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Yong-Geun Oh

Publications and source records attributed to Yong-Geun Oh.

At least 37 records · Page 2Linked to original sources

arXiv2024

Exponential decay estimates and smoothness of the moduli space of pseudoholomorphic curves

In this paper, we examine the dependence of standard gluing process for pseudoholomorphic curves under the change of the length $T$ of the neck-region with respect to the cylindrical metrics associated to the given analytic coordinates near the punctures in the setting of bordered open Riemann surface with boundary punctures. We establish exponential decay of the $T$-derivatives of the $T$-dependent family of glued solutions under the change of the length $T$ of the neck-region in a precise manner. This exponential decay estimate is an important ingredient to prove the smoothness of the Kuranishi structure constructed on the compactified moduli space of pseudoholomorphic curves given in the appendix of the authors' book. We also demonstrate the way how this smoothness follows from the exponential decay.

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arXiv2024

Simplicity of the contactomorphism group of finite regularity

For a given coorientable contact manifold $(M^{2n+1},ξ)$, we consider the group $ \operatorname{Cont}_c^{(r,δ)}(M,α)$ consisting of $C^{r,δ}$ contactomorphisms with compact support which is equipped with $C^{r,δ}$-topology of Hölder regularity $(r,δ)$ for $r \geq 1$ and $0 <δ\leq 1$. We prove that for all Hölder class exponents with $r > n + 2$ or $r = n+1, \, \frac12 < δ\leq 1$ (resp. $r < n+1$ or $r = n+1$ and $ 0< δ<\frac12$), the group is a perfect (and so a simple) group. In particular, $\operatorname{Cont}_c^r(M,ξ)$ is simple for all integer $r \geq 1$. For the case of $\operatorname{Cont}_c^{(r,δ)}(M,α)$ of general Hölder regularity, we prove the simplicity for all pairs $(r,δ)$ leaving only the case of $(r,δ) = (n+1,\frac12)$ open.

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arXiv2024

Fukaya's immersed Lagrangian Floer theory and microlocalization of Fukaya category

Let $\mathfrak{Fuk}(T^*M)$ be the Fukaya category in the Fukaya's immersed Lagrangian Floer theory \cite{fukaya:immersed} which is generated by immersed Lagrangian submanifolds with clean self-intersections. This category is monoidal in that the product of two such immersed Lagrangian submanifolds remains to be a Lagrangian immersion with clean self-intersection. Utilizing this monoidality of Fukaya's immersed Lagrangian Floer theory, we prove the following generation result in this Fukaya category of the cotangent bundle, which is the counterpart of Nadler's generation result \cite{Nadler} for the Fukaya category generated by the exact embedded Lagrangian branes. More specifically we prove that for a given triangulation $\mathcal T = \{τ_{\mathfrak a}\}$ fine enough the Yoneda module $$ \mathcal{Y}_{\mathbb L}: = hom_{\mathfrak{Fuk}(T^*M)}(\cdot, \mathbb L) $$ can be expressed as a twisted complex with terms $hom_{\mathfrak{Fuk}(T^*M)}(α_M(\cdot), L_{τ_{\mathfrak a}*})$ for any curvature-free (aka tatologically unostructed) object $\mathbb L$. Using this, we also extend Nadler's equivalence theorem between the dg category $Sh_c(M)$ of constructible sheaves on $M$ and the triangulated envelope of $\mathfrak{Fuk}^0(T^*M)$ to the one over the Novikov field $\mathbb K$.

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arXiv2024

Lagrangian multi-sections and their toric equivariant mirror

The SYZ conjecture suggests a folklore that "Lagrangian multi-sections are mirror to holomorphic vector bundles". In this paper, we prove this folklore for Lagrangian multi-sections inside the cotangent bundle of a vector space, which are equivariantly mirror to complete toric varieties by the work of Fang-Liu-Treumann-Zaslow. We also introduce the Lagrangian realization problem, which asks whether one can construct an unobstructed Lagrangian multi-section with asymptotic conditions prescribed by a tropical Lagrangian multi-section. We solve the realization problem for tropical Lagrangian multi-sections over a complete 2-dimensional fan that satisfy the so-called $N$-generic condition with $N\geq 3$. As an application, we show that every rank 2 toric vector bundle on the projective plane is mirror to a Lagrangian multi-section.

