arXiv subjects

Yong-Geun Oh

Publications and source records attributed to Yong-Geun Oh.

At least 19 recordsLinked to original sources

arXiv2026

Stratifications associated to generic closed two-forms and stratified $L_\infty$ spaces

Jae-Suk Park and the second-named author introduce the deformation problem of coisotropic submanifolds of a symplectic manifold as the study of Mauer-Cartan moduli problem of an $L_\infty$ algebra attached to the foliation de-Rham complex associated to the null foliation of the corresponding presymplectic structure. The main purpose of the present paper is to extend this study of $L_\infty$ structures to the case of generic closed two-forms on arbitrary smooth manifolds as a stratified $L_\infty$ space. We first prove that there exists a residual subset of closed 2-forms, which we denote by $Z^2_{reg}(M) \subset Z^2(M)$, such that any element $ω$ therefrom admits a Whitney stratification each of whose strata is a presymplectic manifold. We then associate an $L_\infty$ space to each stratum (and to its tubular neighborhood) and glue the collection of $L_\infty$ spaces to a global stratified $L_\infty$ space by the coordinate atlas consisting of $L_\infty$ morphisms, which is a collection of $L_\infty$ morphisms, not necessarily of quasi-isomorphisms.

math.SG↗

arXiv2026

Contact triad connection of contact manifolds in almost contact moving frame

The notion of \emph{contact triad connection} on contact triads $(Q,λ,J_ξ)$ was introduced by Wang and the present author in early 2010's from scratch as the contact analog to the canonical connection of an almost Kähler manifolds. The connection facilitates the study of analysis of the contact instanton equation which ranges from local elliptic priori estimates, for both the interior [OW2,OW3] and the boundary [OY], to the generic perturbation theory of the asymptotic operators [KO], which also encompasses the case of pseudoholomorphic curves on noncompact symplectic manifolds with cylindrical ends [OK]. The main purpose of the present paper is to give a simpler and more canonical construction of the contact triad connection by first giving its characterization in terms of the associated \emph{almost contact structure} and then providing its construction using the almost contact moving frame in the framework of contact geometry.

math.DG↗

arXiv2026

Leafwise de Rham cohomology of generic Reeb foliations

In this paper, we prove that there exists a residual subset of contact forms $λ$ (if any) on any compact connected orientable manifold $M$ for which the foliation de Rham cohomology of the associated Reeb foliation has $H^0(F_λ) \cong \mathbb R$. We also prove the same triviality for a generic choice of contact forms with fixed contact structure $ξ$. This vanishing result of $H^0(F_λ)$ is also equivalent to the statement that the Lie algebra of the group of strict contactomorphisms is isomorphic to the span of Reeb vector fields, and so isomorphic to the 1 dimensional abelian Lie algebra $\mathbb R$. On the other hand, we derive the rank of $H^1( F_λ)$ is infinite whenever $λ$ admits a closed Reeb oribt, i.e., whenever Weinstein's conjecture holds. In particular we prove that $H^1(F_λ)$ is infinite dimensional for all contact form $λ$ in dimension 3, thanks to Taubes' proof of 3-dimensional Weinstein's conjecture.

math.SG↗

arXiv2026

Non-unital monoidal category of contact manifolds and Legendrian correspondence

There are two purposes of the present paper which are interrelated. The first goal is to construct the structure of a non-unital monoidal category $\mathfrak{Cont}$ of contact manifolds, not necessarily coorientable, by developing the contact topology \emph{without contact forms}. The non-unital monoidal product is the functorial contact product $\star$, called star product, introduced in \cite{oh:shelukhin-conjecture}. We prove that the product $\star$ is associative and there exist a collection $α= \{α_{X,Y,Z}\}$ of the \emph{associator} isomorphisms $α_{X,Y,Z}: X \star (Y\star Z) \cong (X \star Y) \star Z$ for $X, \, Y, \, Z \in \mathfrak{Cont}$, that satisfy the pentagon axiom, i.e., that the triples $(\mathfrak{Cont}, \star, α)$ form a nonunital monoidal category. The second goal is to develop the calculus of Legendrian correspondences, which are by definition embedded Legendrian submanifolds of the contact product $Q \star Q'$. Legendrian correspondences will play the role of 1-morphisms in the $2$-categorical structure to be equipped with $\mathfrak{Cont}$ whose two morphisms are contact instanton cohomologies $HI(R_{ab},R'_{ab})$ associated to a pair of Legendrian correspondences $R_{ab}, \, R'_{ab} \in \mathfrak{Leg}(Q_a,Q_b)$. With this future application in mind, we define the composition of Legendrian correspondences and prove that the composition of a generic pair is again embedded and hence canonically becomes a Legendrian correspondence.

