Searcharxiv⌕ Search

arXiv subjects

Xuan Hien Nguyen

Publications and source records attributed to Xuan Hien Nguyen.

At least 19 recordsLinked to original sources
arXiv2026

Failure of a Brunn-Minkowski-type inequality for the Gaussian torsional rigidity

Let $u$ be the torsion function for the Ornstein-Uhlenbeck operator on a bounded domain $Ω\subset \mathbb{R}^n$, i.e., the solution of $Δu - x \cdot \nabla u = -1$ in $Ω$ with $u = 0$ on $\partialΩ$. Let $T_γ(Ω) = \int_Ωu \, dγ$ be the Gaussian torsional rigidity. We prove that the Brunn-Minkowski-type inequality $T_γ((1-t)Ω_0 + tΩ_1)^α \le (1-t) T_γ(Ω_0)^α + t T_γ(Ω_1)^α$ fails for every exponent $α> 0$ and in every dimension $n \ge 2$, for a pair of convex bodies centrally symmetric with respect to the origin, which may be taken smooth with positive curvature. This answers Conjecture 1.4 for any $n \geq 2$, and Question~(Q), of Marín Sola and Salerno in the negative. The mechanism is a first-order lower bound for $T_γ$ at a ball $Ω_0$ under Minkowski perturbations. When the perturbing body $Ω_1$ has small torsion and large mean width, $T_γ((1-t)Ω_0 + tΩ_1)$ increases to first order. Since the Minkowski combination has larger torsion than both endpoints, no exponent can repair the inequality. For $n=1$, convexity with the optimal exponent $1/3$ holds on symmetric intervals by results of the same authors, \cite{MSS26}. We prove that the logarithm of the torsion is neither convex nor concave along Minkowski combinations of symmetric intervals, that no non-zero exponent yields concavity, and that convexity fails for every positive exponent when one set is a union of two intervals or when the sets are reflected off-center intervals.

math.AP↗
arXiv2026

Log-Concavity and Level-Set Horoconvexity of the First Eigenfunction on Horoconvex Domains in the Hyperbolic Plane

Let $Ω\subset\mathbb H^2$ be a bounded smooth horoconvex domain and let $ψ_1>0$ be its first Dirichlet eigenfunction. We prove that \[ \operatorname{Hess}_{\mathbb H^2}(-\logψ_1)>0 \] throughout $Ω$, with no restriction on the diameter or the first eigenvalue. The proof is by contradiction. A degenerate Hessian would yield a shifted translation Killing derivative with a singular interior zero. Then the boundary-zero theorem of Grossi and Provenzano shows that the shifted Killing derivative has exactly two zeros on the boundary. A nodal-domain argument on the surface rules this out. As an application we prove that every superlevel set of $ψ_1$ is horoconvex: every level curve has geodesic curvature at least $1$. The Hessian bound makes the shifted construction available for Killing fields with nonvanishing rotation part, and yields the pointwise inequality $|(\operatorname{Hess} u)^{-1}J\nabla u|\le1$ for $u=-\logψ_1$, where $J$ is rotation by $π/2$; a boundary-zero count for translation fields with arbitrary axis completes the argument.

math.DG↗
arXiv2026

A Large-Diameter Fundamental-Gap Lower Bound for Horoconvex Domains

We prove a large-diameter fundamental-gap lower bound for compact horoconvex domains in real hyperbolic space of curvature \(-1\). The geometric part reduces large horoconvex domains to a fixed-width radial-height problem in all dimensions. The analytic part proves the needed radial-height theorem by comparing the low-energy Dirichlet form with a limiting angular operator on the sphere, while the radial complement is separated by a one-dimensional branch gap and endpoint Green estimates. The result gives the polynomial \(D^{-3}\) scale matching the Nguyen--Stancu--Wei large-diameter upper bound.

math.DG↗
arXiv2024

Log-Concavity and Fundamental Gaps on Surfaces of Positive Curvature

We study the log-concavity of the first Dirichlet eigenfunction of the Laplacian for convex domains. For positively curved surfaces satisfying a condition involving the curvature and its second derivatives, we show that the first eigenfunction is strongly log-concave. Previously, for general convex domains, the log-concavity of the first eigenfunctions were only known when lying in $\mathbb{R}^n$ and $\mathbb{S}^n$. Using this estimate, we establish lower bounds on the fundamental gap of such regions. Furthermore, we study the behavior of these estimates under Ricci flow and other deformations of the metric.

math.DG↗
arXiv2022

Negative curvature constricts the fundamental gap of convex domains

We consider the Laplace-Beltrami operator with Dirichlet boundary conditions on convex domains in a Riemannian manifold $(M^n,g)$, and prove that the product of the fundamental gap with the square of the diameter can be arbitrarily small whenever $M^n$ has even a single tangent plane of negative sectional curvature. In particular, the fundamental gap conjecture strongly fails for small deformations of Euclidean space which introduce any negative curvature. We also show that when the curvature is negatively pinched, it is possible to construct such domains of any diameter up to the diameter of the manifold. The proof is adapted from the argument of Bourni et. al. (Annales Henri Poincaré 2022), which established the analogous result for convex domains in hyperbolic space, but requires several new ingredients.

math.DG↗
arXiv2021

The fundamental gap of horoconvex domains in $\mathbb H^n$

We show that, for horoconvex domains in the hyperbolic space, the product of their fundamental gap with the square of their diameter has no positive lower bound. The result follows from the study of the fundamental gap of geodesic balls as the radius goes to infinity. In the process, we improve the lower bound for the first eigenvalue of balls in hyperbolic space.