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arXiv2024

Corrigendum of "Construction of Kuranishi structures on the moduli spaces of pseudo holomorphic disks I, Surveys in Differential Geometry XXII (2018), 133-190"

This is a corrigendum of Lemma 9.1 of the paper [FOOO3] in the title. This lemma is not correct as pointed out by A. Daemi and a referee of the paper [DF]. The corrigendum does not affect the applications of this lemma in [FOOO3] and other papers and exactly the same proofs as therein apply if one replaces the statement of [FOOO3,Lemma 9.1] by Lemma 2 of the present note.

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arXiv2024

Construction of Kuranishi structures on the moduli spaces of pseudo holomorphic disks: II

This is the second of a series of two articles in which we provide detailed and self-contained account of the construction of a system of Kuranishi structures on the moduli spaces of pseudo holomorphic disks. Using the notion of obstruction bundle data introduced in [FOOO8], we give a systematic way of constructing a system of Kuranishi structures on the moduli spaces of pseudo holomorphic disks which are compatible at the boundary and corners. More specifically, it defines a tree like K-system in the sense of [FOOO6, Definition 21.9]. The method given in this paper does not only simplify the description of the constructions in the earlier literature, but also is designed to provide a systematic utility tool for the construction of a system of Kuranishi structures in the future research. We also establish its uniqueness.

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arXiv2023

Legendrian contact instanton cohomology and its spectral invariants on the one-jet bundle

In the present paper, we develop the Floer-style elliptic Morse theory for the Hamiltonian-perturbed contact action functional attached to the Legendrian links. Motivated by the present authors' construction [OY2] of the a perturbed action functional defined on the Carnot path space introduced in [OY2] as the canonical generating function, we apply a Floer-type theory to the aforementioned functional and associate the Legendrian contact instanton cohomology, denote by $HI^*(J^1B,H;R)$, to each Legendrian submanifold contact isotopic to the zero section of one-jet bundle. Then we give a Floer theoretic construction of Legendrian spectral invariants and establish their basic properties. This theory subsumes the Lagrangian intersection theory and spectral invariants on the cotangent bundle previously developed by the first-named author in [Oh1,Oh2]. The main ingredient for the study is the interplay between the geometric analysis of the Hamiltonian-perturbed contact instantons and the calculus of contact Hamiltonian geometry.

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arXiv2023

Bordered contact instantons and their Fredholm theory and generic transversalities

In this article, we first establish the Fredholm theory for the bordered contact instantons defined on the punctured Riemann surfaces with prescribed asymptotic condition near the boundary punctures. We then prove the generic mapping transversality under the perturbation of Legendrian boundary condition. We also establish their generic (0-jet) evaluation transversality results of their moduli space under the perturbations of CR almost complex structures and of Legendrian boundary conditions. These are fundamental ingredients of the construction of the moduli space of bordered contact instantons and their applications.

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arXiv2023

Geometry of Liouville sectors and the maximum principle

We introduce a new package of Floer data of $λ$-sectorial almost complex structures $J$ and sectorial Hamiltonians $H$ on the Liouville sectors introduced by Ganatra-Pardon-Shende the pairs of which are amenable to the maximum principle for the analysis of pseudoholomorphic curves relevant to the studies of wrapped Fukaya categories and of symplectic cohomology. It is also amenable to the strong maximum principle in addition when paired with cylindrical Lagrangian boundary conditions. The present work answers to a question raised by Ganatra-Pardon-Shende concerning a characterization of almost complex structures and Hamiltonians in that all the relevant confinement results in the studies of wrapped Fukaya category, symplectic cohomology and closed-open (and open-closed) maps between them can be uniformly established via the maximum principle through tensorial calculations, Hamiltonian calculus and sign considerations without making any estimates. Along the way, we prove the existence of a pseudoconvex pair $(ψ,J)$ such that \emph{$J$ is $dλ$-tame and $ψ$ is an exhaustion function of $\operatorname{nbhd}(\partial_\infty M \cup \partial M)$ that also satisfies the equation $-dψ\circ J = λ$ thereon for any Liouville sector with corners $(M,λ)$}.