math.SG↗

arXiv2026

Contact action functional, calculus of variation and canonical generating function of Legendrian submanifolds

In the present paper, we formulate a contact analogue on the one-jet bundle $J^1B$ of Weinstein's observation which reads the classical action functional on the cotangent bundle is a generating function of any Hamiltonian isotope of the zero section. We do this by identifying the correct action functional which is defined on the space of Hamiltonian-translated (piecewise smooth) horizontal curves of the contact distribution, which we call the Carnot path space. Then we give a canonical construction of the Legendrian generating function which is the Legendrian counterpart of Laudenbach-Sikorav's canonical construction of the generating function of Hamiltonian isotope of the zero section on the cotangent bundle which utilizes a finite dimensional approximation of the action functional. Motivated by this construction, we develop a Floer theoretic construction of spectral invariants for the Legendrian submanifolds in the sequel [OY] which is the contact analog to the construction given in [Oh97, Oh99] for the Lagrangian submanifolds in the cotangent bundle.

math.SG↗

arXiv2026

Metriplectic dynamical systems on contact manifolds

Flows on symplectic, Poisson, contact, and metriplectic manifolds are reviewed in order to describe our main result, which is to associate a natural metriplectic dynamical system on the general one-jet bundle $J^1N=T^*N\times \mathbb{R}$, which is at once a (trivial) Poisson manifold and a contact manifold. Unlike the standard contact Hamiltonian system, our metriplectic system is thermodynamically consistent in that $$\dot{H} = 0 \quad\mathrm{and}\quad \dot{S} \geq 0$$ under the flow. Here $H$ is the Hamiltonian, while $S$ is the entropy function which is nothing but the $\mathbb{R}$ coordinate function of $J^1N$. As an example we derive the Duffing equation (autonomous and nonautonomous versions) either as a contact Hamiltonian system or as a metriplectic system. We show that for both systems the Duffing equation is a subsystem of three dimensional systems that contain a thermodynamic component, a form that facilitates asymptotic stability analysis of the relevant equilibrium state.

math.SG↗

arXiv2025

Presymplectic geometry and Liouville sectors with corners and its monoidality

We provide a presymplectic characterization of Liouville sectors introduced by Ganatra-Pardon-Shende in terms of the characteristic foliation of the boundary, which we call Liouville $σ$-sectors. We extend this definition to the case with corners using the presymplectic geometry of null foliations of the coisotropic intersections of transverse coisotropic collection of hypersurfaces which appear in the definition of Liouville sectors with corners. We show that the set of Liouville $σ$-sectors with corners canonically forms a monoid which provides a natural framework of considering the Künneth-type functors in the wrapped Fukaya category. We identify its automorphism group which enables one to give a natural definition of bundles of Liouville sectors. As a byproduct, we affirmatively answer to a question raised in Question 2.6 in [GPS20], which asks about the optimality of their definition of Liouville sectors [GPS20].

math.SG↗

arXiv2025

Contact instantons and Proofs of Weinstein's conjecture and Arnold's chord conjecture

The present paper is a continuation of the study of the interplay between the contact Hamiltonian dynamics and the moduli theory of (perturbed) contact instantons and its applications initiated in [Oh21b, Oh22a]. In this paper we prove Weinstein's conjecture and Arnold's chord conjecture in their full generalities. The two key ingredients lying in the background of our proof of Arnold's chord conjecture are the existence of the fundamental class of the Legendrian contact instanton cohomology modulo bubbling-off, and the evaluation transversality of the moduli space of contact instantons against the level set of conformal exponent function. Our proof of Weinstein's conjecture also utilizes the existence scheme of translated points of a contactomorphism developed in [Oh22a], especially associated to a contact Hamiltonian loop, via the geometric construction of the Legendrianization of contactomorphisms of $(Q,λ)$ in the contact product $M_Q = Q \times Q \times \mathbb R$ and its usage of the $\mathbb Z_2$ anti-contact involutive symmetry.

math.SG↗

arXiv2025

Injectivity radius lower bound of convex sum of tame Riemannian metrics and applications to symplectic topology

