• Journal of the Korean Mathematical Society
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• v.60 no.5
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• pp.1109-1133
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• 2023

We prove that the two-step flag variety 𝓕ℓ(1, n; n+1) carries a non-displaceable and non-monotone Lagrangian Gelfand-Zeitlin fiber diffeomorphic to S³ × T^2n-4 and a continuum family of non-displaceable Lagrangian Gelfand-Zeitlin torus fibers when n > 2.

• The Pure and Applied Mathematics
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• v.19 no.3
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• pp.239-249
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• 2012

For each submanifold X in the sphere $S^n$; we show that the corresponding conormal bundle $N^*X$ is Lagrangian for the Stenzel form on $T^*S^n$. Furthermore, we correspond an austere submanifold X to a special Lagrangian submanifold $N^*X$ in $T^*S^n$. We also discuss austere submanifolds in $S^n$ from isoparametric geometry.

• Bulletin of the Korean Mathematical Society
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• v.44 no.3
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• pp.395-406
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• 2007

Involving the Ricci curvature and the squared mean curvature, we obtain a basic inequality for an integral submanifold of an S-space form. By polarization, we get a basic inequality for Ricci tensor also. Equality cases are also discussed. By giving a very simple proof we show that if an integral submanifold of maximum dimension of an S-space form satisfies the equality case, then it must be minimal. These results are applied to get corresponding results for C-totally real submanifolds of a Sasakian space form and for totally real submanifolds of a complex space form.

• Journal of the Korean Mathematical Society
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• v.34 no.4
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• pp.1065-1087
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• 1997

We prove that there exists a canonical level-preserving isomorphism between the stable Morse homology (or the Morse homology of generating functions) and the Floer homology on the cotangent bundle $T^*M$ for any closed submanifold $N \subset M$ for any compact manifold M.

• Journal of the Korean Mathematical Society
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• v.53 no.4
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• pp.795-834
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• 2016

The main purpose of this paper is to propose a scheme of a proof of the nonsimpleness of the group $Homeo^{\Omega}$ ($D^2$, ${\partial}D^2$) of area preserving homeomorphisms of the 2-disc $D^2$. We first establish the existence of Alexander isotopy in the category of Hamiltonian homeomorphisms. This reduces the question of extendability of the well-known Calabi homomorphism Cal : $Diff^{\Omega}$ ($D^1$, ${\partial}D^2$)${\rightarrow}{\mathbb{R}}$ to a homomorphism ${\bar{Cal}}$ : Hameo($D^2$, ${\partial}D^2$)${\rightarrow}{\mathbb{R}}$ to that of the vanishing of the basic phase function $f_{\underline{F}}$, a Floer theoretic graph selector constructed in [9], that is associated to the graph of the topological Hamiltonian loop and its normalized Hamiltonian ${\underline{F}}$ on $S^2$ that is obtained via the natural embedding $D^2{\hookrightarrow}S^2$. Here Hameo($D^2$, ${\partial}D^2$) is the group of Hamiltonian homeomorphisms introduced by $M{\ddot{u}}ller$ and the author [18]. We then provide an evidence of this vanishing conjecture by proving the conjecture for the special class of weakly graphical topological Hamiltonian loops on $D^2$ via a study of the associated Hamiton-Jacobi equation.