arXiv subjects

Yong-Geun Oh

Publications and source records attributed to Yong-Geun Oh.

At least 55 records · Page 3Linked to original sources

arXiv2021

Contact Hamiltonian dynamics and perturbed contact instantons with Legendrian boundary condition

This is the first of a series of papers in preparation in which we study the Hamiltonian perturbed contact instantons with Legendrian boundary condition and its applications. In this paper, we prove that the asymptotic charge of contact instantons at the punctures under the Legendrian boundary condition vanishes, which eliminates the phenomenon of the appearance of `spiraling cusp instanton along a Reeb core'. This 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. We also extend the a priori elliptic coercive estimates for the contact instantons with boundary, and prove an asymptotic exponential $C^\infty$-convergence result at a puncture under the uniform $C^1$ bound. In a sequel to the present paper, we study the $C^1$ estimates by defining a proper notion of energy for the contact instantons, and develop a Fredholm theory and construct a Gromov-type compactification of the moduli space of contact instantons with Legendrian boundary condition and of finite energy, and apply them to problems in contact topology and dynamics.

math.SG↗

arXiv2021

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

This is the first 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 exponential decay estimate given in [FOOO7]. This article completes the construction of a Kuranishi structure of a single moduli space. This article is an improved version of [FOOO4, Part 4] and its mathematical content is taken from our earlier writing [FOn,FOOO2,FOOO4,FOOO7]. The version 2 contains a correction of typological errors in the statement of Lemma 10.11 and the statement of Proposition 10.4. The proof of them are not changed. (Note these places were written in the same way as version 1 in the version published in Surveys in Differential Geometry.)

math.SG↗

arXiv2020

Holomorphic curves and continuation maps in Liouville bundles

We construct an unwrapped Floer theory for bundles of Liouville sectors. In particular, we construct a compatible collection of unwrapped Fukaya categories of fibers of a Liouville bundle, and prove that the two natural constructions of continuation maps in this setting behave compatibly. These constructions are exploited in [OT19] to construct homotopically coherent actions of Lie groups on wrapped Fukaya categories, thereby proving a conjecture from Teleman's 2014 ICM address.

math.SG↗

arXiv2020

Smooth constructions of homotopy-coherent actions

We prove that, for nice classes of infinite-dimensional smooth groups G, natural constructions in smooth topology and symplectic topology yield homotopically coherent group actions of G. This yields a bridge between infinite-dimensional smooth groups and homotopy theory. The result relies on two computations: One showing that the diffeological homotopy groups of the Milnor classifying space BG are naturally equivalent to the (continuous) homotopy groups, and a second showing that a particular strict category localizes to yield the homotopy type of BG. We then prove a result in symplectic geometry: These methods are applicable to the group of Liouville automorphisms of a Liouville sector. The present work is written with an eye toward [OT19], where our constructions show that higher homotopy groups of symplectic automorphism groups map to Fukaya-categorical invariants, and where we prove a conjecture of Teleman from the 2014 ICM in the Liouville and monotone settings.

math.AT↗

arXiv2019

Lagrangian fibers of Gelfand-Cetlin systems

A Gelfand-Cetlin system is a completely integrable system defined on a partial flag manifold whose image is a rational convex polytope called a Gelfand-Cetlin polytope. Motivated by the study of Nishinou-Nohara-Ueda on the Floer theory of Gelfand-Cetlin systems, we provide a detailed description of topology of Gelfand-Cetlin fibers. In particular, we prove that any fiber over an interior point of a k-dimensional face of the Gelfand-Cetlin polytope is an isotropic submanifold and is diffeomorphic to $(S^1)^k \times N$ for some smooth manifold $N$. We also prove that such $N$'s are exactly the vanishing cycles shrinking to points in the associated toric variety via the toric degeneration. We also devise an algorithm of reading off Lagrangian fibers from the combinatorics of the ladder diagram.

math.SG↗

arXiv2019

A critical point analysis of Landau--Ginzburg potentials with bulk in Gelfand--Cetlin systems

Using the bulk-deformation of Floer cohomology by Schubert cycles and non-Archimedean analysis of Fukaya--Oh--Ohta--Ono's bulk-deformed potential function, we prove that every complete flag manifold $\mathrm{Fl}(n)$ ($n \geq 3$) with a monotone Kirillov--Kostant--Souriau symplectic form carries a continuum of non-displaceable Lagrangian tori which degenerates to a non-torus fiber in the Hausdorff limit. In particular, the Lagrangian $S^3$-fiber in $\mathrm{Fl}(3)$ is non-displaceable, answering the question of which was raised by Nohara--Ueda who computed its Floer cohomology to be vanishing.

