https://neckpinch.com/authors/brian-harvie/New arXiv papers by Brian Harvie, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2509.18026/arxiv:2509.18026Mon, 22 Sep 2025 00:00:00 GMTBrian Harvie, Ye-Kai Wang — We establish several geometric characterizations and rigidity results for 3-dimensional asymptotically locally hyperbolic (ALH) static spaces with horizon boundary. Notably, a 3-dimensional ALH static space with toroidal infinity and strictly non-spherical horizons is isometric to a toroidal Kottler metric. Furthermore, we show that the surface gravity of a static horizon with spherical infinity is bounded below by √3, with equality achieved only by the critical AdS-Schwarzschild metric. Consequently, Poincaré-Einstein fillings of S² × S¹(λ) arising from these spaces have length parameter λ≤ (√3)⁻¹, which supports a recent conjecture of Chang-Yang-Zhang[18]. Finally, static horizons with hyperbolic infinity and non-negative Chruściel-Herzlich mass obey the reverse Riemannian Penrose inequality. In conjunction with work of Ge-Wang-Wu-Xia [29], we use this fact to obtain uniqueness of static ALH graphs with hyperbolic infinities. These results follow from a generalization of the Minkowski inequality in AdS-Schwarzschild space due to Brendle-Hung-Wang [15]. Using optimal coefficients for the sub-static Heintze-Karcher inequality from [24], we construct a new monotone quantity under inverse mean curvature flow (IMCF) in static spaces with negative cosmological constant. Another fundamental tool developed in this paper is a regularity theorem for IMCF in ALH manifolds. Specifically, we prove that a weak solution of IMCF in an ALH 3-manifold with horizon boundary is eventually smooth. This extends the regularity theorem for a spherical infinity due to Shi-Zhu [52].Mean curvature flowOther curvature flowsInverse mean curvature flow3-manifoldsWeak & singular flowsStability & uniquenessGeometric inequalitiesPhysics https://neckpinch.com/papers/2404.08410/arxiv:2404.08410Fri, 12 Apr 2024 00:00:00 GMTBrian Harvie — We prove that a proper weak solution \ Ωₜ \_(0 ≤ t < ∞) to inverse mean curvature flow in ℍⁿ, 3≤ n ≤ 7, is smooth and star-shaped by the time \beginequation* T= (n-1) \log \left( \frac{sinh \left( r_+ \right)}{ sinh \left( r_- \right)} \right), \endequation* where r₊ and r₋ are the geodesic out-radius and in-radius of the initial domain Ω₀. The argument is inspired by the Alexandrov reflection method for extrinsic curvature flows in ℝⁿ due to Chow-Gulliver and uses a result of Li-Wei. In addition to this, our methods establish expanding spheres as the only proper weak IMCF on ℍⁿ ∖ \ 0 \ in all dimensions. As applications, we extend the Minkowski inequalities of Brendle-Hung-Wang and De Lima-Girao to outer-minimizing domains Ω₀ ⊂ ℍⁿ in dimensions 3 ≤ n ≤ 7. From this, we also extend a Penrose-type inequality to balanced asymptotically hyperbolic graphs over the exteriors of outer-minimizing domains of ℍⁿ in these dimensions.Mean curvature flowOther curvature flowsInverse mean curvature flowWeak & singular flowsBoundaries & non-compactGeometric inequalities