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Jaigyoung Choe

Publications and source records attributed to Jaigyoung Choe.

16 recordsLinked to original sources
arXiv2025

Free boundary minimal surfaces in products of balls

In this paper we develop an extremal eigenvalue approach to the problem of construction of free boundary minimal surfaces in the product of Euclidean balls of chosen radii. The extremal problem involves a linear combination of normalized mixed Steklov-Neumann eigenvalues. The problem is motivated by the Schwarz P-surface which is a free boundary minimal surface in a cube. We show that the problem often does not have an absolute maximum in the product case even though it is bounded from above. By imposing a finite group of symmetries on both the surface and on the eigenfunctions we construct at least one free boundary minimal surface in a rectangular prism with arbitrary side lengths. We further show that unless the rectangular prism is a cube there are at least two such surfaces. We also prove that any immersed free boundary minimal surface of genus 0 with one boundary component on each face of a rectangular prism, and that is invariant under the reflections interchanging opposite faces of the prism, is necessarily embedded. Finally we show that for a genus 0 surface with 6 boundary components and suitable reflection symmetries there is a maximizing metric which can be realized by a free boundary minimal immersion into a product of Euclidean balls.

math.DG↗
arXiv2025

Free boundary minimal surfaces and the reflection principle

We show that a minimal surface meeting a sphere at a 90-degree angle can be reflected across the sphere. Using this reflection, we prove the uniqueness that every embedded free boundary minimal annulus in a ball is necessarily the critical catenoid.

math.DG↗
arXiv2022

Generating Axially Symmetric Minimal Hyper-surfaces in R^{1,3}

It is shown that, somewhat similar to the case of classical Baecklund transformations for surfaces of constant negative curvature, infinitely many axially symmetric minimal hypersurfaces in 4-dimensional Minkowski-space can be obtained, in a non-trivial way, from any given one by combining the scaling symmetries of the equations in light cone coordinates with a non-obvious symmetry (the analogue of Bianchis original transformation) - which can be shown to be involutive/correspond to a space-reflection.

math.DG↗
arXiv2021

The periodic Plateau problem and its application

Given a noncompact disconnected complete periodic curve $Γ$ with no self intersection in $\mathbb R^3$, it is proved that there exists a noncompact simply connected periodic minimal surface spanning $Γ$. As an application it is shown that for any tetrahedron $T$ with dihedral angles $\leq90^\circ$ there exist four embedded minimal annuli in $T$ which are perpendicular to $\partial T$ along their boundary. It is also proved that every Platonic solid of $\mathbb R^3$ contains five types of free boundary embedded minimal surfaces of genus zero.

math.DG↗
arXiv2020

The Minimality of Determinantal Varieties

The determinantal variety $Σ_{pq}$ is defined to be the set of all $p\times q$ real matrices with $p\geq q$ whose ranks are strictly smaller than $q$. It is proved that $Σ_{pq}$ is a minimal cone in $\mathbb R^{pq}$ and all its strata are regular minimal submanifolds.

math.DG↗
arXiv2017

Density of a minimal submanifold and total curvature of its boundary

Given a piecewise smooth submanifold $Γ^{n-1} \subset \R^m$ and $p \in \R^m$, we define the {\em vision angle} $Π_p(Γ)$ to be the $(n-1)$-dimensional volume of the radial projection of $Γ$ to the unit sphere centered at $p$. If $p$ is a point on a stationary $n$-rectifiable set $Σ\subset \R^m$ with boundary $Γ$, then we show the density of $Σ$ at $p$ is $\leq$ the density at its vertex $p$ of the cone over $Γ$. It follows that if $Π_p(Γ)$ is less than twice the volume of $S^{n-1}$, for all $p \in Γ$, then $Σ$ is an embedded submanifold. As a consequence, we prove that given two $n$-planes $R^n_1, R^n_2$ in $\R^m$ and two compact convex hypersurfaces $Γ_i$ of $R^n_i, i=1,2$, a nonflat minimal submanifold spanned by $Γ:=Γ_1\cupΓ_2$ is embedded.

math.DG↗
arXiv2016

Mean curvature in manifolds with Ricci curvature bounded from below

Let $M$ be a compact Riemannian manifold of nonnegative Ricci curvature and $Σ$ a compact embedded 2-sided minimal hypersurface in $M$. It is proved that there is a dichotomy: If $Σ$ does not separate $M$ then $Σ$ is totally geodesic and $M\setminusΣ$ is isometric to the Riemannian product $Σ\times(a,b)$, and if $Σ$ separates $M$ then the map $i_*:π_1(Σ)\rightarrow π_1(M)$ induced by inclusion is surjective. This surjectivity is also proved for a compact 2-sided hypersurface with mean curvature $H\geq(n-1)\sqrt{k}$ in a manifold of Ricci curvature $Ric_M\geq-(n-1)k,k>0$, and for a free boundary minimal hypersurface in a manifold of nonnegative Ricci curvature with nonempty strictly convex boundary. As an application it is shown that a compact $n$-dimensional manifold $N$ with the number of generators of $π_1(N)<n$ cannot be minimally embedded in the flat torus $T^{n+1}$.

math.DG↗
arXiv2015

On the area of minimal surfaces in a slab

Consider a non-planar orientable minimal surface S in a slab which is possibly with genus or with more than two boundary components. We show that there exists a catenoidal waist W in the slab whose flux has the same vertical component as S such that Area(S)>= Area(W), provided the intersections of S with horizontal planes have the same orientation.

math.DG↗
arXiv2014

Stable capillary hypersurfaces in a wedge

Let $Σ$ be a compact immersed stable capillary hypersurface in a wedge bounded by two hyperplanes in $\mathbb R^{n+1}$. Suppose that $Σ$ meets those two hyperplanes in constant contact angles and is disjoint from the edge of the wedge. It is proved that if $\partial Σ$ is embedded for $n=2$, or if $\partialΣ$ is convex for $n\geq3$, then $Σ$ is part of the sphere. And the same is true for $Σ$ in the half-space of $\mathbb R^{n+1}$ with connected boundary $\partialΣ$.

math.DG↗
arXiv2013

Classical mechanics of minimal tori in S^3

We formulate a class of minimal tori in S^3 in terms of classical mechanics, reveal a curious property of the Clifford torus, and note that the question of periodicity can be made more explicit in a simple way.

math.DG↗
arXiv2013

New minimal surfaces in S^3 desingularizing the Clifford tori

For each integer m>1 and l>0 we construct a pair of compact embedded minimal surfaces of genus 1+4m(m-1)l. These surfaces desingularize the m Clifford tori meeting each other along a great circle at the angle of π/m. They are invariant under a finite group of screw motions and have no reflection symmetry across a great sphere.

math.DG↗
arXiv2012

Noncommutative Minimal Surfaces

We define noncommutative minimal surfaces in the Weyl algebra, and give a method to construct them by generalizing the well-known Weierstrass-representation.

math.QA↗