, arxiv preprint 32 pages
Solicited for and accepted by Asian Journal of Mathematics.

We study various covering spectra for complete noncompact length spaces with universal covers (including Riemannian manifolds and the pointed Gromov Hausdorff limits of Riemannian manifolds with lower bounds on their Ricci curvature). We relate the covering spectrum to the (marked) shift spectrum of such a space. We define the slipping group generated by elements of the fundamental group whose translative lengths are 0. We introduce a rescaled length, the rescaled covering spectrum and the rescaled slipping group. Applying these notions we prove that certain complete noncompact Riemannian manifolds with nonnegative or positive Ricci curvature have finite fundamental groups. Throughout we suggest further problems both for those interested in Riemannian geometry and those interested in metric space theory. spectrum.

"Nonlinear stability of rotationally symmetric spaces with low regularity"
by Philippe LeFloch and Christina Sormani, arxiv preprint 38 pages

Our main theorem states that if one has a sequence of rotationally symmetric regions of nonnegative scalar curvature with no closed interior minimal surfaces that have spherical boundaries of fixed area with uniform upper bounds on:
• the depth of the region and
• the Hawking mass of the boundary,
then a subsequence converges in the intrinsic flat sense to a region whose metric has $H^1$ regularity with nonnegative scalar curvature (in the generalized sense). In addition
• the area of the boundary is continuous,
• the volume of the region is continuous and
• the Hawking mass of the boundary is continuous.
In order to obtain the continuity of these values we show the metric tensors in a suitable gauge converge in the Sobolev sense.

In addition to this main theorem, we provide a thorough analysis of these limit spaces, proving the Hawking mass is monotone and providing precise estimates on the intrinsic flat distance, the Sobolev distance, and a new notion, the D flat distance, between a region of small Hawking mass and a disk in Euclidean space extending results of mine with Dan Lee. It should be noted that the ideas involving the generalized notions of curvature on a manifold with $H^1$ metric tensors has been explored by LeFloch and collaborators Mardare, Stewart and Rendall. Ordinarily intrinsic flat limit spaces are only countably H^m rectifiable.

Works in Progress:

"Properties of the Intrinsic Flat Distance"
by Christina Sormani
In progress: adding more properties, everything in the files linked here are proven including the Tetrahedral Compactness Theorem:
Jan 2013 preprint including new Arzela-Ascoli and Bolzano-Weierstrass Theorems and some clarifications of other proofs. 63 pages
Aug 2012 preprint including Interval Filling Volumes and Sliced Interval Filling Volumes and Continuity of these notions with a new introduction and a figure. 50 pages.
Feb 2012 preprint of material presented at MIT Feb 15, 2012 including Flat-to-GH convergence, Sliced filling volumes, and the tetrahedral property. 41 pages
A new version will be posted in September with some new proofs and new material.

"Sequences of three dimensional manifolds with positive scalar curvature"
by Jorge Basilio and Christina Sormani
In progress: still adding details to the proofs.

``Big Bang Spacetimes"
by Christina Sormani and Carlos Vega
In progress.

"An Intrinsic Flat Limit of Riemannian manifolds with no Geodesics"
by Jorge Basilio and Christina Sormani
In progress.

For published papers go to my papers website.

Homepage: http://comet.lehman.cuny.edu/sormani