arXiv subjects

Jonghyeon Ahn

Publications and source records attributed to Jonghyeon Ahn.

6 recordsLinked to original sources

arXiv2026

A lower bound for relative symplectic cohomology barcode entropy

In this paper, we continue to study the barcode entropy of relative symplectic cohomology $SH_M(K)$ of a Liouville domain $K$ embedded in a symplectic manifold $M$. This barcode entropy measures the exponential growth rate of the number of not-too-short bars in the persistence module $SH_M(K)$. We prove that this Floer-theoretic invariant admits a nontrivial lower bound in terms of the topological entropy of the Reeb flow on $\partial K$ when the Reeb flow possesses a hyperbolic invariant set. More precisely, we show that the barcode entropy of $SH_M(K)$ is bounded below by the topological entropy of the Reeb flow restricted to a hyperbolic invariant set.

math.SG↗

arXiv2026

Symplectic variations of convex bodies and the mean width

In this work, we study convex bodies in $\RR^{2n}$ with the property that their mean width cannot be infinitesimally decreased by symplectomorphisms. The common theme of our results is that toric symmetry is a preferred feature of convex bodies with this property.

math.SG↗

arXiv2026

On the symplectic capacity and mean width of convex bodies

In this note we consider two topics involving the relationship between the symplectic capacity and the mean width of convex bodies in $\mathbb{R}^{2n}$. We first describe an alternative path from the symplectic Brunn-Minkowski inequality of Artstein-Avidan and Ostrover to another inequality, established by the same authors, that relates the capacity and mean width of convex bodies. This new path is less direct but it relates these inequalities to the quermassintegrals of convex bodies and to the local version of Viterbo's conjecture established by Abbondandolo and Benedetti for domains sufficiently close to the ball. We then consider the problem of identifying convex bodies whose mean width cannot be decreased by natural classes of symplectomorphisms. We state a conjectured characterization of convex bodies whose mean width is already minimal among all their symplectic images. To test this conjecture we identify a simple class of quadratic convex bodies whose mean width can not be decreased by linear symplectic maps near the identity. We then identify a subset of these examples that fail to satisfy the toric conditions of the conjecture, and show that one can find a nonlinear symplectomorphism that decreases their mean width.

math.SG↗

arXiv2026

Barcode entropy and relative symplectic cohomology

In this paper, we study the barcode entropy--the exponential growth rate of the number of not-too-short bars--of the persistence module associated with the relative symplectic cohomology $SH_M(K)$ of a Liouville domain $K$ embedded in a symplectic manifold $M$. Our main result establishes a quantitative link between this Floer-theoretic invariant and the dynamics of the Reeb flow on $\partial K$. More precisely, we show that the barcode entropy of the relative symplectic cohomology $SH_M(K)$ is bounded above by a constant multiple of the topological entropy of the Reeb flow on the boundary of the domain, where the constant depends on the embedding of $K$ into $M$.

math.SG↗

arXiv2025

Comparison of symplectic capacities

In this paper, we compare the symplectic (co)homology capacity with the spectral capacity in the relative case. This result establishes a chain of inequalities of relative symplectic capacities, which is an analogue of the non-relative case. This comparison gives us a criterion for the relative almost existence theorem in terms of heaviness. Also, we investigate a sufficient condition under which the symplectic (co)homology capacity and the first Gutt-Hutchings capacity are equal in both non-relative and relative cases. This condition is less restrictive than the dynamical convexity.

math.SG↗

arXiv2024

$S^1$-equivariant relative symplectic cohomology and relative symplectic capacities

In this paper, we construct an $S^1$-equivariant version of the relative symplectic cohomology developed by Varolgunes. As an application, we construct a relative version of Gutt-Hutchings capacities and a relative version of symplectic (co)homology capacity. We will see that these relative symplectic capacities can detect the diplaceability and the heaviness of a compact subset of a symplectic manifold. We compare the first relative Gutt-Hutchings capacity and the relative symplectic (co)homology capacity and prove that they are equal to each other under a convexity assumption.

math.SG↗