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Eungbeom Yeon

Publications and source records attributed to Eungbeom Yeon.

6 recordsLinked to original sources
arXiv2025

On stationary real matrix Schubert varieties

In this paper, we study when a real matrix Schubert variety is stationary with respect to the first variation. We first show that a necessary condition for its open dense regular part to be a minimal submanifold is that the corresponding partial permutation is vexillary. Among vexillary partial permutations, we establish minimality by a geometric argument when the Rothe diagram is of Grassmannian type and has at most two connected components. We further obtain, as a corollary, the minimality of those varieties that decompose as products of this type. These varieties include all determinantal varieties as well as some new minimal cones.

math.DG↗
arXiv2025

Complete Minimal Surfaces in $\mathbb{R}^4$ with Three Embedded Planar Ends

In this paper, we study complete minimal surfaces in $\mathbb{R}^4$ with three embedded planar ends parallel to those of the union of the Lagrangian catenoid and the plane passing through its waist circle. We show that any complete, oriented, immersed minimal surface in $\mathbb{R}^4$ of finite total curvature with genus $1$ and three such ends must be $J$-holomorphic for some almost complex structure $J$. Under the additional assumptions of embeddedness and at least $8$ symmetries, we prove that the number of symmetries must be either $8$ or $12$, and in each case, the surface is uniquely determined up to rigid motions and scalings. Furthermore, we establish a nonexistence result for genus $g\geq2$ when the surface is embedded and has at least $4(g+1)$ symmetries. Our approach is based on a modification of the method of Costa and Hoffman-Meeks in the setting of $\mathbb{R}^4$, utilizing the generalized Weierstrass representation.

math.DG↗
arXiv2022

Frankel's property for free boundary minimal hypersurfaces in the Riemannian Schwarzschild manifolds

We study the behavior of minimal hypersurfaces in the Schwarzschild $n$-manifolds that intersect the horizon orthogonally along the boundary. We show that a free boundary minimal hypersurface and a totally geodesic hyperplane must intersect when the distance between them is achieved in a bounded region. We also discuss when the Schwarzschild metric is perturbed in a way that its scalar curvature is no longer positive.

math.DG↗
arXiv2021

A new approach to the Fraser-Li conjecture with the Weierstrass representation formula

In this paper, we provide a sufficient condition for a curve on a surface in $\mathbb{R}^3$ to be given by an orthogonal intersection with a sphere. This result makes it possible to express the boundary condition entirely in terms of the Weierstrass data without integration when dealing with free boundary minimal surfaces in a ball $\mathbb{B}^3$. Moreover, we show that the Gauss map of an embedded free boundary minimal annulus is one to one. By using this, the Fraser-Li conjecture can be translated into the problem of determining the Gauss map. On the other hand, we show that the Liouville type boundary value problem in an annulus gives some new insight into the structure of immersed minimal annuli orthogonal to spheres. It also suggests a new PDE theoretic approach to the Fraser-Li conjecture.

math.DG↗
arXiv2020

Characterizations of the plane and the catenoid as capillary surfaces

In this paper we prove that a capillary minimal surface outside the unit ball in $\mathbb {R}^3$ with one embedded end and finite total curvature must be either part of the plane or part of the catenoid. We also prove that a capillary minimal surface outside the unit ball with one end asymptotic to the end of the Enneper's surface and finite total curvature cannot exist if the flux vector vanishes on the first homology calss of the surface. Furthermore, we prove that a capillary minimal surface outside the convex domain bounded by several spheres with one embedded end and finite total curvature must be part of the plane.

math.DG↗