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arXiv2023

Analysis of pseudoholomorphic curves on symplectization: Revisit via contact instantons

In this survey article, we present the analysis of pseudoholomorphic curves $u:(\dot Σ,j) \to (Q \times \mathbb{R}, \widetilde J)$ on the symplectization of contact manifold $(Q,λ)$ as a subcase of the analysis of contact instantons $w:\dot Σ\to Q$, i.e., of the maps $w$ satisfying the equation $$ {\bar{\partial}}^πw = 0, \, d(w^*λ\circ j) = 0 $$ on the contact manifold $(Q,λ)$, which has been carried out by a coordinate-free covariant tensorial calculus. When the analysis is applied to that of pseudoholomorphic curves $u = (w,f)$ with $w = π_Q \circ u$, $f = s\circ u$ on symplectization, the outcome is generally stronger and more accurate than the common results on the regularity presented in the literature in that all of our a priori estimates can be written purely in terms $w$ not involving $f$. The a priori elliptic estimates for $w$ are largely consequences of various Weitzenböck-type formulae with respect to the contact triad connection introduced by Wang and the first author in [OW14], and the estimate for $f$ is a consequence thereof by simple integration of the equation $df = w^*λ\circ j$. We also derive a simple precise tensorial formulae for the linearized operator and for the asymptotic operator that admit a perturbation theory of the operators with respect to (adapted) almost complex structures: The latter has been missing in the analysis of pseudoholomorphic curves on symplectization in the existing literature.

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arXiv2023

Fukaya category of infinite-type surfaces

In this paper, we construct a Fukaya category of any infinite type surface whose objects are gradient sectorial Lagrangians. This class of Lagrangian submanifolds is introduced by one of the authors in [Oh21b] which can serve as an object of a Fukaya category of any Liouville manifold that admits an exhausting proper Morse function, in particular on the Riemann surface of infinite type. We describe a generating set of the Fukaya category in terms of the end structure of the surface when the surface has countably many limit points in its ideal boundary, the latter of which can be described in terms of a subset of the Cantor set. We also show that our Fukaya category is not quasi-equivalent to the limit of the Fukaya category of surfaces of finite type appearing in the literature.

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arXiv2023

Contact instantons with Legendrian boundary condition: a priori estimates, asymptotic convergence and index formula

In Part I, we establish nonlinear ellipticity of the equation of contact instantons with Legendrian boundary condition on punctured Riemann surfaces by proving the a priori elliptic coercive estimates for the contact instantons with Legendrian boundary condition, and prove an asymptotic exponential $C^\infty$-convergence result at a puncture under the uniform $C^1$ bound. We prove that the asymptotic charge of contact instantons at the punctures under the Legendrian boundary condition vanishes. This eliminates the phenomenon of the appearance of spiraling cusp instanton along a Reeb core, which removes the only remaining obstacle towards the compactification and the Fredholm theory of the moduli space of contact instantons in the open string case, which plagues the closed string case. In Part II, we derive an index formula which computes the virtual dimension of the moduli space. These results are the analytic basis for the sequels [Oh21a]-[Oh22b] and [OYb] containing applications to contact topology and contact Hamiltonian dynamics.