Motivated by the aspect of large-scale symplectic topology, we prove that for any pair $g_0, \, g_1$ of complete Riemannian metrics of bounded curvature and \emph{of injectivity radius bounded away from zero}, the convex sum $g_s: = (1-s ) g_0 + s g_1$ also has bounded curvature depending only on the curvature bounds $\|R_{g_i}\|_{C^0}$ of $g_0$ or $g_1$, and that the injectivity radii of $g_s$ have uniform lower bound depending only on the derivative bounds $\|R_{g_i}\|_{C^1} = \|R_{g_i}\|_{C^0} + \|DR_{g_i}\|_{C^0}$. A main technical ingredient to establish the injectivity radius lower bound is an application of the quantitative inverse function theorem. Using these estimates, we prove that each quasi-isometry class of tame metrics is convex and so contractible in strong $C^r$ topology for all finite regularity class of $3 \leq r < \infty$. Using this Riemannian geometry result, we prove that the set of $C^3$-tame almost complex structures inside the same quasi-isometry class associated to the symplectic form $ω$ is contractible.

math.DG↗

arXiv2025

Hypergraph Neural Sheaf Diffusion: A Symmetric Simplicial Set Framework for Higher-Order Learning

The absence of intrinsic adjacency relations and orientation systems in hypergraphs creates fundamental challenges for constructing sheaf Laplacians of arbitrary degrees. We resolve these limitations through symmetric simplicial sets derived directly from hypergraphs, called symmetric simplicial lifting, which encode all possible oriented subrelations within each hyperedge as ordered tuples. This construction canonically defines adjacency via facet maps while inherently preserving hyperedge provenance. We establish that the normalized degree zero sheaf Laplacian on our symmetric simplicial lifting reduces exactly to the traditional graph normalized sheaf Laplacian when restricted to graphs, validating its mathematical consistency with prior graph-based sheaf theory. Furthermore, the induced structure preserves all structural information from the original hypergraph, ensuring that every multi-way relational detail is faithfully retained. Leveraging this framework, we introduce Hypergraph Neural Sheaf Diffusion (HNSD), the first principled extension of neural sheaf diffusion to hypergraphs. HNSD operates via normalized degree zero sheaf Laplacian over symmetric simplicial lifting, resolving orientation ambiguity and adjacency sparsity inherent to hypergraph learning. Experimental evaluations demonstrate HNSDs competitive performance across established benchmarks.

cs.LG↗

arXiv2025

Strict contactomorphisms are scarce

The notion of non-projectible contact forms on a given compact manifold $M$ is introduced by the first-named author in [Ohb], the set of which he also shows is a residual subset of the set of (coorientable) contact forms, both in the case with a fixed contact structure and in the case without it. In this paper, we prove that for any non-projectible contact form $λ$ the set, denoted by $\text{\rm Cont}^{\text{\rm st}}(M,λ)$, consisting of strict contactomorphisms of $λ$ is a a countable disjoint union of real lines $\mathbb R$, one for each connected component.

math.SG↗

arXiv2025

Thermodynamic reduction of contact dynamics

A universal algorithm to derive a macroscopic dynamics from the microscopic dynamical system via the averaging process and symplecto-contact reduction was introduced by Jin-wook Lim and the second-named author in [LO23]. They apply the algorithm to derive non-equilibrium thermodynamics from the statistical mechanics utilizing the relative information entropy as a generating function of the associated thermodynamic equilibrium. In the present paper, we apply this algorithm to the contact Hamiltonian dynamical systems. We describe a procedure of obtaining a discrete set of dynamical invariants of the given contact Hamiltonian system, or more generally of a contact multi-Hamiltonian system in a canonical way by deriving a (finite-dimensional non-equilibrium) thermodynamic system. We call this reduction the thermodynamic reduction of contact dynamics.

math.DS↗

arXiv2025

Contact instantons, anti-contact involution and proof of Shelukhin's conjecture

In this paper, we prove Shelukhin's conjecture on the translated points on any closed contact manifold $(Q,ξ)$ which reads that for any choice of function $H = H(t,x)$ and contact form $λ$ the contactomorphism $ψ_H^1$ carries a translated point in the sense of Sandon, whenever the inequality $$ \|H\| \leq T(λ,M) $$ holds the case. Main geometro-analytical tools are those of bordered contact instantons employed in [Ohc] with Legendrian boundary condition via the Legendrianization of contact diffeomorphisms. Along the way, we utilize the functorial construction of the contact product that carries an involutive symmetry and develop relevant contact Hamiltonian geometry with involutive symmetry. This involutive symmetry plays a fundamental role in our proof in combination with the analysis of contact instantons.

