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Joontae Kim

Publications and source records attributed to Joontae Kim.

14 recordsLinked to original sources
arXiv2026

Floer-theoretic entropy of exact symplectomorphisms

We introduce the notion of a Penner-type class of an exact symplectomorphism on a Liouville domain with an $A_k$-configuration of Lagrangian spheres for $k\geqslant 2$, and prove that such a class has positive Floer-theoretic entropy. As a corollary, we construct infinitely many smoothly trivial symplectic isotopy classes of exact symplectomorphisms with positive Floer-theoretic entropy on any $4n$-dimensional Liouville domain that admits an $A_2$-configuration of Lagrangian spheres. We also prove that the Floer-theoretic entropy of an exact symplectomorphism on a Liouville domain provides a lower bound for its topological entropy.

math.SG↗
arXiv2025

Lagrangian split tori in $S^2 \times S^2$ and billiards

In this paper, we classify up to Hamiltonian isotopy Lagrangian tori that split as a product of circles in $S^2 \times S^2$, when the latter is equipped with a non-monotone split symplectic form. We show that this classification is equivalent to a problem of mathematical billiards in rectangles. We give many applications, among others: (1) answering a question on Lagrangian packing numbers raised by Polterovich--Shelukhin, (2) studying the topology of the space of Lagrangian tori, and (3) determining which split tori are images under symplectic ball embeddings of Chekanov or product tori in $\mathbb{R}^4$.

math.SG↗
arXiv2024

Volume growth via real Lagrangians in Milnor fibers of Brieskorn polynomials

In this paper we study the volume growth in the component of fibered twists in Milnor fibers of Brieskorn polynomials. We obtain a uniform lower bound of the volume growth for a class of Brieskorn polynomials using a Smith inequality for involutions in wrapped Floer homology. To this end, we investigate a family of real Lagrangians in those Milnor fibers whose topology can be systematically described in terms of the join construction.

math.SG↗
arXiv2023

On the topology of Lagrangian fillings of the standard Legendrian sphere

In this paper we study the uniqueness of Lagrangian fillings of the standard Legendrian sphere $\mathcal{L}_0$ in the standard contact sphere $(S^{2n-1}, ξ_{\text st})$. We show that every exact Maslov zero Lagrangian filling $L$ of $\mathcal{L}_0$ in a Liouville filling of $(S^{2n-1}, ξ_{\text st})$ is a homology ball. If we restrict ourselves to real Lagrangian fillings, then $L$ is diffeomorphic to the $n$-ball for $n \geq 6$.

math.SG↗
arXiv2021

The Chekanov torus in $S^2\times S^2$ is not real

We prove that the count of Maslov index 2 $J$-holomorphic discs passing through a generic point of a real Lagrangian submanifold in a closed spherically monotone symplectic manifold must be even. As a corollary, we exhibit a genuine real symplectic phenomenon in terms of involutions, namely that the Chekanov torus $\mathbb{T}_{\text{Chek}}$ in $S^2\times S^2$, which is a monotone Lagrangian torus not Hamiltonian isotopic to the Clifford torus $\mathbb{T}_{\text{Clif}}$, can be seen as the fixed point set of a smooth involution, but not of an antisymplectic involution.

math.SG↗
arXiv2021

Equivariant wrapped Floer homology and symmetric periodic Reeb orbits

The aim of this article is to apply a Floer theory to study symmetric periodic Reeb orbits. We define positive equivariant wrapped Floer homology using a (anti-)symplectic involution on a Liouville domain and investigate its algebraic properties. By a careful analysis of index iterations, we obtain a non-trivial lower bound on the minimal number of geometrically distinct symmetric periodic Reeb orbits on a certain class of real contact manifolds. This includes non-degenerate real dynamically convex starshaped hypersurfaces in $\mathbb{R}^{2n}$ which are invariant under complex conjugation. As a result, we give a partial answer to the Seifert conjecture on brake orbits in the contact setting.

math.SG↗
arXiv2020

Remarks on the systoles of symmetric convex hypersurfaces and symplectic capacities

In this note we study the systoles of convex hypersurfaces in $\mathbb{R}^{2n}$ invariant under an anti-symplectic involution. We investigate a uniform upper bound of the ratio between the systole and the symmetric systole of the hypersurfaces using symplectic capacities from Floer theory. We discuss various concrete examples in which the ratio can be understood explicitly.

math.SG↗
arXiv2020

Unknottedness of real Lagrangian tori in $S^2\times S^2$

We prove the Hamiltonian unknottedness of real Lagrangian tori in the monotone $S^2\times S^2$, namely any real Lagrangian torus in $S^2\times S^2$ is Hamiltonian isotopic to the Clifford torus $\mathbb{T}_{\text{Clif}}$. The proof is based on a neck-stretching argument, Gromov's foliation theorem, and the Cieliebak-Schwingenheuer criterion.

math.SG↗
arXiv2020

Bifurcations of symmetric periodic orbits via Floer homology

We give criteria for the existence of bifurcations of symmetric periodic orbits in reversible Hamiltonian systems in terms of local equivariant Lagrangian Rabinowitz Floer homology. As an example, we consider the family of the direct circular orbits in the rotating Kepler problem and observe bifurcations of torus-type orbits. Our setup is motivated by numerical work of Hénon on Hill's lunar problem.

math.DS↗
arXiv2020

Uniqueness of real Lagrangians up to cobordism

We prove that a real Lagrangian submanifold in a closed symplectic manifold is unique up to cobordism. We then discuss the classification of real Lagrangians in $\mathbb{C} P^2$ and $S^2\times S^2$. In particular, we show that a real Lagrangian in $\mathbb{C} P^2$ is unique up to Hamiltonian isotopy and that a real Lagrangian in $S^2\times S^2$ is either Hamiltonian isotopic to the antidialgonal sphere or Lagrangian isotopic to the Clifford torus.

math.SG↗
arXiv2019

On the topology of real Lagrangians in toric symplectic manifolds

We explore the topology of real Lagrangian submanifolds in a toric symplectic manifold which come from involutive symmetries on its moment polytope. We establish a real analog of the Delzant construction for those real Lagrangians, which says that their diffeomorphism type is determined by combinatorial data. As an application, we realize all possible diffeomorphism types of connected real Lagrangians in toric symplectic del Pezzo surfaces.

math.SG↗
arXiv2017

Volume growth in the component of fibered twists

For a Liouville domain $W$ whose boundary admits a periodic Reeb flow, we can consider the connected component $[τ] \in π_0(\text{Symp}^c(\widehat W))$ of fibered twists. In this paper, we investigate an entropy-type invariant, called the slow volume growth, of the component $[τ]$ and give a uniform lower bound of the growth using wrapped Floer homology. We also show that $[τ]$ has infinite order in $π_0(\text{Symp}^c(\widehat W))$ if there is an admissible Lagrangian $L$ in $W$ whose wrapped Floer homology is infinite dimensional. We apply our results to fibered twists coming from the Milnor fibers of $A_k$-type singularities and complements of a symplectic hypersurface in a real symplectic manifold. They admit so-called real Lagrangians, and we can explicitly compute wrapped Floer homology groups using a version of Morse-Bott spectral sequences.

math.SG↗