We prove that pseudo-holomorphic discs attached to a maximal totally real submanifold inherit their regularity from the regularity of the submanifold and of the almost complex structure. The proof is based on the computation of an explicit lower bound for the Kobayashi metric in almost complex manifolds, which also yie…
Floer theory for Lagrangians in open symplectic manifolds with smooth divisors.
problem Floer homology for Lagrangians in open symplectic manifolds with smooth divisors.
method Compactification of moduli space of pseudo-holomorphic discs, using a stronger boundary condition than stable maps.
result Established fundamental properties of the compactification as a topological space.
n this paper we define an invariant of a pair of 6 dimensional symplectic %optional manifold with vanishing 1st Chern class and its Lagrangian submanifold with vanishing Maslov index. This invariant is a function on the set of the path connected components of the bounding cochains (solution of A infinity version of Mau…
In this paper we use continuous family of multisections of the moduli space of pseudo holomorphic discs to partially improve, in the case of real coefficient, the construction of Lagrangian Floer cohomology of which the author developed jointly with Oh-Ohta-Ono. Namely we associate cyclically symmetric filtered A infin…
Gromov has shown how to construct holomorphic maps of the plane to a complex manifold with prescribed values on a lattice. In the present paper, a similar interpolation theorem for pseudo-holomorphic maps from the cylinder S to an almost-complex manifold (M,J) is proved. Properties of the space of pseudo-holomorphic ma…
Projective surfaces metrisability linked to pseudo-holomorphic curves existence.
problem Metrisability of projective surfaces.
method Equivalence between metrisability and pseudo-holomorphic curves existence.
result Metrisability of projective surfaces is equivalent to the existence of pseudo-holomorphic curves.
Defines a new operation in string topology using pseudo-holomorphic disks.
problem Defines a new operation in string topology.
method Uses virtual fundamental chains of moduli spaces of pseudo-holomorphic disks.
result Defines a Maurer-Cartan element of a Lie bracket operation in string topology.
Equivalence found between certain embeddings in complex and Euclidean spaces.
problem Finding embeddings in complex and Euclidean spaces.
method Establishing an equivalence between pseudo-holomorphic embeddings and CR regular embeddings.
result Existence of pseudo-holomorphic embeddings is equivalent to CR regular embeddings.
Constructs Kuranishi structures on pseudo holomorphic disk moduli spaces.
problem Building a system of Kuranishi structures on moduli spaces of pseudo holomorphic disks.
method Using exponential decay estimate from [FOOO7], completes the construction of a Kuranishi structure for a single moduli space.
result Detailed and self-contained account of Kuranishi structures construction completed.
In \cite{FOinteger}, Fukaya and Ono outlined a way of counting pseudo-holomorphic curves in a general compact symplectic manifold to obtain integer valued invariants. This paper contains the details of Fukaya and Ono's suggested construction for any compact symplectic manifold and a large class of exploded manifolds.
Extends Gromov invariant to Calabi-Yau 3-folds.
problem Counting embedded curves in Calabi-Yau 3-folds.
method Detailed study of bifurcations of moduli spaces of embedded pseudo-holomorphic curves.
result Integer-valued virtual count of embedded curves defined.
We prove that proper pseudo-holomorphic maps between strictly pseudoconvex regions in almost complex manifolds extend to the boundary. The key point is that the Jacobian is far from zero near the boundary, and the proof is mainly based on an almost complex analogue of the scaling method. We also establish a link betwee…
A smooth, compact 4-manifold with a Riemannian metric and b^(2+) > 0 has a non-trivial, closed, self-dual 2-form. If the metric is generic, then the zero set of this form is a disjoint union of circles. On the complement of this zero set, the symplectic form and the metric define an almost complex structure; and the la…
Two new proofs of Gromov's non-squeezing theorem using curve reparametrization and gradient bounds.
problem Gromov's non-squeezing theorem in symplectic geometry.
method Reparametrization of pseudo-holomorphic curves and application of mean value inequality or Gromov-Schwarz lemma.
result Uniform bounds on the gradient of pseudo-holomorphic curves leading to compactness of moduli space.
We prove two theorems on the removal of singularities on the boundary of a pseudo-holomorphic curve. In one theorem, we need no apriori assumption on the area of the curve. The proof uses a doubling argument with the goal of converting curves with boundary to curves without boundary. Our method is new and geometric and…
This part 2 discusses virtual fundamental chain and cycle technique for K-systems.
problem Foundation of virtual fundamental chain and cycle technique for K-systems.
method Consider a system of spaces with Kuranishi structures and their simultaneous perturbations.
result Discuss the virtual fundamental chain and cycle technique for K-systems.
We consider any pseudo holomorphic integral 2-cycle in an arbitrary almost complex manifold and perform a blow up analysis at an arbitrary point. Building upon a pseudo algebraic blow up (previously introduced by the author) we prove a geometric rate of decay for the mass ratio towards the limiting density, with an exp…
Proves connection between curve moduli and Feynman diagrams.
problem Understanding the relationship between symplectic and elliptic structures.
method Establishes homeomorphism between moduli spaces of curves and Feynman diagrams.
result Moduli spaces determine A∞ structures in both models. The paper studies chambered invariants of real Cauchy-Riemann operators on Riemann surfaces.
problem Counting pseudo-holomorphic curves in symplectic Calabi-Yau 3-folds.
method Constructs three chambered invariants: nBl, n1,2, n2,1, defined by counting solutions to ADHM vortex equations and pseudo-holomorphic sections of bundles. result Conjectures a relationship between n1,2 and n2,1 and symplectic invariants. This is the first part of a trilogy where we apply the theory of virtual manifold/orbifolds developed by the first named author and Tian to study the Gromov-Witten moduli spaces. In this paper, we resolve the main analytic issue arising from the lack of differentiability of $PSL(2, \C)$-action on spaces of W1,p-m…
The study quantizes energy for curves in symplectic manifolds.
problem Quantization of energy for pseudo-holomorphic curves.
method Extending Topping's theorem to almost everywhere immersed submanifolds and using it to prove energy quantization.
result Energy quantization for pseudo-holomorphic curves of all genus.