math.SG↗

arXiv2024

Rational contact instantons and Legendrian Fukaya category

This is the first of a series of papers in preparation on the Fukaya-type $A_\infty$ category generated by tame Legendrian submanifolds, called the Legendrian contact instanton Fukaya category (abbreviated as the Legendrian CI Fukaya category) and its applications to contact dynamics and topology. In the present paper, we give the construction of an $A_\infty$ category whose objects are Legendrian links and whose structure maps are defined by the moduli spaces of finite energy contact instantons on tame contact manifolds in the sense of [Oh21b]. In a sequel [KO], jointed by Jongmyeong Kim, we will explain the relationships with various previous results in the literature concerning Rabinowitz Fukaya categories [CF09, CFO10], [GGV], [BJK] on the Liouville manifolds with ideal boundary of contact manifolds, and the Floer theory of Lagrangian cobordism [CDRGG20], [EES05].

math.SG↗

arXiv2024

Perturbation theory of asymptotic operators of contact instantons and pseudoholomorphic curves on symplectization

In this paper, we first provide precise tensorial formulae for the asymptotic operators of contact instantons $w:\dot Σ\to Q$ and of pseudoholomorphic curves $u:(\dotΣ,j) \to (Q \times \mathbb R, \widetilde J)$ on the symplectization of contact manifold $(Q,λ)$. The formula exhibits explicit dependence on the compatible pair $(λ,J)$ of the given contact triad $(Q,λ, J)$. Then based on this, we present a perturbation theory of the asymptotic operator under the change of compatible CR almost complex structure $J$ for given contact form $λ$. This perturbation theory has been missing in the literature on the study of pseudoholomorphic curves on symplectization. Through this systematic tensorial approach, we also provide the study of finer asymptotic behavior at the punctures, such as convergence of tangent plane of contact instantons, in terms of the eigenvalues and eigenvectors of the asymptotic operator, which also simplifies the corresponding study of pseudoholomorphic curves on symplectization in the literature.

math.SG↗

arXiv2024

Geometry and analysis of contact instantons and entanglement of Legendrian links I

The purposes of the present paper are two-fold. Firstly we further develop the interplay between the contact Hamiltonian geometry and the geometric analysis of Hamiltonian-perturbed contact instantons with the Legendrian boundary condition, which is initiated by the present author in \cite{oh:contacton-Legendrian-bdy}. We introduce the class of \emph{tame contact manifolds} $(M,λ)$, which includes compact ones but not necessarily compact, and establish uniform a priori $C^0$-estimates for the contact instantons. Then we study the problem of estimating the Reeb-untangling energy of one Legendrian submanifold from another, and formulate a particularly designed parameterized moduli space for the study of the problem. We establish the Gromov-Floer-Hofer type convergence result for contact instantons of finite energy and construct its compactification of the moduli space, first by defining the correct energy and then by proving uniform a priori energy bounds in terms of the oscillation of the relevant contact Hamiltonian. Secondly, as an application of this geometry and analysis of contact instantons, we prove that the \emph{self Reeb-untangling energy} of a compact Legendrian submanifold $R$ in any tame contact manifold $(M,λ)$ is greater than that of the period gap $T_λ(M,R)$ of the Reeb chords of $R$. This is an optimal result in general. In a sequel \cite{oh:shelukhin-conjecture}, we also prove Shelukhin's conjecture specializing to the Legendrianization of contactomorphisms of closedcoorientable contact manifold $(Q,ξ)$ and utilizing its $\mathbb Z_2$-symmetry as the fixed point set of anti-contact involution to overcome the \emph{nontameness} of contact product $M = Q \times Q \times \mathbb R$.

math.SG↗

arXiv2024

Continuous and coherent actions on wrapped Fukaya categories

We establish the continuous functoriality of wrapped Fukaya categories with respect to Liouville automorphisms, yielding a way to probe the homotopy type of the automorphism group of a Liouville sector. These methods prove Liouville and monotone cases of a conjecture of Teleman from the 2014 ICM. In the case of a cotangent bundle, we show that the Abouzaid equivalence between the wrapped category and the infinity-category of local systems intertwines our action with the action of diffeomorphisms of the zero section. In particular, our methods yield a typically non-trivial map from the rational homotopy groups of Liouville automorphisms to the rational string topology algebra of the zero section.

math.SG↗

arXiv2024

Infinity-categorical universal properties of quotients and localizations of A-infinity-categories

We show that certain hands-on A-infinity-categorical constructions satisfy desirable universal properties in the infinity-category of A-infinity categories. For sufficiently cofibrant A-infinity categories, two models for quotients of A-infinity categories (as constructed by Lyubashenko-Manzyuk and Lyubashenko-Ovisienko), and a model for localizations (as used by Ganatra-Pardon-Shende), satisfy the relevant universal properties. We apply the results here in a companion work to prove a Liouville version of a conjecture of Teleman from the 2014 ICM.

math.CT↗