math.DG↗
arXiv2020

Ancient solutions for flow by powers of the curvature in $\mathbb R^2$

We construct a new compact convex embedded ancient solution of the $κ^α$ flow in $\mathbb R^2$, $α\in(\frac12,1)$ that lies between two parallel lines. Using this solution we classify all convex ancient solutions of the $κ^α$ flow in $\mathbb R^2$, for $α\in(\frac23,1)$. Moreover, we show that any non-compact convex embedded ancient solution of the $κ^α$ flow in $\mathbb R^2$, $α\in(\frac12,1)$ must be a translating solution.

math.DG↗
arXiv2019

Explicit fundamental gap estimates for some convex domains in $\mathbb H^2$

Motivated by an example of Shih, we compute the fundamental gap of a family of convex domains in the hyperbolic plane $\mathbb H^2$, showing that for some of them $λ_2 - λ_1 < \frac{3π^2}{D^2}$, where $D$ is the diameter of the domain and $λ_1$, $λ_2$ are the first and second Dirichlet eigenvalues of the Laplace operator on the domain. The result contrasts with what is known in $\mathbb R^n $ or $\mathbb S^n$, where $λ_2 - λ_1 \geq \frac{3 π^2}{D^2}$ for convex domains. We also show that the fundamental gap of the example in Shih's article is still greater than $\tfrac 32 \frac{π^2}{D^2}$, even though the first eigenfunction of the Laplace operator is not log-concave.

math.DG↗
arXiv2017

A survey of closed self-shrinkers with symmetry

In this paper, we survey known results on closed self-shrinkers for mean curvature flow and discuss techniques used in recent constructions of closed self-shrinkers with classical rotational symmetry. We also propose new existence and uniqueness problems for closed self-shrinkers with bi-rotational symmetry and provide numerical evidence for the existence of new examples.

math.DG↗
arXiv2017

Shrinking doughnuts via variational methods

We use variational methods and a modified curvature flow to give an alternative proof of the existence of a self-shrinking torus under mean curvature flow. As a consequence of the proof, we establish an upper bound for the weighted energy of our shrinking doughnuts.

math.DG↗
arXiv2016

Mean curvature flow of an entire graph evolving away from the heat flow

We present two initial graphs over the entire $\mathbb{R}^n$, $n \geq 2$ for which the mean curvature flow behaves differently from the heat flow. In the first example, the two flows stabilize at different heights. With our second example, the mean curvature flow oscillates indefinitely while the heat flow stabilizes. These results highlight the difference between dimensions $n \geq 2$ and dimension $n=1$, where Nara-Taniguchi proved that entire graphs in $C^{2,α}(\mathbb{R})$ evolving under curve shortening flow converge to solutions to the heat equation with the same initial data.

math.DG↗
arXiv2015

Finite topology self-translating surfaces for the mean curvature flow in $\mathbb R^3$

Finite topology self translating surfaces to mean curvature flow of surfaces constitute a key element for the analysis of Type II singularities from a compact surface, since they arise in a limit after suitable blow-up scalings around the singularity. We find in $\mathbb R^3$ a surface $M$ orientable, embedded and complete with finite topology (and large genus) with three ends asymptotically paraboloidal, such that the moving surface $Σ(t) = M + te_z$ evolves by mean curvature flow. This amounts to the equation $H_M = ν\cdot e_z$ where $H_M$ denotes mean curvature, $ν$ is a choice of unit normal to $M$, and $e_z$ is a unit vector along the $z$-axis. The surface $M$ is in correspondence with the classical 3-end Costa-Hoffmann-Meeks minimal surface with large genus, which has two asymptotically catenoidal ends and one planar end, and a long array of small tunnels in the intersection region resembling a periodic Scherk surface. This example is the first non-trivial one of its kind, and it suggests a strong connection between this problem and the theory of embedded, complete minimal surfaces with finite total curvature.

math.AP↗
arXiv2014

Doubly periodic self-translating surfaces for the mean curvature flow

We construct new examples of self-translating surfaces for the mean curvature flow from a periodic configuration with finitely many grim reaper cylinders in each period. Because this work is an extension of the author's article on the desingularization of a finite family of grim reaper cylinders, we simply discuss the ideas of the construction and here prove only that the periodic configuration has the necessary flexibility. These examples show that self-translating surfaces do not necessarily have quadratic volume growth rate in contrast to self-shrinking surfaces.

math.DG↗
arXiv2014

Construction of Complete Embedded Self-Similar Surfaces under Mean Curvature Flow. Part III

We present new examples of complete embedded self-similar surfaces under mean curvature by gluing a sphere and a plane. These surfaces have finite genus and are the first examples of self-shrinkers in $\mathbb R^3$ that are not rotationally symmetric. The strategy for the construction is to start with a family of initial surfaces by desingularizing the intersection of a sphere and a plane, then solve a perturbation problem to obtain a one parameter family of self-similar surfaces. Although we start with surfaces asymptotic to a plane at infinity, the constructed self-similar surfaces are asymptotic to cones at infinity.

math.DG↗
arXiv2007

Construction of Complete Embedded Self-Similar Surfaces under Mean Curvature Flow. Part II

We study the Dirichlet problem associated to the equation for self-similar surfaces for graphs over the Euclidean plane with a disk removed. We show the existence of a solution provided the boundary conditions on the boundary circle are small enough and satisfy some symmetries. This is the second step towards the construction of new examples of complete embedded self similar surfaces under mean curvature flow.

math.DG↗