arXiv subjects

Christina Sormani

Publications and source records attributed to Christina Sormani.

At least 19 recordsLinked to original sources

arXiv2026

Existence of uniform Temple charts and applications to null distance

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 $(N,\hat{d}_τ)$, where $\hat{d}_τ$ is the null distance of Sormani and Vega defined using a weak temporal function $τ$. In particular, we prove that $(N, \hat{d}_τ)$ is a rectifiable metric space, where the causal structure is locally encoded by $τ$ and $\hat{d}_τ$. 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.

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arXiv2026

Geometric Stability of the Schoen-Yau Zero Mass Theorem

In 1979, Schoen and Yau proved their famous Positive Mass Theorem which is a combination of a comparison theorem: {\em a three dimensional asymptotically flat Riemannian manifold with nonnegative scalar curvature has nonnegative ADM mass}, and a rigidity theorem: {\em if such a manifold has zero ADM mass then it is isometric to Euclidean space}. Here we review results and open questions on the geometric stability of their zero mass rigidity theorem: {\em if such a manifold has almost zero mass, how close is its geometry to that of Euclidean space}? We review the geometry of these spaces, examples of sequences of such spaces with mass approaching zero, and a variety of geometric notions of convergence. Although there has been much progress, it is still an open question (even in dimension three): exactly which geometric notion of convergence works best to capture the geometric stability of this famous rigidity theorem.

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arXiv2026

Gromov's Compactness Theorem for the Intrinsic Timed-Hausdorff Distance

The intrinsic timed-Hausdorff distance between timed-metric spaces, first introduced by Sakovich--Sormani, yields a weak notion of convergence for space-times. In this paper we prove a 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 convergence of metric spaces. We also obtain a new Arzelà--Ascoli theorem for real valued uniformly bounded Lipschitz functions on Gromov--Hausdorff converging compact metric spaces. Moreover, we establish the triangle inequality for the intrinsic timed-Hausdorff distance.

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arXiv2025

Geometric Convergence to an Extreme Limit Space with nonnegative scalar curvature

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.

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arXiv2025

Introducing Various Notions of Distances between Space-Times

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.

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arXiv2023

From Varadhan's Limit to Eigenmaps: A Guide to the Geometric Analysis behind Manifold Learning

We present an overview of the history of the heat kernel and eigenfunctions on Riemannian manifolds and how the theory has lead to modern methods of analyzing high dimensional data via eigenmaps and other spectral embeddings. We begin with Varadhan's Theorem relating the heat kernel to the distance function on a Riemannian manifold. We then review various theorems which bound the heat kernel on classes of Riemannian manifolds. Next we turn to eigenfunctions, the Sturm-Liouville Decomposition of the heat kernel using eigenfunctions, and various theorems which bound eigenfunctions on classes of Riemannian manifolds. We review various notions of convergence of Riemannian manifolds and which classes of Riemannian manifolds are compact with respect to which notions of convergence. We then present Bérard-Besson-Gallot's heat kernel embeddings of Riemannian manifolds and the truncation of those embeddings. Finally we turn to Applications of Spectral embeddings to the Dimension Reduction of data sets lying in high dimensional spaces reviewing, in particular, the work of Belkin-Niyogi and Coifman-Lafon. We also review the Spectral Theory of Graphs and the work of Dodziuk and Chung and others. We close with recent theorems of Portegies and of the first author controlling truncated spectral embeddings uniformly on key classes of Riemannian manifolds. Throughout we provide many explicitly computed examples and graphics and attempt to provide as complete a set of references as possible. We hope that this article is accessible to both pure and applied mathematicians working in Geometric Analysis and their doctoral students.

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arXiv2023

An Extreme Limit with Nonnegative Scalar Curvature

In 2014, Gromov vaguely conjectured that a sequence of manifolds with nonnegative scalar curvature should have a subsequence which converges in some weak sense to a limit space with some generalized notion of nonnegative scalar curvature. The conjecture has been made precise at an IAS Emerging Topics meeting: requiring that the sequence be three dimensional with uniform upper bounds on diameter and volume, and a positive uniform lower bound on MinA, which is the minimum area of a closed minimal surface in the manifold. Here we present a sequence of warped product manifolds with warped circles over standard spheres, that have circular fibres over the poles whose length diverges to infinity, that satisfy the hypotheses of this IAS conjecture. We prove this sequence converges in the $W^{1,p}$ sense for $p<2$ to an extreme limit space that has nonnegative scalar curvature in the distributional sense as defined by Lee-LeFloch and that the total distributional scalar curvature converges. This paper only requires expertise in smooth Riemannian Geometry, smooth minimal surfaces, and Sobolev Spaces. In a second paper, requiring expertise in metric geometry, the first two authors prove intrinsic flat and Gromov-Hausdorff convergence of our sequence to this extreme limit space and investigate its geometric properties.

