https://neckpinch.com/authors/artemis-a-vogiatzi/New arXiv papers by Artemis A. Vogiatzi, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2602.20865arxiv:2602.20865Tue, 24 Feb 2026 00:00:00 GMTHuy The Nguyen, Artemis A. Vogiatzi — We study curve shortening flow in high codimension for arcs with free boundary meeting a fixed smooth barrier orthogonally. We prove dilation-invariant curvature and higher-derivative estimates up to the boundary using a Stahl-type localised maximum principle and an adapted cut-off. Using a reflected Gaussian entropy and blow-up analysis, Type I boundary singularities yield a shrinking semicircle model after reflection. Type II blow-ups give a Grim Reaper translator, which is ruled out under a free-boundary entropy bound <2. Hence in the low-entropy regime the flow either converges to the orthogonal chord or has only semicircle boundary singularities.Curve shorteningSelf-shrinkers & translatorsSingularitiesFree boundary & capillary https://arxiv.org/abs/2408.08022arxiv:2408.08022Thu, 15 Aug 2024 00:00:00 GMTArtemis A. Vogiatzi — We prove a sharp quartic curvature pinching for the mean curvature flow in S^(n+m), m≥2, which generalises Pu's work on the convergence of submanifolds in S^(n+m) to a round point. Using a blow up argument, we prove a codimension and a cylindrical estimate, where in regions of high curvature, the submanifold becomes approximately codimension one, quantitatively, and is weakly convex and moves by translation or is a self shrinker. With a decay estimate, the rescaling converges smoothly to a totally geodesic limit in infinite time, without using Stampacchia iteration or integral analysis.Mean curvature flowSelf-shrinkers & translatorsSingularitiesCurvature pinchingLong-time existence & convergence https://arxiv.org/abs/2311.01407arxiv:2311.01407Thu, 02 Nov 2023 00:00:00 GMTArtemis A. Vogiatzi — We study the mean curvature flow of smooth m-dimensional compact submanifolds with quadratic pinching in the Riemannian manifold ℂPⁿ. Our main focus is on the case of high codimension, k≥ 2. We establish a codimension estimate that shows in regions of high curvature, the submanifold becomes approximately codimension one in a quantifiable way. This estimate enables us to prove at a singular time of the flow, there exists a rescaling that converges to a smooth codimension-one limiting flow in Euclidean space. Under a cylindrical type pinching, we show that this limiting flow is weakly convex and moves by translation. These estimates allow us to analyse the behaviour of the flow near singularities and establish the existence of the limiting flow. Lastly, we prove a decay estimate that shows that the rescaling converges smoothly to a totally geodesic limit in infinite time. This behaviour is only possible if the dimension of the submanifold is even. Our approach relies on the preservation of the quadratic pinching condition along the flow and a gradient estimate that controls the mean curvature in regions of high curvature. This result generalises the work of Pipoli and Sinestrari on the mean curvature flow of submanifolds of the complex projective space.Mean curvature flowSingularitiesCurvature pinchingLong-time existence & convergence https://arxiv.org/abs/2303.00414arxiv:2303.00414Wed, 01 Mar 2023 00:00:00 GMTArtemis A. Vogiatzi, Huy T. Nguyen — We study the mean curvature flow of smooth n-dimensional compact submanifolds with quadratic pinching in a Riemannian manifold 𝓝^(n+m). Our main focus is on the case of high codimension, m≥ 2. We establish a codimension estimate that shows in regions of high curvature, the submanifold becomes approximately codimension one in a quantifiable way. This estimate enables us to prove at a singular time of the flow, there exists a rescaling that converges to a smooth codimension-one limiting flow in Euclidean space. Under a cylindrical type pinching, this limiting flow is weakly convex and moves by translation. Our approach relies on the preservation of the quadratic pinching condition along the flow and a gradient estimate that controls the mean curvature in regions of high curvature. These estimates allow us to analyse the behaviour of the flow near singularities and establish the existence of the limiting flow.Mean curvature flowSingularitiesCurvature pinching