math.SG↗

arXiv2019

Asymptotic behavior of Vianna's exotic Lagrangian tori $T_{a,b,c}$ in $\mathbb{CP}^2$ as $a+b+c \to \infty$

In this paper, we study various asymptotic behavior of the infinite family of monotone Lagrangian tori $T_{a,b,c}$ in $\mathbb{CP}^2$ associated to Markov triples $(a,b,c)$ described in \cite{Vi14}. We first prove that the Gromov capacity of the complement $\mathbb{CP}^2 \setminus T_{a,b,c}$ is greater than or equal to $\frac13$ of the area of the complex line for all Markov triple $(a,b,c)$. We then prove that there is a representative of the family $\{T_{a,b,c}\}$ whose loci completely miss a metric ball of nonzero size and in particular the loci of the union of the family is not dense in $\mathbb{CP}^2$.

math.SG↗

arXiv2019

A wrapped Fukaya category of knot complement

This is the first of a series of two articles where we construct a version of wrapped Fukaya category $\mathcal W\mathcal F(M\setminus K;H_{g_0})$ of the cotangent bundle $T^*(M \setminus K)$ of the knot complement $M \setminus K$ of a compact 3-manifold $M$, and do some calculation for the case of hyperbolic knots $K \subset M$. For the construction, we use the wrapping induced by the kinetic energy Hamiltonian $H_{g_0}$ associated to the cylindrical adjustment $g_0$ on $M \setminus K$ of a smooth metric $g$ defined on $M$. We then consider the torus $T = \partial N(K)$ as an object in this category and its wrapped Floer complex $CW^*(ν^*T;H_{g_0})$ where $N(K)$ is a tubular neighborhood of $K \subset M$. We prove that the quasi-equivalence class of the category and the quasi-isomorphism class of the $A_\infty$ algebra $CW^*(ν^*T;H_{g_0})$ are independent of the choice of cylindrical adjustments of such metrics depending only on the isotopy class of the knot $K$ in $M$. In a sequel [BKO], we give constructions of a wrapped Fukaya category $\mathcal W\mathcal F(M\setminus K;H_h)$ for hyperbolic knot $K$ and of $A_\infty$ algebra $CW^*(ν^*T;H_h)$ directly using the hyperbolic metric $h$ on $M \setminus K$, and prove a formality result for the asymptotic boundary of $(M \setminus K, h)$.

math.SG↗

arXiv2019

Formality of Floer complex of the ideal boundary of hyperbolic knot complement

This is a sequel to the authors' article [BKO](arXiv:1901.02239). We consider a hyperbolic knot $K$ in a closed 3-manifold $M$ and the cotangent bundle of its complement $M \setminus K$. We equip $M \setminus K$ with a hyperbolic metric $h$ and its cotangent bundle $T^*(M \setminus K)$ with the induced kinetic energy Hamiltonian $H_h = \frac{1}{2} |p|_h^2$ and Sasakian almost complex structure $J_h$, and associate a wrapped Fukaya category to $T^*(M\setminus K)$ whose wrapping is given by $H_h$. We then consider the conormal $ν^*T$ of a horo-torus $T$ as its object. We prove that all non-constant Hamiltonian chords are transversal and of Morse index 0 relative to the horo-torus $T$, and so that the structure maps satisfy $\widetilde{\mathfrak m}^k = 0$ unless $k \neq 2$ and an $A_\infty$-algebra associated to $ν^*T$ is reduced to a noncommutative algebra concentrated to degree 0. We prove that the wrapped Floer cohomology $HW(ν^*T; H_h)$ with respect to $H_h$ is well-defined and isomorphic to the Knot Floer cohomology $HW(\partial_\infty(M \setminus K))$ that was introduced in [BKO] for arbitrary knot $K \subset M$. We also define a reduced cohomology, denoted by $\widetilde{HW}^d(\partial_\infty(M \setminus K))$, by modding out constant chords and prove that if $\widetilde{HW}^d(\partial_\infty(M \setminus K))\neq 0$ for some $d \geq 1$, then $K$ cannot be hyperbolic. On the other hand, we prove that all torus knots have $\widetilde{HW}^1(\partial_\infty(M \setminus K)) \neq 0$.

math.SG↗

arXiv2019

Deformations of Coisotropic Submanifolds in Jacobi Manifolds

In this paper, we attach an $L_\infty$-algebra to any coisotropic submanifold in a Jacobi manifold. Our construction generalizes and unifies analogous constructions by Oh-Park (symplectic case), Cattaneo-Felder (Poisson case), Lê-Oh (locally conformal symplectic case). As a new special case, we attach an $L_\infty$-algebra to any coisotropic submanifold in a contact manifold. The $L_\infty$-algebra of a coisotropic submanifold $S$ governs the (formal) deformation problem of $S$.

