arXiv subjects

Dongyeong Ko

Publications and source records attributed to Dongyeong Ko.

7 recordsLinked to original sources

arXiv2026

Free boundary minimal annuli in convex balls

We construct a $4$-parameter family of properly embedded genus-zero surfaces with at most two boundary components in a Riemannian $3$-ball. The construction is a free boundary analog of the canonical $5$-parameter family of surfaces on the $3$-sphere by Marques-Neves in the proof of Willmore conjecture. In the Euclidean unit ball, this family realizes the critical catenoid as a Simon-Smith min-max limit. As applications, we give an upper bound of Almgren-Pitts $4$-width $\omega_4(\mathbb{B}^3)$ by the area of the critical catenoid. Moreover, we prove the existence of at least three free boundary minimal annuli in any compact $3$-manifold with nonnegative Ricci curvature and strictly convex boundary.

math.DG↗

arXiv2026

Capillary minimal slicing and scalar curvature rigidity

We develop minimal slicing via capillary hypersurfaces to understand positive scalar curvature metric on manifolds with boundary. The method provides rigidity statements once the regularity of minimizers of capillary area functional holds. In particular, in dimension $4$, we prove following comparison and rigidity statement: given a compact Riemannian $4$-manifold $(M^4,g)$ with a mean convex boundary whose boundary is diffeomorphic to boundary of a connected convex domain in $\mathbb R^4$, if the scalar curvature is non-negative and the scaled mean curvature comparison holds along the boundary, then $M$ is isometric to the Euclidean domain.

math.DG↗

arXiv2025

Regularity of Cohomogeneity two equivariant isotopy minimization problems and minimal hypersurfaces with large first Betti number on spheres

We prove the regularity of cohomogeneity two equivariant isotopy minimization problems. Based on this, we develop cohomogeneity two equivariant min-max theory for minimal hypersurfaces proposed by Pitts and Rubinstein in 1988. As an application, for $g \ge 1$ and $4 \le n+1 \le 7$, we construct minimal hypersurfaces $Σ_{g}^{n}$ on round spheres $\mathbb{S}^{n+1}$ with $(SO(n-1) \times \mathbb{D}_{g+1})$-symmetry. For sufficiently large $g$, $Σ_{g}^{n}$ is a sequence of minimal hypersurfaces with arbitrarily large Betti numbers of topological type $\#^{2g} (S^{1} \times S^{n-1})$ or $\#^{2g+2} (S^{1} \times S^{n-1})$, which converges to a union of $\mathbb{S}^{n}$ and a Clifford hypersurface $\sqrt{\frac{1}{n}}\mathbb{S}^{1} \times \sqrt{\frac{n-1}{n}} \mathbb{S}^{n-1}$ or $\sqrt{\frac{2}{n}}\mathbb{S}^{2} \times \sqrt{\frac{n-2}{n}} \mathbb{S}^{n-2}$. In particular, for dimensions $5$ and $6$, $Σ_{g}^{n}$ has a topological type $\#^{2g} (S^{1} \times S^{n-1})$.

math.DG↗

arXiv2025

Scalar curvature comparison and rigidity of $3$-dimensional weakly convex domains

For a compact Riemannian $3$-manifold $(M^{3}, g)$ with mean convex boundary which is diffeomorphic to a weakly convex compact domain in $\mathbb{R}^{3}$, we prove that if scalar curvature is nonnegative and the scaled mean curvature comparison $H^{2}g \ge H_{0}^{2} g_{Eucl}$ holds, then $(M,g)$ is flat. Our result is a smooth analog of Gromov's dihedral rigidity conjecture and an effective version of extremity results on weakly convex balls in $\mathbb R^3$. More generally, we prove the comparison and rigidity theorem for several classes of manifold with corners. Our proof uses capillary minimal surfaces with prescribed contact angle together with the construction of foliation with nonnegative mean curvature and with prescribed contact angles.

math.DG↗

arXiv2023

Existence and Morse Index of two free boundary embedded geodesics on Riemannian 2-disks with convex boundary

We prove that a free boundary curve shortening flow on closed surfaces with a strictly convex boundary remains noncollapsed for a finite time in the sense of the reflected chord-arc profile introduced by Langford-Zhu. This shows that such flow converges to free boundary embedded geodesic in infinite time, or shrinks to a round half-point on the boundary. As a consequence, we prove the existence of two free boundary embedded geodesics on a Riemannian $2$-disk with a strictly convex boundary. Moreover, we prove that there exists a simple closed geodesic with Morse Index $1$ and $2$. This settles the free boundary analog of Grayson's theorem.

math.DG↗

arXiv2023

Min-max construction of two capillary embedded geodesics on Riemannian $2$-disks

In this paper, we prove the existence of two capillary embedded geodesics with a contact angle $θ\in (0,π/2)$ on Riemannian $2$-disks with strictly convex boundary, where the absence of a simple closed geodesic loop based on a point of boundary is given. In particular, our condition contains the cases of Riemannian $2$-disks with strictly convex boundary, nonnegative Gaussian curvature and total geodesic curvature lower bound $π$ of the boundary. Moreover, by providing examples, we prove that our total geodesic curvature condition is sharp to admit a capillary embedded geodesic with a contact angle $θ\in (0,π/2)$ under the nonnegative interior Gaussian curvature condition. We also prove the existence of Morse Index $1$ and $2$ capillary embedded geodesics for generic metric under the assumptions above.

math.DG↗

arXiv2023

Morse Index bound of simple closed geodesics on 2-spheres and strong Morse Inequalities

We give a Morse-theoretic characterization of simple closed geodesics on Riemannian $2$-spheres. On any Riemannian $2$-sphere endowed with a generic metric, we show there exists a simple closed geodesic with Morse index $1$, $2$ and $3$. In particular, for an orientable Riemannian surface we prove strong Morse inequalities for the length functional applied to the space of simple closed curves.

math.DG↗