arXiv subjects

Christina Sormani

Publications and source records attributed to Christina Sormani.

At least 37 records · Page 2Linked to original sources

arXiv2018

Oberwolfach Report: Spacetime Intrinsic Flat Convergence

This Oberwolfach report describes published work with Carlos Vega introducing the null distance on a spacetime, and announces unpublished applications of this work jointly with Carlos Vega and with Anna Sakovich applying the null distance to define a spacetime intrinsic flat convergence (SF convergence). The announced work with Vega concerns convergence of sequences of big bang spacetimes and the announced work with Anna Sakovich concerns convergence of future maximal developments of initial data sets. Both were originally announced at the 2018 JMM. This report lists some open questions that will be discussed at the Oberwolfach meeting this coming August.

gr-qc↗

arXiv2016

Scalar Curvature and Intrinsic Flat Convergence

Herein we present open problems and survey examples and theorems concerning sequences of Riemannian manifolds with uniform lower bounds on scalar curvature and their limit spaces. Examples of Gromov and of Ilmanen which naturally ought to have certain limit spaces do not converge with respect to smooth or Gromov-Hausdorff convergence. Thus we focus here on the notion of Intrinsic Flat convergence, developed jointly with Wenger. This notion has been applied successfully to study sequences that arise in General Relativity. Gromov has suggested it should be applied in other settings as well. We first review intrinsic flat convergence, its properties, and its compactness theorems, before presenting the applications and the open problems.

math.MG↗

arXiv2016

Null distance on a spacetime

Given a time function $τ$ on a spacetime $M$, we define a `null distance function', $\hat{d}_τ$, built from and closely related to the causal structure of $M$. In basic models with timelike $\nabla τ$, we show that 1) $\hat{d}_τ$ is a definite distance function, which induces the manifold topology, 2) the causal structure of $M$ is completely encoded in $\hat{d}_τ$ and $τ$. In general, $\hat{d}_τ$ is a conformally invariant pseudometric, which may be indefinite. We give an `anti-Lipschitz' condition on $τ$, which ensures that $\hat{d}_τ$ is definite, and show this condition to be satisfied whenever $τ$ has gradient vectors $\nabla τ$ almost everywhere, with $\nabla τ$ locally `bounded away from the light cones'. As a consequence, we show that the cosmological time function of [1] is anti-Lipschitz when `regular', and hence induces a definite null distance function. This provides what may be interpreted as a canonical metric space structure on spacetimes which emanate from a common initial singularity, e.g. a `big bang'.

math.DG↗

arXiv2015

Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space

The rigidity of the Positive Mass Theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We study the stability of this statement for spaces that can be realized as graphical hypersurfaces in Euclidean space. We prove (under certain technical hypotheses) that if a sequence of complete asymptotically flat graphs of nonnegative scalar curvature has mass approaching zero, then the sequence must converge to Euclidean space in the pointed intrinsic flat sense. The appendix includes a new Gromov-Hausdorff and intrinsic flat compactness theorem for sequences of metric spaces with uniform Lipschitz bounds on their metrics.

math.DG↗

arXiv2014

Intrinsic Flat Convergence of Covering Spaces

We examine the limits of covering spaces and the covering spectra of oriented Riemannian manifolds, $M_j$, which converge to a nonzero integral current space, $M_\infty$, in the intrinsic flat sense. We provide examples demonstrating that the covering spaces and covering spectra need not converge in this setting. In fact we provide a sequence of simply connected $M_j$ diffeomorphic to $\mathbb{S}^4$ that converge in the intrinsic flat sense to a torus $\mathbb{S}^1\times\mathbb{S}^3$. Nevertheless, we prove that if the $δ$-covers, $\tilde{M}_j^δ$, have finite order $N$, then a subsequence of the $\tilde{M}_j^δ$ converge in the intrinsic flat sense to a metric space, $M^δ_\infty$, which is the disjoint union of covering spaces of $M_\infty$.

math.MG↗

arXiv2014

Intrinsic Flat Arzela-Ascoli Theorems

One of the most powerful theorems in metric geometry is the Arzela-Ascoli Theorem which provides a continuous limit for sequences of equicontinuous functions between two compact spaces. This theorem has been extended by Gromov and Grove-Petersen to sequences of functions with varying domains and ranges where the domains and the ranges respectively converge in the Gromov-Hausdorff sense to compact limit spaces. However such a powerful theorem does not hold when the domains and ranges only converge in the intrinsic flat sense due to the possible disappearance of points in the limit. In this paper two Arzela-Ascoli Theorems are proven for intrinsic flat converging sequences of manifolds: one for uniformly Lipschitz functions with fixed range whose domains are converging in the intrinsic flat sense, and one for sequences of uniformly local isometries between spaces which are converging in the intrinsic flat sense. A basic Bolzano-Weierstrass Theorem is proven for sequences of points in such sequences of spaces. In addition it is proven that when a sequence of manifolds has a precompact intrinsic flat limit then the metric completion of the limit is the Gromov-Hausdorff limit of regions within those manifolds. Applications and suggested applications of these results are described in the final section of this paper.

