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Huy The Nguyen

Publications and source records attributed to Huy The Nguyen.

At least 19 recordsLinked to original sources
arXiv2026

Allard Regularity for Abelian Yang--Mills--Higgs Equation

We study solutions to the self-dual Abelian Yang--Mills--Higgs (YMH) equations in the singular limit $\e \to 0 $, where the associated self-dual Ginzburg--Landau type energy \begin{align*} E_\e\begin{pmatrix}u\\ A\end{pmatrix} = \int_M \left( |\nabla^A u|^2 + \e^2 |F_A|^2 + \frac{(1 - |u|^2)^2}{4\e^2} \right) \mathrm{dvol}_g \end{align*} exhibits concentration along codimension-two sets. Using techniques inspired by Allard's regularity theory, we construct approximate solutions concentrating near a minimal submanifold and analyse their perturbations via a linearised operator projected orthogonally to gauge and translational zero modes. By working in Fermi coordinates and enforcing Coulomb gauge conditions, we derive uniform Lipschitz and curvature estimates for the solutions and obtain Hölder regularity for the scalar and connection components. These results establish a geometric framework for understanding vortex sheet formation and provide a regularity theory for the limiting defect set in the context of Abelian gauge theories.

math.DG↗
arXiv2026

High Codimension Curve Shortening Flow with Free Boundary

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.

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arXiv2025

High Codimension Mean Curvature Flow with Surgery

We construct a mean curvature flow with surgery for submanifolds of arbitrary codimension. The theory applies to closed submanifolds satisfying a natural quadratic pinching condition, which serves as the high-codimension analogue of 2-convexity and is preserved under the flow in dimensions $n \geq 8$. Our results therefore are in line with the current state-of-the-art in codimension one (where at present 2-convexity is required for surgery). Central to our analysis is a collection of new a priori estimates for the second fundamental form, uniform across surgeries, which yield a precise description of high-curvature regions and permit controlled surgeries. This provides the first notion of mean curvature flow through singularities with topological control in higher codimensions. As a consequence we obtain a sharp classification: Every closed quadratically 2-convexity submanifold is diffeomorphic either to $\mathbb{S}^n$ or to a finite connected sum of $\mathbb{S}^{n-1}$-bundles over $\mathbb{S}^1$.

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arXiv2024

Noncompact self-shrinkers for mean curvature flow with arbitrary genus

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.

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arXiv2023

Second order estimates for transition layers and a curvature estimate for the parabolic Allen-Cahn

The parabolic Allen-Cahn equation is a semilinear partial differential equation linked to the mean curvature flow by a singular perturbation. We show an improved convergence property of the parabolic Allen-Cahn equation to the mean curvature flow, which is the parabolic analogue of the improved convergence property of the elliptic Allen-Cahn to minimal surfaces by Wang-Wei and Chodosh-Mantoulidis. More precisely, we show if the phase-transition level sets are converging in $C^2$, then they converge in $C^{2,θ}$. As an application, we obtain a curvature estimate for parabolic Allen-Cahn equation, which can be viewed as a diffused version of Brakke's and White's regularity theorem for mean curvature flow

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arXiv2023

Quantization of the Energy for the inhomogeneous Allen-Cahn mean curvature

We consider the varifold associated to the Allen--Cahn phase transition problem in $\mathbb R^{n+1}$(or $n+1$-dimensional Riemannian manifolds with bounded curvature) with integral $L^{q_0}$ bounds on the Allen--Cahn mean curvature (first variation of the Allen--Cahn energy) in this paper. It is shown here that there is an equidistribution of energy between the Dirichlet and Potential energy in the phase field limit and that the associated varifold to the total energy converges to an integer rectifiable varifold with mean curvature in $L^{q_0}, q_0 > n$. The latter is a diffused version of Allard's convergence theorem for integer rectifiable varifolds.

