arXiv subjects

C. Sormani

Publications and source records attributed to C. Sormani.

7 recordsLinked to original sources

arXiv2024

Monotone Sequences of Metric Spaces with Compact Limits

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.

math.MG↗

arXiv2022

The Null distance encodes causality

A Lorentzian manifold endowed with a time function, $\tau$, can be converted into a metric space using the null distance, $\hat{d}_\tau$, defined by Sormani and Vega. We show that if the time function is a proper regular cosmological time function as studied by Andersson, Galloway and Howard, and also by Wald and Yip, or if, more generally, it satisfies the anti-Lipschitz condition of Chru\'sciel, Grant and Minguzzi, then the causal structure is encoded by the null distance in the following sense: $$ \hat{d}_\tau(p,q)=\tau(q)-\tau(p) \iff q \textrm{ lies in the causal future of } p. $$ As a consequence, in dimension $n+1$, $n\ge 2$, we prove that if there is a bijective map between two such spacetimes, $F: M_1\to M_2$, which preserves the cosmological time function, $ \tau_2(F(p))= \tau_1(p)$ for any $ p \in M_1$, and preserves the null distance, $\hat{d}_{\tau_2}(F(p),F(q))=\hat{d}_{\tau_1}(p,q)$ for any $p,q\in M_1$, then there is a Lorentzian isometry between them, $F_*g_1=g_2$. This yields a canonical procedure allowing us to convert such spacetimes into unique metric spaces with causal structures and time functions. This will be applied in our upcoming work to define Spacetime Intrinsic Flat Convergence.

math.DG↗

arXiv2021

SWIF Convergence of Smocked Metric Spaces

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.

math.MG↗

arXiv2019

Sequences of three dimensional manifolds with positive scalar curvature

We develop two new methods of constructing sequences of manifolds with positive scalar curvature that converge in the Gromov-Hausdorff and Intrinsic Flat sense to limit spaces with "pulled regions". The examples created rigorously within using these methods were announced by us a few years ago and have influenced the statements of some of Gromov's conjectures concerning sequences of manifolds with positive scalar curvature. Both methods extend the notion of "sewing along a curve" developed in prior work of the authors with Dodziuk to create limits that are pulled string spaces. The first method allows us to sew any compact set in a fixed initial manifold to create a limit space in which that compact set has been scrunched to a single point. The second method allows us to edit a sequence of regions or curves in a sequence of distinct manifolds.

math.MG↗

arXiv2017

Sewing Riemannian Manifolds with Positive Scalar Curvature

We explore to what extent one may hope to preserve geometric properties of three dimensional manifolds with lower scalar curvature bounds under Gromov-Hausdorff and Intrinsic Flat limits. We introduce a new construction, called sewing, of three dimensional manifolds that preserves positive scalar curvature. We then use sewing to produce sequences of such manifolds which converge to spaces that fail to have nonnegative scalar curvature in a standard generalized sense. Since the notion of nonnegative scalar curvature is not strong enough to persist alone, we propose that one pair a lower scalar curvature bound with a lower bound on the area of a closed minimal surface when taking sequences as this will exclude the possibility of sewing of manifolds.

math.DG↗

arXiv2015

Properties of the Intrinsic Flat Distance

Here we explore a variety of properties of intrinsic flat convergence. We introduce the sliced filling volume and interval sliced filling volume and explore the relationship between these notions, the tetrahedral property and the disappearance of points under intrinsic flat convergence. We prove two new Gromov-Hausdorff and intrinsic flat compactness theorems including the Tetrahedral Compactness Theorem. Much of the work in this paper builds upon Ambrosio-Kirchheim's Slicing Theorem combined with an adapted version Gromov's Filling Volume.

math.DG↗

arXiv2011

The Intrinsic Flat Distance between Riemannian Manifolds and other Integral Current Spaces

Inspired by the Gromov-Hausdorff distance, we define the intrinsic flat distance between oriented $m$ dimensional Riemannian manifolds with boundary by isometrically embedding the manifolds into a common metric space, measuring the flat distance between them and taking an infimum over all isometric embeddings and all common metric spaces. This is made rigorous by applying Ambrosio-Kirchheim's extension of Federer-Fleming's notion of integral currents to arbitrary metric spaces. We prove the intrinsic flat distance between two compact oriented Riemannian manifolds is zero iff they have an orientation preserving isometry between them. Using the theory of Ambrosio-Kirchheim, we study converging sequences of manifolds and their limits, which are in a class of metric spaces that we call integral current spaces. We describe the properties of such spaces including the fact that they are countably $\mathcal{H}^m$ rectifiable spaces and present numerous examples.

math.DG↗