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arXiv2022

Volume Above Distance Below

Given a pair of metric tensors $g_1 \ge g_0$ on a Riemannian manifold, $M$, it is well known that $\operatorname{Vol}_1(M) \ge \operatorname{Vol}_0(M)$. Furthermore one has rigidity: the volumes are equal if and only if the metric tensors are the same $g_1=g_0$. Here we prove that if $g_j \ge g_0$ and $\operatorname{Vol}_1(M)\to \operatorname{Vol}_0(M)$ then $(M,g_j)$ converge to $(M,g_0)$ in the volume preserving intrinsic flat sense. Well known examples demonstrate that one need not obtain smooth, $C^0$, Lipschitz, or even Gromov-Hausdorff convergence in this setting. Our theorem may also be applied as a tool towards proving other open conjectures concerning the geometric stability of a variety of rigidity theorems in Riemannian geometry. To complete our proof, we provide a novel way of estimating the intrinsic flat distance between Riemannian manifolds which is interesting in its own right.

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arXiv2021

Lorentzian area and volume estimates for integral mean curvature bounds

In the present paper we establish area and volume estimates for spacetimes satisfying the strong energy condition in terms of the area and the $L^n$-norm of the second fundamental form or the mean curvature of an initial Cauchy hypersurface. We believe that these estimates will lay some of the groundwork in establishing new convergence results for Cauchy developments $(M_j, g_j)$ of suitably converging initial data $(Σ_j ,h_j ,K_j )$.

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arXiv2021

Positive scalar curvature on $\mathbf{Pin}^\pm$- and $\mathbf{Spin}^c$-manifolds

It is well-known that spin structures and Dirac operators play a crucial role in the study of positive scalar curvature metrics (psc-metrics) on compact manifolds. Here we consider a class of non-spin manifolds with "almost spin" structure, namely those with spin$^c$ or pin$^\pm$-structures. It turns out that in those cases (under natural assumptions on such a manifold $M$), the index of a relevant Dirac operator completely controls existence of a psc-metric which is $S^1$- or $C_2$-invariant near a "special submanifold" $B$ of $M$. This submanifold $B\subset M$ is dual to the complex (respectively, real) line bundle $L$ which determines the spin$^c$ or pin$^\pm$ structure on $M$. We also show that these manifold pairs $(M,B)$ can be interpreted as "manifolds with fibered singularities" equipped with "well-adapted psc-metrics". This survey is based on our recent work as well as on our joint work with Paolo Piazza.

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arXiv2021

Stability of the Spacetime Positive Mass Theorem in Spherical Symmetry

The rigidity statement of the positive mass theorem asserts that an asymptotically flat initial data set for the Einstein equations with zero ADM mass, and satisfying the dominant energy condition, must arise from an embedding into Minkowski space. In this paper we address the question of what happens when the mass is merely small. In particular, we formulate a conjecture for the stability statement associated with the spacetime version of the positive mass theorem, and give examples to show how it is basically sharp if true. This conjecture is then established under the assumption of spherical symmetry in all dimensions. More precisely, it is shown that a sequence of asymptotically flat initial data satisfying the dominant energy condition, without horizons except possibly at an inner boundary, and with ADM masses tending to zero must arise from isometric embeddings into a sequence of static spacetimes converging to Minkowski space in the pointed volume preserving intrinsic flat sense. The difference of second fundamental forms coming from the embeddings and initial data must converge to zero in $L^p$, $1\leq p<2$. In addition some minor tangential results are also given, including the spacetime version of the Penrose inequality with rigidity statement in all dimensions for spherically symmetric initial data, as well as symmetry inheritance properties for outermost apparent horizons.

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arXiv2020

Smocked Metric Spaces and their Tangent Cones

We introduce the notion of a smocked metric spaces and explore the balls and geodesics in a collection of different smocked spaces. We find their rescaled Gromov-Hausdorff limits and prove these tangent cones at infinity exist, are unique, and are normed spaces. We close with a variety of open questions suitable for advanced undergraduates, masters students, and doctoral students.

