https://neckpinch.com/authors/huy-the-nguyen/New arXiv papers by Huy The Nguyen, 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/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 https://arxiv.org/abs/2110.06027arxiv:2110.06027Tue, 12 Oct 2021 00:00:00 GMTReto Buzano, Huy The Nguyen, Mario B. Schulz — In his lecture notes on mean curvature flow, Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. Here, we employ min-max techniques to give a rigorous existence proof for these surfaces. Conjecturally, the self-shrinkers that we obtain have precisely one (asymptotically conical) end. We confirm this for large genus via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity. Finally, we provide numerical evidence for a further family of noncompact self-shrinkers with odd genus and two asymptotically conical ends.Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compactMinimal & CMC surfacesNumerics & visualization 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