math.DG↗

arXiv2018

Displacement energy of compact Lagrangian submanifold from open subset

We prove that for any compact Lagrangian submanifold intersecting an open subset $U$ in tame symplectic manifold $(M,ω)$, the Hofer displacement energy of $L$ from $U$ is positive, provided $L \cap U \neq \emptyset$. We also give an explicitlower bound in terms of an $ε$-regularity type invariant for pseudo-holomorphic curves relative to $L$.

math.SG↗

arXiv2018

Lagrangian fibers of Gelfand-Cetlin systems

Motivated by the study of Nishinou-Nohara-Ueda on the Floer thoery of Gelfand-Cetlin systems over complex partial flag manifolds, we provide a complete description of the topology of Gelfand-Cetlin fibers. We prove that all fibers are \emph{smooth} isotropic submanifolds and give a complete description of the fiber to be Lagrangian in terms of combinatorics of Gelfand-Cetlin polytope. Then we study (non-)displaceability of Lagrangian fibers. After a few combinatorial and numercal tests for the displaceability, using the bulk-deformation of Floer cohomology by Schubert cycles, we prove that every full flag manifold $\mathcal{F}(n)$ ($n \geq 3$) with a monotone Kirillov-Kostant-Souriau symplectic form carries a continuum of non-displaceable Lagrangian tori which degenerates to a non-torus fiber in the Hausdorff limit. In particular, the Lagrangian $S^3$-fiber in $\mathcal{F}(3)$ is non-displaceable the question of which was raised by Nohara-Ueda who computed its Floer cohomology to be vanishing.

math.SG↗

arXiv2017

Analysis of Contact Cauchy-Riemann maps I: a priori $C^k$ estimates and asymptotic convergence

In the present article, we develop the analysis of the following nonlinear elliptic system of equations $$ \bar\partial^πw = 0, \, d(w^*λ\circ j) = 0 $$ first introduced by Hofer, associated to each given contact triad $(M,λ,J)$ on a contact manifold $(M,ξ)$. We directly work with this elliptic system on the contact manifold without involving the symplectization process. We establish the local a priori $C^k$ coercive pointwise estimates for all $k \geq 2$ in terms of $\|dw\|_{C^0}$ by doing tensorial calculations on contact manifold itself using the contact triad connection introduced by present the authors. Equipping the punctured Riemann surface $(\dot Σ,j)$ with a cylindrical Kähler metric and isothermal coordinates near every puncture, we prove the asymptotic (subsequence) convergence to the `spiraling' instantons along the `rotating' Reeb orbit for any solution $w$, not necessarily for $w^*λ\circ j$ being exact (i.e., allowing non-zero `charge' $Q \neq 0$), with bounded gradient $\|d w\|_{C^0} < C$ and finite $π$-harmonic energy. For nondegenerate contact forms, we employ the `three-interval method' to prove the exponential convergence to a closed Reeb orbit when $Q = 0$. (The Morse-Bott case using this method is treated in a sequel (arXiv:1311.6196).)

math.SG↗

arXiv2017

Exact Lagrangian submanifolds, Lagrangian spectral invariants and Aubry-Mather theory

We construct graph selectors for compact exact Lagrangians in the cotangent bundle of an orientable, closed manifold. The construction combines Lagrangian spectral invariants developed by Oh and results by Abouzaid about the Fukaya category of a cotangent bundle. We also introduce the notion of Lipschitz-exact Lagrangians and prove that these admit an appropriate generalization of graph selector. We then, following Bernard-Oliveira dos Santos, use these results to give a new characterization of the Aubry and Mane sets of a Tonelli Hamiltonian and to generalize a result of Arnaud on Lagrangians invariant under the flow of such Hamiltonians.

math.SG↗

arXiv2017

Kuranishi structure, Pseudo-holomorphic curve, and virtual fundamental chain: Part 2

This article is the second part of the article we promised to write at the end of Section 1 of [FOOO15] (arXiv:1209.4410). (Part I appeared in [Part I] (arXiv:1503.07631).) We discuss the foundation of the virtual fundamental chain and cycle technique, especially its version that appeared in [FOn] and also in Section A1, Section 7.5 [FOOO4], Section 12 [FOOO7], [Fu2]. This article is independent of our earlier writing [FOOO15]. We also do not assume that the readers have any knowledge on the pseudo-holomorphic curve. In this second part, we consider a system of spaces with Kuranishi structures (abbreviated as a K-system) and its simultaneous perturbations.