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arXiv2022

Gluing theories of contact instantons and of pseudoholomoprhic curves in SFT

We develop the gluing theory of contact instantons in the context of open strings and in the context of closed strings \emph{with vanishing charge}, for example in the symplectization context. This is one of the key ingredients for the study of (virtually) smooth moduli space of (bordered) contact instantons needed for the construction of contact instanton Floer cohomology and more generally for the construction of Fukaya-type category of Legendrian submanifolds in contact manifold $(M,ξ)$. As an application, we apply the gluing theorem to give the construction of (cylindrical) Legendrian contact instanton homology that enters in our solution to Sandon's question for the nondegenerate case. We also apply this gluing theory to that of moduli spaces of holomorphic buildings arising in Symplectic Field Theory (SFT) by canonically lifting the former to that of the latter.

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arXiv2022

Analysis of contact Cauchy-Riemann maps III: energy, bubbling and Fredholm theory

In [OW2,OW3], the authors studied the nonlinear elliptic system $$ \overline{\partial}^πw = 0, \, d(w^*λ\circ j) = 0 $$ without involving symplectization for each given contact triad $(Q,λ, J)$, and established the a priori $W^{k,2}$ elliptic estimates and proved the asymptotic (subsequence) convergence of the map $w: \dot Σ\to Q$ for any solution, called a contact instanton, on $\dot Σ$ under the hypothesis $\|w^*λ\|_{C^0} < \infty$ and $d^πw \in L^2 \cap L^4$. The asymptotic limit of a contact instanton is a `spiraling' instanton along a `rotating' Reeb orbit near each puncture on a punctured Riemann surface $\dot Σ$. Each limiting Reeb orbit carries a `charge' arising from the integral of $w^*λ\circ j$. In this article, we further develop analysis of contact instantons, especially the $W^{1,p}$ estimate for $p > 2$ (or the $C^1$-estimate), which is essential for the study of compactfication of the moduli space and the relevant Fredholm theory for contact instantons. In particular, we define a Hofer-type off-shell energy $E^λ(j,w)$ for any pair $(j,w)$ with a smooth map $w$ satisfying $d(w^*λ\circ j) = 0$, and develop the bubbling-off analysis and prove an $ε$-regularity result. We also develop the relevant Fredholm theory and carry out index calculations (for the case of vanishing charge).

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arXiv2022

Geometric analysis of perturbed contact instantons with Legendrian boundary conditions

In the present article, we provide analytic foundation of the following nonlinear elliptic system, called the \emph{Hamiltonian-perturbed contact instanton equation}, $$ (du - X_H \otimes γ)^{π(0,1)} = 0, \quad d(e^{g_{H, u}}u^*(λ+ H \otimes γ)\circ j) = 0 $$ associated to a contact triad $(M,λ,J)$ and contact Hamiltonian $H$ and its boundary value problem under the Legendrian boundary condition. (1) We identify the correct choice of the action functional for perturbed contact Hamiltonian trajectories which provides a gradient structure for the system and derive its first variation formula. (2) We identify the correct choice of the energy for the bubbling analysis for the finite energy solutions for the equation. (3) We develop elliptic regularity theory for the solution, called \emph{perturbed contact instantons}: We first establish a global $W^{2,2}$ bound by the Hamiltonian calculus and the harmonic theory of the vector-valued one form $d_Hu : = du - X_H(u)\otimes γ$ and its relevant Weitzenböck formulae utilizing the contact triad connection of the contact triad $(M,λ, J)$. Then we establish $C^{k,α}$-estimates by an alternating boot-strap argument between the $π$-component of $d_Hu$ and the Reeb-component of $d_Hu$. Along the way, we also establish the boundary regularity theorem of $W^{1,4}$-weak solutions of perturbed contact instanton equation under the weak Legendrian boundary condition. (4) Based on this regularity theory, we prove an asymptotic $C^\infty$ convergence result at a puncture under the hypothesis of finite energy.