math.DG↗
arXiv2021

Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery

We study mean curvature flow of $n$-dimensional submanifolds of $S_K^{n+\ell}$, the round $(n+\ell)$-sphere of sectional curvature $K>0$, under the quadratic curvature pinching condition $|A|^{2} < \frac{1}{n-2}|H|^{2} + 4K$ when $n\geq 8$, $|A|^{2} < \frac{4}{3n}|H|^{2}+\frac{n}{2}K$ when $n=7$, and $|A|^2<\frac{3(n+1)}{2n(n+2)}|H|^2+\frac{2n(n-1)}{3(n+1)}K$ 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 $S_K^{n+\ell}$ satisfying $|A|^2<\frac{2n}{3}K$ 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 $S_K^{n+1}$ satisfying the pinching condition is diffeomorphic either to the sphere $S^n$ or to the connected sum of a finite number of handles $S^1\times S^{n-1}$. The results are sharp when $n\geq 8$ due to hypersurface counterexamples.

math.DG↗
arXiv2020

Brakke Regularity for the Allen-Cahn Flow

In this paper we prove an analogue of the Brakke's $\varepsilon$-regularity theorem for the parabolic Allen-Cahn equation. In particular, we show uniform $C^{2,α}$ regularity for the transition layers converging to smooth mean curvature flows as $\varepsilon\rightarrow0$. The proof utilises Allen-Cahn versions of the monotonicity formula, parabolic Lipschitz approximation and blowups. A corresponding gap theorem for entire eternal solutions of the parabolic Allen-Cahn is also obtained. As an application of the regularity theorem, we give an affirmative answer to a question of Ilmanen that there is no cancellation in $\mathbf {BV}$ convergence in the mean convex setting.

math.AP↗
arXiv2020

Sharp pinching estimates for mean curvature flow in the sphere

We prove a suite of asymptotically sharp quadratic curvature pinching estimates for mean curvature flow in the sphere which generalize Simons' rigidity theorem for minimal hypersurfaces. We then obtain derivative estimates for the second fundamental form which we utilize, via a compactness argument, to obtain a convexity estimate. Together, the convexity and cylindrical estimates yield a partial classification of singularity models. We also obtain new rigidity results for ancient solutions.

math.DG↗
arXiv2020

Quadratically pinched hypersurfaces of the sphere via mean curvature flow with surgery

We study mean curvature flow in $\mathbb S_K^{n+1}$, the round sphere of sectional curvature $K>0$, under the quadratic curvature pinching condition $|A|^{2} < \frac{1}{n-2} H^{2} + 4 K$ when $n\ge 4$ and $|A|^{2} < \frac{3}{5}H^{2}+\frac{8}{3}K$ when $n=3$. This condition is related to a famous theorem of Simons, which states that the only minimal hypersurfaces satisfying $\vert A\vert^2<nK$ are the totally geodesic hyperspheres. It is related to but distinct from two-convexity. Notably, in contrast to two-convexity, it allows the mean curvature to change sign. We show that the pinching condition is preserved by mean curvature flow, and obtain a cylindrical estimate and corresponding pointwise derivative estimates for the curvature. As a result, we find that the flow becomes either uniformly convex or quantitatively cylindrical in regions of high curvature. This allows us to apply the surgery apparatus developed by Huisken and Sinestrari. We conclude that any smoothly, properly, isometrically immersed hypersurface $\mathcal{M}$ of $\mathbb S_K^{n+1}$ satisfying the pinching condition is diffeomorphic to $\mathbb S^n$ or the connected sum of a finite number of copies of $\mathbb S^1\times \mathbb S^{n-1}$. If $\mathcal M$ is embedded, then it bounds a 1-handlebody. The results are sharp when $n\ge 4$.

math.DG↗
arXiv2020

Convexity Estimates for High Codimension Mean Curvature Flow

We consider the evolution by mean curvature of smooth $n$-dimensional submanifolds in $\mathbb{R}^{n+k}$ which are compact and quadratically pinched. We will be primarily interested in flows of high codimension, the case $k\geq 2$. We prove that our submanifold is asymptotically convex, that is the first eigenvalue of the second fundamental form in the principal mean curvature direction blows up at a strictly slower rate than the mean curvature vector. We use this convexity estimate to show that at a singular time of the flow, there exists a rescaling that converges to a smooth codimension-one limiting flow which is convex and moves by translation.

