Searcharxiv⌕ Search

arXiv subjects

Keomkyo Seo

Publications and source records attributed to Keomkyo Seo.

13 recordsLinked to original sources
arXiv2025

Sharp lower bounds for the first eigenvalue of Steklov-type eigenvalue problems on a compact surface

Let $Ω$ be a compact surface with smooth boundary and the geodesic curvature $k_g \ge {c > 0}$ along $\partial Ω$ for some constant $c \in \mathbb{R}$. We prove that, if the Gaussian curvature satisfies $K \ge -α$ for a constant $α\ge 0$, then the first eigenvalue $σ_1$ of the Steklov-type eigenvalue problem satisfies \[ σ_1 + \fracα{σ_1} \ge c. \] Moreover, equality holds if and only if $Ω$ is a Euclidean disk of radius $\frac{1}{c}$ and $α= 0$. Furthermore, we obtain a sharp lower bound for the first eigenvalue of the fourth-order Steklov-type eigenvalue problem on $Ω$.

math.DG↗
arXiv2025

Free boundary minimal annuli in $S^2_+\times S^1$

Let $M$ be a compact 3-dimensional Riemannian manifold with nonnegative Ricci curvature and a nonempty boundary $\partial M$. Fraser and Li \cite{Fraser&Li} established a compactness theorem for the space of compact, properly embedded minimal surfaces of fixed topological type in $M$ with a free boundary on $\partial M$, assuming that $\partial M$ is strictly convex with respect to the inward unit normal. In this paper, we show that the strict convexity condition on $\partial M$ cannot be relaxed.

math.DG↗
arXiv2021

Chen's conjecture on biharmonic submanifolds in Riemannian manifolds

We study biharmonic hypersurfaces and biharmonic submanifolds in a Riemannian manifold. One of interesting problems in this direction is Chen's conjecture which says that any biharmonic submanifold in a Euclidean space is minimal. From the invariant equation for biharmonic submanifolds, we derive a fundamental identity involving the mean curvature vector field, and using this, we prove Chen's conjecture on biharmonic submanifolds in a Euclidean space. More generally, it is proved that any biharmonic submanifold in a space form of nonpositively sectional curvature is minimal. Furthermore we provide affirmative partial answers to the generalized Chen's conjecture and Balmuş-Montaldo-Oniciuc conjecture.

math.DG↗
arXiv2021

Free boundary constant mean curvature surfaces in a strictly convex three-manifold

Let $C$ be a strictly convex domain in a $3$-dimensional Riemannian manifold with sectional curvature bounded above by a constant and let $Σ$ be a constant mean curvature surface with free boundary in $C$. We provide a pinching condition on the length of the traceless second fundamental form on $Σ$ which guarantees that the surface is homeomorphic to either a disk or an annulus. Furthermore, under the same pinching condition, we prove that if $C$ is a geodesic ball of $3$-dimensional space forms, then $Σ$ is either a spherical cap or a Delaunay surface.

math.DG↗
arXiv2020

Radial symmetry and partially overdetermined problems in a convex cone

We obtain the radial symmetry of the solution to a partially overdetermined boundary value problem in a convex cone in space forms by using the maximum principle for a suitable subharmonic function $P$ and integral identities. In dimension $2$, we prove Serrin-type results for partially overdetermined problems outside a convex cone. Furthermore, we obtain a Rellich identity for an eigenvalue problem with mixed boundary conditions in a cone.

math.DG↗
arXiv2012

Optimal isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space

Let $Σ$ be a $k$-dimensional complete proper minimal submanifold in the Poincaré ball model $B^n$ of hyperbolic geometry. If we consider $Σ$ as a subset of the unit ball $B^n$ in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold $Σ$ and the ideal boundary $\partial_\infty Σ$, say $\rvol(Σ)$ and $\rvol(\partial_\infty Σ)$, respectively. Using this concept, we prove an optimal linear isoperimetric inequality. We also prove that if $\rvol(\partial_\infty Σ) \geq \rvol(\mathbb{S}^{k-1})$, then $Σ$ satisfies the classical isoperimetric inequality. By proving the monotonicity theorem for such $Σ$, we further obtain a sharp lower bound for the Euclidean volume $\rvol(Σ)$, which is an extension of Fraser and Schoen's recent result \cite{FS} to hyperbolic space. Moreover we introduce the Möbius volume of $Σ$ in $B^n$ to prove an isoperimetric inequality via the Möbius volume for $Σ$.