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arXiv2020

Smooth Convergence Away from Singular Sets

We consider sequences of metrics, $g_j$, on a Riemannian manifold, $M$, which converge smoothly on compact sets away from a singular set $S\subset M$, to a metric, $g_\infty$, on $M\setminus S$. We prove theorems which describe when $M_j=(M, g_j)$ converge in the Gromov-Hausdorff sense to the metric completion, $(M_\infty,d_\infty)$, of $(M\setminus S, g_\infty)$. To obtain these theorems, we study the intrinsic flat limits of the sequences. A new method, we call hemispherical embedding, is applied to obtain explicit estimates on the Gromov-Hausdorff and Intrinsic Flat distances between Riemannian manifolds with diffeomorphic subdomains. Seven years after the publication of this paper in CAG, Brian Allen discovered a counter example to the published statement of Theorem 1.3. Note that Theorem 4.6 (which is the key theorem cited in other papers) remains correct. We have added an hypothesis to correct the statement of Theorem 1.3 and its consequences. This v4 includes corrections in blue, an erratum at the end of the introduction, and Brian Allen's example in an appendix. An erratum is also being sent to the journal.

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arXiv2020

Contrasting Various Notions of Convergence in Geometric Analysis

We explore the distinctions between $L^p$ convergence of metric tensors on a fixed Riemannian manifold versus Gromov-Hausdorff, uniform, and intrinsic flat convergence of the corresponding sequence of metric spaces. We provide a number of examples which demonstrate these notions of convergence do not agree even for two dimensional warped product manifolds with warping functions converging in the $L^p$ sense. We then prove a theorem which requires $L^p$ bounds from above and $C^0$ bounds from below on the warping functions to obtain enough control for all these limits to agree.

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arXiv2020

Relating Notions of Convergence in Geometric Analysis

We relate $L^p$ convergence of metric tensors or volume convergence to a given smooth metric to Intrinsic Flat and Gromov-Hausdorff convergence for sequences of Riemannian manifolds. We present many examples of sequences of conformal metrics which demonstrate that these notions of convergence do not agree in general even when the sequence is conformal, $g_j=f_j^2g_0$, to a fixed manifold. We then prove a theorem demonstrating that when sequences of metric tensors on a fixed manifold $M$ are bounded, $(1-1/j)g_0 \le g_j \le K g_0$, and either the volumes converge, $\operatorname{Vol}_j(M)\rightarrow \operatorname{Vol}_0(M)$, or the metric tensors converge in the $L^p$ sense, then the Riemannian manifolds $(M,g_j)$ converge in the measured Gromov-Hausdorff and volume preserving Intrinsic Flat sense to $(M,g_0)$.

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arXiv2020

Bartnik's mass and Hamilton's Modified Ricci Flow

We provide estimates on the Bartnik mass of constant mean curvature (CMC) surfaces which are diffeomorphic to spheres and have positive mean curvature. We prove that the Bartnik mass is bounded from above by the Hawking mass and a new notion we call the asphericity mass. The asphericity mass is defined by applying Hamilton's modified Ricci flow and depends only upon the restricted metric of the surface and not on its mean curvature. The theorem is proven by studying a class of asymptotically flat Riemannian manifolds foliated by surfaces satisfying Hamilton's modified Ricci flow with prescribed scalar curvature. Such manifolds were first constructed by the first author in her dissertation conducted under the supervision of M.T. Wang. We make a further study of this class of manifolds bounding the ADM masses of such manifolds and analyzing the rigid case when the Hawking mass of the inner surface of the manifold agrees with its ADM mass. New in 2020: After this paper was published, Hyun-Chul Jang observed that we dropped a term in our calculations. Tracking the consequences throughout, we see that we need only slightly change the definition of the asphericity mass and then all statements of our theorems, propositions, and lemmas remain the same as the published version with slight revisions to the proofs. Pengzi Miao observed we needed an assumption on Gauss curvature in Theorem 1. We include these corrections and also some clarifications where they are needed in blue. Both Hyun-Chul Jang and Pengzi Miao have approved of our corrections and we have sent an erratum to the journal.

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arXiv2018

An intrinsic flat limit of Riemannian manifolds with no geodesics

In this paper we produce a sequence of Riemannian manifolds $M_j^m$, $m \ge 2$, which converge in the intrinsic flat sense to the unit $m$-sphere with the restricted Euclidean distance. This limit space has no geodesics achieving the distances between points, exhibiting previously unknown behavior of intrinsic flat limits. In contrast, any compact Gromov-Hausdorff limit of a sequence of Riemannian manifolds is a geodesic space. Moreover, if $m\geq3$, the manifolds $M_j^m$ may be chosen to have positive scalar curvature.

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arXiv2018

Geometrostatic Manifolds of Small ADM Mass

We bound the locations of outermost minimal surfaces in geometrostatic manifolds whose ADM mass is small relative to the separation between the black holes and prove the Intrinsic Flat Stability of the Positive Mass Theorem in this setting.

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