https://neckpinch.com/authors/mat-langford/New arXiv papers by Mat Langford, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2603.06949arxiv:2603.06949Fri, 06 Mar 2026 00:00:00 GMTTheodora Bourni, Nathan Burns, Mat Langford — We establish a sharp rate of convergence for a free-boundary curve shortening flow in a convex domain in ℝ² which converges in finite time to a round half-point.Curve shorteningSingularitiesStability & uniquenessFree boundary & capillary https://arxiv.org/abs/2510.12037arxiv:2510.12037Tue, 14 Oct 2025 00:00:00 GMTMat Langford, James McCoy — In a recent article, a localization of the Huisken–Stampacchia iteration method was developed, and used to establish localizations of the well-known "umbilic", "convexity" and "cylindrical" estimates for hypersurfaces evolving in Euclidean space by mean curvature flow. Here, we adapt the methods developed there to treat more general (fully nonlinear) flows, establishing localizations of asymptotically sharp curvature pinching estimates for hypersurfaces evolving by one-homogeneous functions of curvature under very general conditions. We also briefly describe how the method can be adapted to treat the deformation of hypersurfaces in curved ambient spaces (by suitable speed functions), which is fundamental for many important applications of such flows.Mean curvature flowOther curvature flowsCurvature pinchingHomogeneous spaces https://arxiv.org/abs/2404.01525arxiv:2404.01525Mon, 01 Apr 2024 00:00:00 GMTMat Langford, Yuxing Liu, George McNamara — Given any non-central interior point o of the unit disc D, the diameter L through o is the union of two linear arcs emanating from o which meet ∂D orthogonally, the shorter of them stable and the longer unstable (under these boundary conditions). In each of the two half discs bounded by L, we construct a convex eternal solution to curve shortening flow which fixes o and meets ∂D orthogonally, and evolves out of the unstable critical arc at t=-∞ and into the stable one at t=+∞. We then prove that these two (congruent) solutions are the only non-flat convex ancient solutions to the curve shortening flow satisfying the specified boundary conditions. We obtain analogous conclusions in the "degenerate" case o∈∂D as well, although in this case the solution contracts to the point o at a finite time with asymptotic shape that of a half Grim Reaper, thus providing an interesting example for which an embedded flow develops a collapsing singularity.Curve shorteningAncient solutionsSelf-shrinkers & translatorsSingularities(Non)collapsingStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2310.02649arxiv:2310.02649Wed, 04 Oct 2023 00:00:00 GMTPaul Bryan, Mat Langford, Jonathan J. Zhu — We prove that curve shortening flow on the round sphere displays sharp chord-arc improvement, precisely as in the planar setting (Andrews and Bryan, Comm. Anal. Geom., 2011). As in the planar case, the sharp estimate implies control on the curvature, resulting in a direct and efficient proof that simple spherical curves either contract to round points (in finite time) or converge to great circles (in infinite time).Curve shorteningSingularities https://arxiv.org/abs/2303.09078arxiv:2303.09078Thu, 16 Mar 2023 00:00:00 GMTSathya Rengaswami, Mat Langford — We construct O(1)× O(n)-invariant ancient "pancake" solutions to a large and natural class of fully nonlinear curvature flows. We then establish that these are the unique O(n)-invariant ancient solutions to the corresponding flow which sweep out a slab by carrying out a fine asymptotic analysis for this class. This extends the main results of to a surprisingly general class of flows.Other curvature flowsAncient solutions https://arxiv.org/abs/2302.14258arxiv:2302.14258Tue, 28 Feb 2023 00:00:00 GMTMat Langford, Jonathan J. Zhu — We introduce a reflected chord-arc profile for curves with orthogonal boundary condition and obtain a chord-arc estimate for embedded free boundary curve shortening flows in a convex planar domain. As a consequence, we are able to prove that any such flow either converges in infinite time to a (unique) "critical chord", or contracts in finite time to a "round half-point" on the boundary.Curve shorteningSingularitiesBoundaries & non-compactFree boundary & capillary https://arxiv.org/abs/2302.04964arxiv:2302.04964Thu, 09 Feb 2023 00:00:00 GMTTheodora Bourni, Timothy Buttsworth, Ramiro Lafuente, Mat Langford — For each n≥ 3, we construct a 'pancake-like', O(2)× O(n-1)-invariant ancient Ricci flow with positive curvature operator and bounded "girth", and we determine its asymptotic limits backwards in time. This solution is new even in dimension three. The construction hinges on the Ricci flow invariance of certain conditions on the curvature and its spatial derivatives under this symmetry regime, whose proof does not follow from Hamilton's tensor maximum principle.Ricci flowAncient solutions3-manifoldsSphere theorems https://arxiv.org/abs/2211.12639arxiv:2211.12639Tue, 22 Nov 2022 00:00:00 GMTTheodora Bourni, Mat Langford, Stephen Lynch — We make rigorous an old idea of using mean curvature flow to prove a theorem of Richard Hamilton on the compactness of proper hypersurfaces with pinched, bounded curvature.Mean curvature flowLimit spaces & RCD https://arxiv.org/abs/2204.06710arxiv:2204.06710Thu, 14 Apr 2022 00:00:00 GMTTheodora Bourni, Mat Langford — We prove that there exists, in every dimension, a unique (modulo rotations about the origin and time translations) convex ancient mean curvature flow in the ball with free boundary on the sphere.Mean curvature flowAncient solutionsFree boundary & capillary https://arxiv.org/abs/2111.06051arxiv:2111.06051Thu, 11 Nov 2021 00:00:00 GMTTheodora Bourni, Mat Langford — We classify convex ancient curve shortening flows in the disc with free boundary on the circle.Curve shorteningAncient solutionsFree boundary & capillary https://arxiv.org/abs/2109.03651arxiv:2109.03651Wed, 08 Sep 2021 00:00:00 GMTMat Langford, Stephen Lynch, Huy The Nguyen — We study mean curvature flow of n-dimensional submanifolds of S_K^(n+ℓ), the round (n+ℓ)-sphere of sectional curvature K>0, under the quadratic curvature pinching condition |A|² < 1/(n-2)|H|² + 4K when n≥ 8, |A|² < 4/3n|H|²+n/2K when n=7, and |A|²<(3(n+1))/(2n(n+2))|H|²+(2n(n-1))/(3(n+1))K when n=5 or 6. This condition is related to a theorem of Li and Li [Arch. Math., 58:582–594, 1992] which states that the only n-dimensional minimal submanifolds of S_K^(n+ℓ) satisfying |A|²<2n/3K are the totally geodesic n-spheres. We prove the existence of a suitable mean curvature flow with surgeries starting from initial data satisfying the pinching condition. As a result, we conclude that any smoothly, properly immersed submanifold of S_Kⁿ⁺¹ satisfying the pinching condition is diffeomorphic either to the sphere Sⁿ or to the connected sum of a finite number of handles S¹× Sⁿ⁻¹. The results are sharp when n≥ 8 due to hypersurface counterexamples.Mean curvature flowSurgeryCurvature pinchingMinimal & CMC surfaces