Brian Allen

activity

20152023

most citedProperties of the Null Distance and Spacetime Convergence

25 citations · 102 across the 18 of their papers we have counts for

collaborators

20 papers

gr-qc2023

Null Distance and Gromov-Hausdorff Convergence of Warped Product Spacetimes

Brian Allen

What is the analogous notion of Gromov-Hausdorff convergence for sequences of spacetimes? Since a Lorentzian manifold is not inherently a metric space, one cannot simply use the tr…

math.DG2023

On the Stability of Llarull's Theorem in Dimension Three

Brian Allen, Edward Bryden, Demetre Kazaras

Llarull's Theorem states that any Riemannian metric on the $n$ -sphere which has scalar curv{\-}ature greater than or equal to $n (n - 1)$ , and whose distance function is bounded below…

math.DG2022★ 1 cited

Stability of the positive mass theorem and torus rigidity theorems under integral curvature bounds

Brian Allen, Edward Bryden, Demetre Kazaras

Work of D. Stern and Bray-Kazaras-Khuri-Stern provide differential-geometric identities which relate the scalar curvature of Riemannian 3-manifolds to global invariants in terms of…

math.DG2021★ 1 cited

Sobolev Inequalities and Convergence For Riemannian Metrics and Distance Functions

Brian Allen, Edward Bryden

If one thinks of a Riemannian metric, $g_{1}$ , analogously as the gradient of the corresponding distance function, $d_{1}$ , with respect to a background Riemannian metric, $g_{0}$ , then…

math.DG2021★ 1 cited

From $L^{p}$ Bounds To Gromov-Hausdorff Convergence Of Riemannian Manifolds

Brian Allen

In this paper we provide a way of taking $L^{p}$ , $p > \frac{m}{2}$ bounds on a $m -$ dimensional Riemannian metric and transforming that into Hölder bounds for the corresponding dist…

math.DG2020★ 17 cited

Almost non-negative scalar curvature on Riemannian manifolds conformal to tori

Brian Allen

In this article we reduce the geometric stability conjecture for the scalar torus rigidity theorem to the conformal case via the Yamabe problem. Then we are able to prove the case…

math.DG2020★ 2 cited

Intrinsic Flat Stability of Manifolds with Boundary where Volume Converges and Distance is Bounded Below

Brian Allen, Raquel Perales

Given a compact, connected, and oriented manifold with boundary $M$ and a sequence of smooth Riemannian metrics defined on it, $g_{j}$ , we prove volume preserving intrinsic flat conv…

math.MG2020★ 1 cited

Volume Above Distance Below

Brian Allen, Raquel Perales, Christina Sormani

Given a pair of metric tensors $g_{1} \geq g_{0}$ on a Riemannian manifold, $M$ , it is well known that $Vol_{1} (M) \geq Vol_{0} (M)$ . Furthermore one has rigidit…

math.MG2019

Relating Notions of Convergence in Geometric Analysis

Brian Allen, Christina Sormani

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 manifold…

math.DG2019★ 25 cited

Properties of the Null Distance and Spacetime Convergence

Brian Allen, Annegret Burtscher

The null distance for Lorentzian manifolds was recently introduced by Sormani and Vega. Under mild assumptions on the time function of the spacetime, the null distance gives rise t…

math.DG2018

Sobolev bounds and convergence of Riemannian manifolds

Brian Allen, Edward Bryden

We consider sequences of compact Riemannian manifolds with uniform Sobolev bounds on their metric tensors, and prove that their distance functions are uniformly bounded in the Höld…

math.DG2018★ 5 cited

Sobolev stability of the PMT and RPI using IMCF

Brian Allen

We study the Sobolev stability of the Positive Mass Theorem (PMT) and the Riemannian Penrose Inequality (RPI) in the case where a region of a sequence of manifolds $M_{i}^{3}$ can be f…

