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Sung-Hong Min

Publications and source records attributed to Sung-Hong Min.

4 recordsLinked to original sources
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↗
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↗
arXiv2010

Embeddedness of proper minimal submanifolds in homogeneous spaces

We prove the three embeddedness results as follows. $({\rm i})$ Let $Γ_{2m+1}$ be a piecewise geodesic Jordan curve with $2m+1$ vertices in $\mathbb{R}^n$, where $m$ is an integer $\geq2$. Then the total curvature of $Γ_{2m+1}<2mπ$. In particular, the total curvature of $Γ_5<4π$ and thus any minimal surface $Σ\subset \mathbb{R}^n$ bounded by $Γ_5$ is embedded. Let $Γ_5$ be a piecewise geodesic Jordan curve with $5$ vertices in $\mathbb{H}^n$. Then any minimal surface $Σ\subset \mathbb{H}^n$ bounded by $Γ_5$ is embedded. If $Γ_5$ is in a geodesic ball of radius $\fracπ{4}$ in $\mathbb{S}^n_+$, then $Σ\subset \mathbb{S}^n_+$ is also embedded. As a consequence, $Γ_5$ is an unknot in $\mathbb{R}^3$, $\mathbb{H}^3$ and $\mathbb{S}^3_+$. $({\rm ii})$ Let $Σ$ be an $m$-dimensional proper minimal submanifold in $\mathbb{H}^n$ with the ideal boundary $\partial_{\infty} Σ= Γ$ in the infinite sphere $\mathbb{S}^{n-1}=\partial_\infty \mathbb{H}^n$. If the M{ö}bius volume of $Γ$ $\widetilde{\vol}(Γ) < 2\vol(\mathbb{S}^{m-1})$, then $Σ$ is embedded. If $\widetilde{\vol}(Γ) = 2\vol(\mathbb{S}^{m-1})$, then $Σ$ is embedded unless it is a cone. $({\rm iii})$ Let $Σ$ be a proper minimal surface in $\hr$. If $Σ$ is vertically regular at infinity and has two ends, then $Σ$ is embedded.

math.DG↗