math.MG↗

arXiv2014

Stability of the Positive Mass Theorem for Rotationally Symmetric Riemannian Manifolds

We study the stability of the Positive Mass Theorem using the Intrinsic Flat Distance. In particular we consider the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces whose boundaries are either outermost minimal hypersurfaces or are empty. We prove that a sequence of these manifolds whose ADM masses converge to zero must converge to Euclidean space in the pointed Intrinsic Flat sense. In fact we provide explicit bounds on the Intrinsic Flat Distance between annular regions in the manifold and annular regions in Euclidean space by constructing an explicit filling manifold and estimating its volume. In addition, we include a variety of propositions that can be used to estimate the Intrinsic Flat distance between Riemannian manifolds without rotationally symmetry. Conjectures regarding the Intrinsic Flat stability of the Positive Mass Theorem in the general case are proposed in the final section.

math.DG↗

arXiv2014

Various Covering Spectra for Complete Metric Spaces

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.

math.DG↗

arXiv2014

The nonlinear stability of rotationally symmetric spaces with low regularity

We consider rotationally symmetric spaces with low regularity, which we regard as integral currents spaces or manifolds with Sobolev regularity and are assumed to have nonnegative scalar curvature. Relying on the flat distance and on Sobolev norms, we establish several nonlinear stability estimates about the ``distance'' between a rotationally symmetric manifold and the Euclidian space, which are stated in terms of the ADM mass of the manifold. Importantly, we make explicit the dependencies and scales involved in this problem, particularly the ADM mass, the depth, and the CMC reference hypersurface. Several notions of independent interest are introduced in the course of our analysis, including the notion of depth of a manifold and a scaled version of the flat-distance, the D-flat distance as we call it, which involves the diameter of the manifold. Finally we prove a compactness theorem for sequences of regions with uniformly bounded depth, whose outer boundaries have fixed area and an upper bound on Hawking mass.

gr-qc↗

arXiv2013

Sequences of Open Riemannian Manifolds with Boundary

We consider sequences of open Riemannian manifolds with boundary that have no regularity conditions on the boundary. To define a reasonable notion of a limit of such a sequence, we examine "$δ$ inner regions" which avoid the boundary by a distance $δ$. We prove Gromov-Hausdorff compactness theorems for sequences of these "$δ$ inner regions". We then build "glued limit spaces" out of the Gromov-Hausdorff limits of these $δ$ interior regions and study the properties of these glued limit spaces. Our main applications assume the sequence is noncollapsing and has nonnegative Ricci curvature. We include open questions.

math.MG↗

arXiv2012

The Tetrahedral Property and a new Gromov-Hausdorff Compactness Theorem

We present the Tetrahedral Compactness Theorem which states that sequences of Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the Gromov-Hausdorff sense to a countably $\mathcal{H}^m$ rectifiable metric space of the same dimension. The tetrahedral property depends only on distances between points in spheres, yet we show it provides a lower bound on the volumes of balls. The proof is based upon intrinsic flat convergence and a new notion called the sliced filling volume of a ball.

math.DG↗

arXiv2011

How Riemannian Manifolds Converge: A Survey

This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic notions which have been applied to study sequences of submanifolds in Euclidean space: Hausdorff convergence of sets, flat convergence of integral currents, and weak convergence of varifolds. We next describe a variety of intrinsic notions of convergence which have been applied to study sequences of compact Riemannian manifolds: Gromov-Hausdorff convergence of metric spaces, convergence of metric measure spaces, Instrinsic Flat convergence of integral current spaces, and ultralimits of metric spaces. We close with a speculative section addressing possible notions of intrinsic varifold convergence, convergence of Lorentzian manifolds and area convergence.

math.DG↗

arXiv2011

Near-equality of the Penrose Inequality for rotationally symmetric Riemannian manifolds