math.DG↗
arXiv2011

Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature

We give an estimate of the first eigenvalue of the Laplace operator on a complete noncompact stable minimal hypersurface $M$ in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface $M$ has sufficiently small total scalar curvature then $M$ has only one end. We also obtain a vanishing theorem for $L^2$ harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover we provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional curvature to be stable.

math.DG↗
arXiv2010

Spacelike capillary surfaces in the Lorentz-Minkowski space

For a compact spacelike constant mean curvature surface with nonempty boundary in the three-dimensional Lorentz-Minkowski space, we introduce a rotation index of the lines of curvature at the boundary umbilic point, which was developed by Choe \cite{Choe}. Using the concept of the rotation index at the interior and boundary umbilic points and applying the Poincaré-Hopf index formula, we prove that a compact immersed spacelike disk type capillary surface with less than $4$ vertices in a domain of $\Bbb L^3$ bounded by (spacelike or timelike) totally umbilic surfaces is part of a (spacelike) plane or a hyperbolic plane. Moreover we prove that the only immersed spacelike disk type capillary surface inside de Sitter surface in $\Bbb L^3$ is part of (spacelike) plane or a hyperbolic plane.

math.DG↗
arXiv2010

$L^2$ harmonic 1-forms on minimal submanifolds in hyperbolic space

In this paper, we prove the nonexistence of $L^2$ harmonic 1-forms on a complete super stable minimal submanifold $M$ in hyperbolic space under the assumption that the first eigenvalue $λ_1 (M)$ for the Laplace operator on $M$ is bounded below by $(2n-1)(n-1)$. Moreover, we provide sufficient conditions for minimal submanifolds in hyperbolic space to be super stable.

math.DG↗
arXiv2010

Stable minimal hypersurfaces in the hyperbolic space

In this paper we give an upper bound of the first eigenvalue of the Laplace operator on a complete stable minimal hypersurface $M$ in the hyperbolic space which has finite $L^2$-norm of the second fundamental form on $M$. We provide some sufficient conditions for minimal hypersurface of the hyperbolic space to be stable. We also describe stability of catenoids and helicoids in the hyperbolic space. In particular, it is shown that there exists a family of stable higher-dimensional catenoids in the hyperbolic space.

math.DG↗
arXiv2010

Rigidity of minimal submanifolds in hyperbolic space

We prove that if an $n$-dimensional complete minimal submanifold $M$ in hyperbolic space has sufficiently small total scalar curvature then $M$ has only one end. We also prove that for such $M$ there exist no nontrivial $L^2$ harmonic 1-forms on $M$.

math.DG↗
arXiv2009

Regularity of soap film-like surfaces spanning graphs in a Riemannian manifold

Let $M$ be an $n$-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant $-κ^2$. Using the cone total curvature $TC(Γ)$ of a graph $Γ$ which was introduced by Gulliver and Yamada Math. Z. 2006, we prove that the density at any point of a soap film-like surface $Σ$ spanning a graph $Γ\subset M$ is less than or equal to $\frac{1}{2π}\{TC(Γ) - κ^{2}\area(p\mbox{$\times\hspace*{-0.178cm}\times$}Γ)\}$. From this density estimate we obtain the regularity theorems for soap film-like surfaces spanning graphs with small total curvature. In particular, when $n=3$, this density estimate implies that if \begin{eqnarray*} TC(Γ) < 3.649π+ κ^2 \inf_{p\in M} \area({p\mbox{$\times\hspace*{-0.178cm}\times$}Γ}), \end{eqnarray*} then the only possible singularities of a piecewise smooth $(\mathbf{M},0,δ)$-minimizing set $Σ$ is the $Y$-singularity cone. In a manifold with sectional curvature bounded above by $b^2$ and diameter bounded by $π/b$, we obtain similar results for any soap film-like surfaces spanning a graph with the corresponding bound on cone total curvature.

math.DG↗