https://neckpinch.com/authors/brian-allen/New arXiv papers by Brian Allen, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/1808.07841/arxiv:1808.07841Thu, 23 Aug 2018 00:00:00 GMTBrian 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 M³ᵢ can be foliated by a smooth solution of Inverse Mean Curvature Flow (IMCF) which is uniformly controlled for time t ∈ [0,T]. In particular, we consider a sequence of regions of manifolds U_Tⁱ⊂ Mᵢ³, foliated by a IMCF, Σₜ, such that if ∂U_Tⁱ = Σ₀ⁱ ∪ Σ_Tⁱ and m_H(Σ_Tⁱ) → 0 then U_Tⁱ 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_Tⁱ)−m_H(Σ₀ⁱ) → 0 and m_H(Σ_Tⁱ) → m >0 then we show that U_Tⁱ 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.Mean curvature flowOther curvature flowsInverse mean curvature flowStability & uniquenessGeometric inequalitiesPhysics https://neckpinch.com/papers/1807.08822/arxiv:1807.08822Mon, 23 Jul 2018 00:00:00 GMTBrian 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_Tⁱ⊂ Mᵢ³ are foliated by a smooth solution to Inverse Mean Curvature Flow (IMCF) which may not be uniformly controlled near the boundary. Then if ∂U_Tⁱ = Σ₀ⁱ ∪ Σ_Tⁱ, m_H(Σ_Tⁱ) → 0 and extra technical conditions are satisfied we show that U_Tⁱ converges to a flat annulus with respect to Sormani-Wenger Intrinsic Flat (SWIF) convergence.Mean curvature flowOther curvature flowsInverse mean curvature flowScalar curvatureStability & uniquenessPhysics https://neckpinch.com/papers/1708.02535/arxiv:1708.02535Tue, 08 Aug 2017 00:00:00 GMTBrian 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 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.Mean curvature flowOther curvature flowsInverse mean curvature flowStability & uniquenessLong-time existence & convergenceRotational symmetry & warped productsGeometric inequalitiesPhysics https://neckpinch.com/papers/1707.09388/arxiv:1707.09388Fri, 28 Jul 2017 00:00:00 GMTBrian 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 M³ 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_Tⁱ⊂ Mᵢ³, foliated by a smooth solution to IMCF which is uniformly controlled, and if ∂U_Tⁱ = Σ₀ⁱ ∪ Σ_Tⁱ and m_H(Σ_Tⁱ) → 0 then U_Tⁱ converges to a topological annulus portion of hyperbolic space with respect to L² metric convergence. If instead m_H(Σ_Tⁱ)−m_H(Σ₀ⁱ) → 0 and m_H(Σ_Tⁱ) → m >0 then we show that U_Tⁱ converges to a topological annulus portion of the Anti-deSitter Schwarzschild metric with respect to L² metric convergence.Mean curvature flowOther curvature flowsInverse mean curvature flowStability & uniquenessBoundaries & non-compactGeometric inequalitiesPhysics https://neckpinch.com/papers/1705.00591/arxiv:1705.00591Mon, 01 May 2017 00:00:00 GMTBrian 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 M³ 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_Tⁱ⊂ Mᵢ³, foliated by a smooth solution to IMCF which is uniformly controlled, and if ∂U_Tⁱ = Σ₀ⁱ ∪ Σ_Tⁱ and m_H(Σ_Tⁱ) → 0 then U_Tⁱ converges to a flat annulus with respect to L² metric convergence. If instead m_H(Σ_Tⁱ)−m_H(Σ₀ⁱ) → 0 and m_H(Σ_Tⁱ) → m >0 then we show that U_Tⁱ converges to a topological annulus portion of the Schwarzschild metric with respect to L² metric convergence.Mean curvature flowOther curvature flowsInverse mean curvature flowStability & uniquenessBoundaries & non-compactGeometric inequalitiesPhysics https://neckpinch.com/papers/1610.01211/arxiv:1610.01211Tue, 04 Oct 2016 00:00:00 GMTBrian 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 investigate Inverse Mean Curvature Flow (IMCF) of non-compact hypersurfaces in hyperbolic space. Specifically, we look at bounded graphs over horospheres in ℍⁿ⁺¹ and show long time existence of the flow as well as asymptotic convergence to horospheres.Mean curvature flowOther curvature flowsInverse mean curvature flowBoundaries & non-compactLong-time existence & convergence