arXiv subjects

Brian Allen

Publications and source records attributed to Brian Allen.

28 records · Page 2Linked to original sources

arXiv2020

Properties of the Null Distance and Spacetime Convergence

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 to an intrinsic, conformally invariant metric that induces the manifold topology. We show when warped products of low regularity and globally hyperbolic spacetimes endowed with the null distance are (local) integral current spaces. This metric and integral current structure sets the stage for investigating convergence analogous to Riemannian geometry. Our main theorem is a general convergence result for warped product spacetimes relating uniform, Gromov--Hausdorff and Sormani--Wenger intrinsic flat convergence of the corresponding null distances. In addition, we show that non-uniform convergence of warping functions in general leads to distinct limiting behavior, such as limits that disagree.

math.DG↗

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.

math.MG↗

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 $(M,g_0)$.

math.MG↗

arXiv2019

Sobolev bounds and convergence of Riemannian manifolds

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 Hölder sense. This is done by establishing a general trace inequality on Riemannian manifolds which is an interesting result on its own. We provide examples demonstrating how each of our hypotheses are necessary. In the Appendix by the first author with Christina Sormani, we prove that sequences of compact integral current spaces without boundary (including Riemannian manifolds) that have uniform Hölder bounds on their distance functions have subsequences converging in the Gromov--Hausdorff (GH) sense. If in addition they have a uniform upper bound on mass (volume) then they converge in the Sormani--Wenger Intrinsic Flat (SWIF) sense to a limit whose metric completion is the GH limit. We provide an example of a sequence developing a cusp demonstrating why the SWIF and GH limits may not agree.

math.DG↗

arXiv2019

Sobolev stability of the PMT and RPI using IMCF

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 $M^3_i$ can be foliated by a smooth solution of Inverse Mean Curvature Flow (IMCF) which is uniformly controlled for time $t \in [0,T]$. In particular, we consider a sequence of regions of manifolds $U_T^i\subset M_i^3$, foliated by a IMCF, $Σ_t$, such that if $\partial U_T^i = Σ_0^i \cup Σ_T^i$ and $m_H(Σ_T^i) \rightarrow 0$ then $U_T^i$ converges in $W^{1,2}$ to a flat annulus or in the hyperbolic setting it converges to a annulus portion of hyperbolic space. If instead $m_H(Σ_T^i)-m_H(Σ_0^i) \rightarrow 0$ and $m_H(Σ_T^i) \rightarrow m >0$ then we show that $U_T^i$ converges in $W^{1,2}$ to a topological annulus portion of the Schwarzschild metric or in the Hyperbolic case to a topological annulus portion of the Anti-de~Sitter Schwarzschild metric.

math.DG↗

arXiv2018

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

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 investigate Inverse Mean Curvature Flow (IMCF) of non-compact hypersurfaces in hyperbolic space. Specifically, we look at bounded graphs over horospheres in $\mathbb{H}^{n+1}$ and show long time existence of the flow as well as asymptotic convergence to horospheres.

math.DG↗

arXiv2018

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

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 be foliated by a smooth solution of Inverse Mean Curvature Flow (IMCF) which is uniformly controlled. We consider a sequence of regions of asymptotically hyperbolic manifolds $U_T^i\subset M_i^3$, foliated by a smooth solution to IMCF which is uniformly controlled, and if $\partial U_T^i = Σ_0^i \cup Σ_T^i$ and $m_H(Σ_T^i) \rightarrow 0$ then $U_T^i$ converges to a topological annulus portion of hyperbolic space with respect to $L^2$ metric convergence. If instead $m_H(Σ_T^i)-m_H(Σ_0^i) \rightarrow 0$ and $m_H(Σ_T^i) \rightarrow m >0$ then we show that $U_T^i$ converges to a topological annulus portion of the Anti-deSitter Schwarzschild metric with respect to $L^2$ metric convergence.

math.DG↗

arXiv2018

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

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 in warped product manifolds is related to the conformal vector field on the conformal metric and use this to gain control of the flow in order to show long time existence. Connections are made to recent results of the author on stability of the positive mass theorem (PMT) and the Riemannian Penrose inequality (RPI) where long time existence of IMCF is an important assumption.

math.DG↗

arXiv2017

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

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 foliated by a smooth solution of Inverse Mean Curvature Flow (IMCF) which is uniformly controlled. We consider a sequence of regions of asymptotically flat manifolds $U_T^i\subset M_i^3$, foliated by a smooth solution to IMCF which is uniformly controlled, and if $\partial U_T^i = Σ_0^i \cup Σ_T^i$ and $m_H(Σ_T^i) \rightarrow 0$ then $U_T^i$ converges to a flat annulus with respect to $L^2$ metric convergence. If instead $m_H(Σ_T^i)-m_H(Σ_0^i) \rightarrow 0$ and $m_H(Σ_T^i) \rightarrow m >0$ then we show that $U_T^i$ converges to a topological annulus portion of the Schwarzschild metric with respect to $L^2$ metric convergence.

math.DG↗

arXiv2016

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

We investigate Inverse Mean Curvature Flow (IMCF) of non-compact hypersurfaces in hyperbolic space. Specifically, we look at bounded graphs over horospheres in $\mathbb{H}^{n+1}$ and show long time existence of the flow. Along the way many important local estimates as well as global estimates are obtained. In addition, we develop a useful family of cutoff functions for IMCF as well as a non-compact ODE maximum principle at infinity which are integral tools used throughout the document.

math.DG↗