math.DG↗
arXiv2020

Evolving Pinched Submanifolds of the Sphere by Mean Curvature Flow

In this paper, we prove convergence of the high codimension mean curvature flow in the sphere to either a round point or a totally geodesic sphere assuming a pinching condition between the norm squared of the second fundamental form and the norm squared of the mean curvature and the background curvature of the sphere. We show that this pinching is sharp for dimension $n\geq 4$ but is not sharp for dimension $n=2,3$. For dimension $n=2$ and codimension $2$, we consider an alternative pinching condition which includes the normal curvature of the normal bundle. Finally, we sharpen the Chern-do Carmo-Kobayashi curvature condition for surfaces in the four sphere - this curvature condition is sharp for minimal surfaces and we conjecture it to be sharp for curvature flows in the sphere.

math.DG↗
arXiv2018

Cylindrical Estimates for High Codimension Mean Curvature Flow

We study high codimension mean curvature flow of a submanifold $\mathcal{M}^n$ of dimension $n$ in Euclidean space $\mathbb{R}^{n+k}$ subject to the quadratic curvature condition $ |A|^{2}\leq c_n |H|^{2}, c _n = \min\{ \frac{4}{3n} , \frac{1}{n-2}\}$. This condition extends the notion of two-convexity for hypersurfaces to high codimension submanifolds. We analyse singularity formation in the mean curvature flow of high codimension by directly proving a pointwise gradient estimate. We then show that near a singularity the surface is quantitatively cylindrical.

math.DG↗
arXiv2018

Global Conformal Invariants of Submanifolds

The goal of the present paper is to investigate the algebraic structure of global conformal invariants of submanifolds. These are defined to be conformally invariant integrals of geometric scalars of the tangent and normal bundle. A famous example of a global conformal invariant is the Willmore energy of a surface. In codimension one we classify such invariants, showing that under a structural hypothesis (more precisely we assume the integrand depends separately on the intrinsic and extrinsic curvatures, and not on their derivatives) the integrand can only consist of an intrinsic scalar conformal invariant, an extrinsic scalar conformal invariant and the Chern-Gauss-Bonnet integrand. In particular, for codimension one surfaces, we show that the Willmore energy is the unique global conformal invariant, up to the addition of a topological term (the Gauss curvature, giving the Euler Characteristic by the Gauss Bonnet Theorem). A similar statement holds also for codimension two surfaces, once taking into account an additional topological term given by the Chern-Gauss-Bonnet integrand of the normal bundle. We also discuss existence and properties of natural higher dimensional (and codimensional) generalizations of the Willmore energy.

math.DG↗
arXiv2017

Pinched Ancient Solutions to the High Codimension Mean Curvature Flow

We study solutions of high codimension mean curvature flow defined for all negative times, usually referred to as ancient solutions. We show that any compact ancient solution whose second fundamental form satisfies a certain natural pinching condition must be a family of shrinking spheres. Andrews and Baker have shown that initial submanifolds satisfying this pinching condition, which generalises the notion of convexity, converge to round points under the flow. As an application, we use our result to simplify their proof.

math.DG↗
arXiv2017

The Chern-Gauss-Bonnet formula for singular non-compact four-dimensional manifolds

We generalise the classical Chern-Gauss-Bonnet formula to a class of 4-dimensional manifolds with finitely many conformally flat ends and singular points. This extends results of Chang-Qing-Yang in the smooth case. Under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the singularities, we obtain a new Chern-Gauss-Bonnet formula with error terms that can be expressed as isoperimetric deficits. This is the first such formula in a dimension higher than two which allows the underlying manifold to have isolated branch points or conical singularities.

math.DG↗
arXiv2017

The higher-dimensional Chern-Gauss-Bonnet formula for singular conformally flat manifolds

In a previous article, we generalised the classical four-dimensional Chern-Gauss-Bonnet formula to a class of manifolds with finitely many conformally flat ends and singular points, in particular obtaining the first such formula in a dimension higher than two which allows the underlying manifold to have isolated conical singularities. In the present article, we extend this result to all even dimensions $n\geq 4$ in the case of a class of conformally flat manifolds.

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