This article is the sequel to our previous paper [LS] dealing with the near-equality case of the Positive Mass Theorem. We study the near-equality case of the Penrose Inequality for the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature whose boundaries are outermost minimal hypersurfaces. Specifically, we prove that if the Penrose Inequality is sufficiently close to being an equality on one of these manifolds, then it must be close to a Schwarzschild space with an appended cylinder, in the sense of Lipschitz Distance. Since the Lipschitz Distance bounds the Intrinsic Flat Distance on compact sets, we also obtain a result for Intrinsic Flat Distance, which is a more appropriate distance for more general near-equality results, as discussed in [LS]

math.DG↗

arXiv2009

Weak convergence and cancellation

In this article, we study the relationship between the weak limit of a sequence of integral currents in a metric space and the possible Hausdorff limit of the sequence of supports. Due to cancellation, the weak limit is in general supported in a strict subset of the Hausdorff limit. We exhibit sufficient conditions in terms of topology of the supports which ensure that no cancellation occurs and that the support of the weak limit agrees with the Hausdorff limit of the supports. We use our results to prove countable $\hm^m$-rectifiability of the Gromov-Hausdorff limit of sequences of Lipschitz manifolds $M_n$ all of which are $λ$-linearly locally contractible up to some scale $r_0$. In an appendix, written by Raanan Schul and the second author, it is shown that the Gromov-Hausdorff limit need not be countably $\hm^m$-rectifiable if the $M_n$ have a common local geometric contractibility function which is only concave (and not linear). We also relate our results to work of Cheeger-Colding on the limits of noncollapsing sequences of manifolds with nonnegative Ricci curvature.

math.DG↗

arXiv2008

Conjugate Points in Length Spaces

In this paper we extend the concept of a conjugate point in a Riemannian manifold to complete length spaces (also known as geodesic spaces). In particular, we introduce symmetric conjugate points and ultimate conjugate points. We then generalize the long homotopy lemma of Klingenberg to this setting as well as the injectivity radius estimate also due to Klingenberg which was used to produce closed geodesics or conjugate points on Riemannian manifolds. Our versions apply in this more general setting. We next focus on ${\rm CBA}(κ)$ spaces, proving Rauch-type comparison theorems. In particular, much like the Riemannian setting, we prove an Alexander-Bishop theorem stating that there are no ultimate conjugate points less than $π$ apart in a ${\rm CBA}(1)$ space. We also prove a relative Rauch comparison theorem to precisely estimate the distance between nearby geodesics. We close with applications and open problems.

math.MG↗

arXiv2008

The Cut-off Covering Spectrum

We introduce the $R$ cut-off covering spectrum and the cut-off covering spectrum of a complete length space or Riemannian manifold. The spectra measure the sizes of localized holes in the space and are defined using covering spaces called $δ$ covers and $R$ cut-off $δ$ covers. They are investigated using $δ$ homotopies which are homotopies via grids whose squares are mapped into balls of radius $δ$. On locally compact spaces, we prove that these new spectra are subsets of the closure of the length spectrum. We prove the $R$ cut-off covering spectrum is almost continuous with respect to the pointed Gromov-Hausdorff convergence of spaces and that the cut-off covering spectrum is also relatively well behaved. This is not true of the covering spectrum defined in our earlier work which was shown to be well behaved on compact spaces. We close by analyzing these spectra on Riemannian manifolds with lower bounds on their sectional and Ricci curvature and their limit spaces.

math.MG↗

arXiv2006

The Topology of Open Manifolds with Nonnegative Ricci Curvature

We survey all results concerning the topology of complete noncompact Riemannian manifolds with nonnegative Ricci curvature that have no additional conditions other than restrictions to the dimension, volume growth or diameter growth of the manifold. We will also present relevant examples and list open problems.

math.DG↗

arXiv2006

Convergence and the Length Spectrum

The author defines and analyzes the $1/k$ length spectra, $L_{1/k}(M)$, whose union, over all $k\in \NN$ is the classical length spectrum. These new length spectra are shown to converge in the sense that $\lim_{i\to\infty} L_{1/k}(M_i) \subset \{0\}\cup L_{1/k}(M)$ as $M_i\to M$ in the Gromov-Hausdorff sense. Energy methods are introduced to estimate the shortest element of $L_{1/k}$, as well as a concept called the minimizing index which may be used to estimate the length of the shortest closed geodesic of a simply connected manifold in any dimension. A number of gap theorems are proven, including one for manifolds, $M^n$, with $Ricci\ge (n-1)$ and volume close to $Vol(S^n)$. Many results in this paper hold on compact length spaces in addition to Riemannian manifolds.

math.MG↗