math.DG2018

Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem

Brian Allen

We study the stability of the Positive Mass Theorem (PMT) in the case where a sequence of regions of manifolds with positive scalar curvature $U_{T}^{i} \subset M_{i}^{3}$ are foliated by a…

math.DG2018★ 13 cited

Warped Tori with Almost Non-Negative Scalar Curvature

Brian Allen, Lisandra Hernandez-Vazquez, Davide Parise +2

For sequences of warped product metrics on a $3$ -torus satisfying the scalar curvature bound $R_{j} \geq - \frac{1}{j}$ , uniform upper volume and diameter bounds, and a uniform lower…

math.MG2018★ 13 cited

Contrasting Various Notions of Convergence in Geometric Analysis

Brian Allen, Christina Sormani

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 corre…

math.DG2017★ 2 cited

Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds

Brian Allen

In this paper we study Inverse Mean Curvature Flow (IMCF) on manifolds that are conformal to a warped product manifold. To this end, we show how the gradient conformal vector field…

math.DG2017★ 8 cited

Stability of the Positive Mass Theorem and Riemannian Penrose Inequality for Asymptotically Hyperbolic Manifolds Foliated by Inverse Mean Curvature Flow

Brian Allen

We study the stability of the Positive Mass Theorem (PMT) and the Riemannian Penrose Inequality (RPI) in the case where a region of an asymptotically hyperbolic manifold $M^{3}$ can…

math.DG2017★ 10 cited

Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem and Riemannian Penrose Inequality Under $L^{2}$ Metric Convergence

Brian Allen

We study the stability of the Positive Mass Theorem (PMT) and the Riemannian Penrose Inequality (RPI) in the case where a region of an asymptotically flat manifold $M^{3}$ can be fol…

math.DG2016★ 3 cited

ODE Maximum Principle at Infinity and Non-Compact Solutions of IMCF in Hyperbolic Space

Brian Allen

In this work we extend the ODE Maximum principle of Hamilton to non-compact hypersurfaces using the Omari-Yau maximum principle at infinity. As an application of this result, we in…

math.DG2015

Long Time Existence of Non-Compact Inverse Mean Curvature Flow in Hyperbolic Space

Brian Allen

We investigate Inverse Mean Curvature Flow (IMCF) of non-compact hypersurfaces in hyperbolic space. Specifically, we look at bounded graphs over horospheres in $H^{n + 1}$ a…

Null Distance and Gromov-Hausdorff Convergence of Warped Product Spacetimes

On the Stability of Llarull's Theorem in Dimension Three

Stability of the positive mass theorem and torus rigidity theorems under integral curvature bounds

Sobolev Inequalities and Convergence For Riemannian Metrics and Distance Functions

From Lp Bounds To Gromov-Hausdorff Convergence Of Riemannian Manifolds

Almost non-negative scalar curvature on Riemannian manifolds conformal to tori

Intrinsic Flat Stability of Manifolds with Boundary where Volume Converges and Distance is Bounded Below

Volume Above Distance Below

Relating Notions of Convergence in Geometric Analysis

Properties of the Null Distance and Spacetime Convergence

Sobolev bounds and convergence of Riemannian manifolds

Sobolev stability of the PMT and RPI using IMCF

Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem

Warped Tori with Almost Non-Negative Scalar Curvature

Contrasting Various Notions of Convergence in Geometric Analysis

Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds

Stability of the Positive Mass Theorem and Riemannian Penrose Inequality for Asymptotically Hyperbolic Manifolds Foliated by Inverse Mean Curvature Flow

Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem and Riemannian Penrose Inequality Under L2 Metric Convergence

ODE Maximum Principle at Infinity and Non-Compact Solutions of IMCF in Hyperbolic Space

Long Time Existence of Non-Compact Inverse Mean Curvature Flow in Hyperbolic Space

From $L^{p}$ Bounds To Gromov-Hausdorff Convergence Of Riemannian Manifolds

Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem and Riemannian Penrose Inequality Under $L^{2}$ Metric Convergence