https://neckpinch.com/authors/theodora-bourni/New arXiv papers by Theodora Bourni, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.12817arxiv:2610.12817Thu, 08 Oct 2026 00:00:00 GMTTheodora Bourni, Nathan Burns, Marcus Flook — We construct compact, embedded solutions to the mean curvature flow in ℝⁿ⁺¹ that arise from various non-compact initial data and exist for a finite time.Mean curvature flowSingularitiesBoundaries & non-compact 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/2602.12186arxiv:2602.12186Thu, 12 Feb 2026 00:00:00 GMTTheodora Bourni, José M. Espinar, Aakash Mishra — We develop an Aleksandrov reflection framework for a large class of expanding curvature flows in hyperbolic space, with inverse mean curvature flow serving as a model case. The method applies to the level-set formulation of the flow, and as a consequence we obtain graphical and Lipschitz estimates. Using these estimates, we show that solutions become star-shaped and therefore converge exponentially fast to an umbilic hypersurface at infinity. We also extend these results to the non-compact setting in two cases. First, assuming the asymptotic boundary of the solution consists of a single point, we show that the flow becomes a graph over a horosphere with uniform gradient bounds and converges to a limiting horosphere. Second, assuming the asymptotic boundary consists of two points, we prove that the flow eventually becomes a global graph over a hyperbolic cylinder with uniform gradient bounds; this is achieved through an explicit cylindrical barrier construction analogous to the horospherical one.Mean curvature flowOther curvature flowsInverse mean curvature flowWeak & singular flowsBoundaries & non-compactLong-time existence & convergence https://arxiv.org/abs/2602.08850arxiv:2602.08850Mon, 09 Feb 2026 00:00:00 GMTTheodora Bourni, Giada Franz — We establish rigidity results for ancient solutions to the free boundary mean curvature flow in manifolds with convex boundary. In particular, we show that any free boundary minimal hypersurface of Morse index I admits an I-parameter family of ancient solutions that emanate from it. Moreover, among ancient solutions that backward converge exponentially fast to the minimal hypersurface, these exhaust all possibilities. Additionally, we construct a smooth free boundary mean convex foliation around an unstable free boundary minimal hypersurface that enables us to provide a more detailed geometric description of mean-convex ancient solutions that backward converge to that minimal surface.Mean curvature flowAncient solutionsStability & uniquenessFree boundary & capillaryLong-time existence & convergenceMinimal & CMC surfaces https://arxiv.org/abs/2404.08824arxiv:2404.08824Fri, 12 Apr 2024 00:00:00 GMTTheodora Bourni, Nathan Burns, Spencer Catron — We classify convex ancient curve shortening flows with free boundary on general bounded convex domains.Curve shorteningAncient solutionsFree boundary & capillary https://arxiv.org/abs/2309.00712arxiv:2309.00712Fri, 01 Sep 2023 00:00:00 GMTTheodora Bourni, Benjamin Richards — We construct a translating solution to anisotropic curve shortening flow and show that for a given anisotropic factor g:S¹→ℝ_+, and a given direction and speed, this translator is unique. We then construct an ancient compact solution to anisotropic curve shortening flow, and show that this solution, along with the appropriate translating solution, are the unique solutions to anisotropic curve shortening flow that lie in a slab of a given width and no smaller.Curve shorteningAncient solutionsSelf-shrinkers & translators https://arxiv.org/abs/2303.17385arxiv:2303.17385Thu, 30 Mar 2023 00:00:00 GMTTheodora Bourni, Martin Reiris — We construct a slingshot, that is a compact, embedded solution to curve shortening flow that comes out of a non compact curve and exists for a finite time.Curve shorteningSingularitiesBoundaries & non-compact 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/2201.02134arxiv:2201.02134Thu, 06 Jan 2022 00:00:00 GMTTheodora Bourni, Alexander Mramor — In this note we construct new nonplanar ancient (in fact, eternal) solutions to the curve shortening flow in ℝ³, built out of translating grim reapers laying in perpendicular planes.Curve shorteningAncient solutionsSelf-shrinkers & translators 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