math.SG↗

arXiv2017

Spectral invariants with bulk, quasimorphisms and Lagrangian Floer theory

In this paper we first develop various enhancements of the theory of spectral invariants of Hamiltonian Floer homology and of Entovi-Polterovich theory of spectral symplectic quasi-states and quasimorphisms by incorporating \emph{bulk deformations}, i.e., deformations by ambient cycles of symplectic manifolds, of the Floer homology and quantum cohomology. Essentially the same kind of construction is independently carried out by Usher [Us4] in a slightly less general context. Then we explore various applications of these enhancements to the symplectic topology, especially new construction of symplectic quasi-states, quasimorphisms and new Lagrangian intersection results on toric manifolds. The most novel part of this paper is to use open-closed Gromov-Witten theory (operator $\frak q$ in [FOOO1] and its variant involving closed orbits of periodic Hamiltonian system) to connect spectral invariants (with bulk deformation), symplectic quasi-states, quasimorphism to the Lagrangian Floer theory (with bulk deformation). We use this open-closed Gromov-Witten theory to produce new examples. Especially using the calculation of Lagrangian Floer homology with bulk deformation in [FOOO3,FOOO4], we produce examples of compact toric manifolds $(M,ω)$ which admits uncountably many independent quasimorphisms $\widetilde{\operatorname{Ham}}(M,ω) \to \mathbb R$. We also obtain a new intersection result of Lagrangian submanifolds on $S^2 \times S^2$ discovered in [FOOO6]. Many of these applications were announced in [FOOO3,FOOO4,FOOO6].

math.SG↗

arXiv2017

Analysis of contact Cauchy-Riemann maps II: canonical neighborhoods and exponential convergence for the Morse-Bott case

This is a sequel to the papers [OW1], [OW2]. In [OW1], the authors introduced a canonical affine connection on $M$ associated to the contact triad $(M,λ,J)$. In [OW2], they used the connection to establish a priori $W^{k,p}$-coercive estimates for maps $w: \dot Σ\to M$ satisfying $\overline{\partial}^πw= 0, \, d(w^*λ\circ j) = 0$ \emph{without involving symplectization}. We call such a pair $(w,j)$ a contact instanton. In this paper, we first prove a canonical neighborhood theorem of the locus $Q$ foliated by closed Reeb orbits of a Morse-Bott contact form. Then using a general framework of the three-interval method, we establish exponential decay estimates for contact instantons $(w,j)$ of the triad $(M,λ,J)$, with $λ$ a Morse-Bott contact form and $J$ a CR-almost complex structure adapted to $Q$, under the condition that the asymptotic charge of $(w,j)$ at the associated puncture vanishes. We also apply the three-interval method to the symplectization case and provide an alternative approach via tensorial calculations to exponential decay estimates in the Morse-Bott case for the pseudoholomorphic curves on the symplectization of contact manifolds. This was previously established by Bourgeois [Bou] (resp. by Bao [Ba]), by using special coordinates, for the cylindrical (resp. for the asymptotically cylindrical) ends. The exponential decay result for the Morse-Bott case is an essential ingredient in the set-up of the moduli space of pseudoholomorphic curves which plays a central role in contact homology and symplectic field theory (SFT).

math.SG↗

arXiv2016

Canonical connection on contact manifolds

We introduce a canonical affine connection on the contact manifold $(Q,ξ)$, which is associated to each contact triad $(Q,λ,J)$ where $λ$ is a contact form and $J:ξ\to ξ$ is an endomorphism with $J^2 = -id$ compatible to $dλ$. We call it the \emph{contact triad connection} of $(Q,λ,J)$ and prove its existence and uniqueness. The connection is canonical in that the pull-back connection $ϕ^*\nabla$ of a triad connection $\nabla$ becomes the triad connection of the pull-back triad $(Q, ϕ^*λ, ϕ^*J)$ for any diffeomorphism $ϕ:Q \to Q$ satisfying $ϕ^*λ= λ$ (sometimes called a strict contact diffeomorphism). It also preserves both the triad metric $$ g_{(λ,J)} = dλ(\cdot, J\cdot) + λ\otimes λ$$ and $J$ regarded as an endomorphism on $TQ = \mathbb R\{X_λ\}\oplus ξ$, and is characterized by its torsion properties and the requirement that the contact form $λ$ be holomorphic in the $CR$-sense. In particular, the connection restricts to a Hermitian connection $\nabla^π$ on the Hermitian vector bundle $(ξ,J,g_ξ)$ with $g_ξ= dλ(\cdot, J\cdot)|_ξ$, which we call the \emph{contact Hermitian connection} of $(ξ,J,g_ξ)$. These connections greatly simplify tensorial calculations in the sequels \cite{oh-wang1}, \cite{oh-wang2} performed in the authors' analytic study of the map $w$, called contact instantons, which satisfy the nonlinear elliptic system of equations $\overline{\partial}^πw = 0, \, d(w^*λ\circ j) = 0$ in the contact triad $(Q,λ,J)$.

math.SG↗