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arXiv2022

Partial collapsing degeneration of Floer trajectories and adiabatic gluing

We study partial collapsing degeneration of Hamiltonian-perturbed Floer trajectories for an adiabatic $\varepsilon$-family and its reversal adiabatic gluing, as the prototype of the partial collapsing degeneration of $2$-dimensional (perturbed) $J$-holomorphic maps to $1$-dimensional gradient segments. We consider the case when the Floer equations are $S^1$-invariant on parts of their domains whose adiabatic limits have positive lengths as $\varepsilon \to 0$, which we call thimble-flow-thimble configurations. The main gluing theorem we prove also applies to the case with Lagrangian boundaries such as in the problem of recovering holomorphic disks out of pearly configurations. In particular, our gluing theorem gives rise to a new direct proof of the chain isomorphism property between the Morse-Bott version of Lagrangian intersection Floer complex of $L$ by Fukaya-Oh-Ohta-Ono and the pearly complex of $L$ by Lalonde and Biran-Cornea (for monotone Lagrangian submanifolds). It also provides another proof of the present authors' earlier proof of the isomorphism property of the PSS map without involving the target rescaling and the scale-dependent gluing. (This is a rewritten version of our previous arXiv posting, arXiv:1103.3525.)

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arXiv2022

Pseudoholomoprhic curves on the $\mathfrak{LCS}$-fication of contact manifolds

For each contact diffeomorphism $ϕ: (Q,ξ) \to (Q,ξ)$ of $(Q,ξ)$, we equip its mapping torus $M_ϕ$ with a \emph{locally conformal symplectic} form of Banyaga's type, which we call the \emph{$\text{\rm lcs}$ mapping torus} of contact diffeomorphism $ϕ$. In the present paper, we consider the product $Q \times S^1= M_{id}$ (corresponding to $ϕ= id$) and develop basic analysis of the associated $J$-holomorphic curve equation, which has the form $$ \overline{\partial}^πw = 0, \quad w^*λ\circ j = f^*dθ$$ for the map $u = (w,f): \dot Σ\to Q \times S^1$ for the $λ$-compatible almost complex structure $J$ and a punctured Riemann surface $(\dot Σ, j)$. In particular, $w$ is a \emph{contact instanton} in the sense of [OW2, OW3]. We develop a scheme of treating the non-vanishing charge by introducing the notion of \emph{charge class} in $H^1(\dot Σ,\mathbb Z)$ and develop the geometric framework for the study of pseudoholomorphic curves, a correct choice of energy and the definition of moduli spaces towards the construction of compactification of the moduli space on the $\mathfrak{lcs}$-fication of $(Q,λ)$ (more generally on arbitrary locally conformal symplectic manifolds).

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arXiv2022

Construction of a linear K-system in Hamiltonian Floer theory

The notion of linear K-system is introduced by the present authors as an abstract model arising from the structure of compactified moduli spaces of solutions to Floer's equation in the book [FOOO14]. The purpose of the present article is to provide a geometric realization of the linear K-system associated with solutions to Floer's equation in the Morse-Bott setting. Immediate consequences (when combined with the abstract theory from [FOOO14]) are construction of Floer cohomology for periodic Hamiltonian system on general compact symplectic manifold without any restriction, and construction of an isomorphism over the Novikov ring between the Floer cohomology and the singular cohomology of the underlying symplectic manifold. The present article utilizes various analytical results on pseudo-holomorphic curves established in our earlier papers and books. However the paper itself is geometric in nature, and does not presume the readers' prior knowledge much on Kuranishi structure and its construction but assumes only the elementary part of thereof, and results from [FOOO11] and [FOOO,Chapter 8] on its construction, and the standard knowledge on Hamiltonian Floer theory. We explain the general procedure of the construction of a linear K-system by explaining in detail the inductive steps of ensuring the compatibility conditions for the system of Kuranishi structures leading to a linear K-system for the case of Hamiltonian Floer theory. A certain minor errors are corrected in this version. (to appear in the special volume dedicated to Claude Viterbo of Journal of Fixed Point Theory and Applications.)

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