New flow category for contact manifolds from Reeb orbits.
problem No direct problem stated; focuses on new construction.
method Adapting Kuranishi charts to contact setting, associating flow category based on Reeb orbits and pseudo-holomorphic buildings.
result Lifts contact homology and associates flow bimodule to exact symplectic cobordisms.
We prove that any compact almost complex manifold (M,J) of real dimension 2m admits a pseudo-holomorphic embedding in a Euclidean space of dimension 4m+2, endowed with a suitable non-standard almost complex structure. Moreover, we give a necessary and sufficient condition, expressed in terms of the Segre class…
Study pseudo-holomorphic disks in real analytic hypersurfaces using exterior differential systems.
problem Existence of pseudo-holomorphic disks in non-integrable real analytic hypersurfaces.
method Theory of exterior differential systems.
result Non-existence of certain equivalent structures in the non-integrable case.
Smoothly isotopic 3-discs in 4-sphere become identical after 5-dimensional push.
problem Smooth isotopy of 3-discs in 4-sphere.
method Pushing 3-discs into 5-dimensional space.
result Isotopic 3-discs in 4-sphere become identical after 5-dimensional push.
The paper proves conditions for the existence of holomorphic discs in Kähler manifolds.
problem Existence of holomorphic discs for higher A∞ operations. method Showing existence of minimal discs with specific properties implies existence of holomorphic discs.
result Minimal discs in Kähler manifolds with certain boundary conditions are holomorphic.
We establish a product formula for Gromov-Witten invariants for closed, connected, relatively semi-positive Hamiltonian fibrations over any symplectic base. Furthermore, we show that the fibration projection induces a locally trivial (orbi-)fibration map from the moduli space of pseudo-holomorphic maps with marked poin…
New result on energy-genus bounds for 6D symplectic Calabi-Yau manifolds.
problem Energy-genus bounds for pseudo-holomorphic curves in almost complex manifolds.
method Compactness and regularity theorems for J-holomorphic currents.
result Established a new result in dimension 6 for symplectic Calabi-Yau 6-manifolds.
Study minimal discs in metric spaces with a quadratic isoperimetric inequality.
problem Understanding the geometry of minimal discs in metric spaces.
method Associate a compact metric space to each minimal disc, controlling its properties by the isoperimetric inequality.
result The geometry of the associated space can control the shapes of curves and the original space's topology.
We prove that the envelope of meromorphy of any imbedded symplectic sphere in CP2 coincides with the whole CP2. As a tool for the proof we use the Gromov theory of pseudo-holomorphic curves. Several results in this subject, such as adjunction formula, smoothness of moduli space in the neighborhood of a cusp-curve…
Study on the topology of ordered disc configurations, revealing nontrivial homotopy classes.
problem Topology of ordered disc configurations and their homotopy types.
method Analysis of ordered configuration spaces of hard discs, focusing on homotopy types and nontrivial classes.
result Exhibit nontrivial classes in π_{n-3} for all n, and their persistence in deformed ambient discs.
Classifies homotopy ribbon discs for certain slice knots.
problem Characterizing homotopy ribbon discs for specific slice knots.
method Classifies Γ-homotopy ribbon slice discs up to topological ambient isotopy. result In the infinite cyclic case, there is a unique equivalence class of such slice discs. For the Baumslag-Solitar group, there are at most two equivalence classes of Γ-homotopy ribbon discs. Researchers solve a Plateau problem for maximal surfaces in pseudo-hyperbolic spaces.
problem Finding maximal surfaces with given boundary curves in pseudo-hyperbolic spaces.
method Defined and proved the existence of unique solutions using asymptotic Plateau problem and analysis of pseudo-holomorphic curves.
result Existence and uniqueness of maximal surfaces with specified boundary conditions.
New knots found with tough, unsliceable discs.
problem Finding tough knots that can't be sliced smoothly.
method Constructed infinitely many knots with non-approximable slice discs.
result Smoothly sliceable knots have non-approximable slice discs.
Classifies ancient flows in a disc with boundary.
problem Ancient convex flows in a disc with boundary.
method Classifies flows using curve shortening.
result Ancient convex flows in a disc are classified.
This is the first part of the article we promised at the end of [FOOO13, Section 1]. We discuss the foundation of the virtual fundamental chain and cycle technique, especially its version appeared in [FOn] and also in [FOOO4, Section A1, Section 7.5], [FOOO7, Section 12], [Fu2]. In Part 1, we focus on the construction …
Holomorphic discs cover a ball in complex space.
problem Covering a ball in complex space with holomorphic discs.
method Showed a nonsingular holomorphic foliation by complete discs.
result The open unit ball in complex space admits a foliation by complete discs.
Circular disc can be tiled with up to 3 congruent pieces, showing symmetry.
problem Tiling a circular disc with congruent pieces.
method Proving the existence of a k-fold rotational symmetry for k≤3. result First nontrivial estimate on minimum number of tiles for certain tiling configurations.
The paper explores isometric models and Busemann functions for Funk and Hilbert discs.
problem Exploring isometric models and Busemann functions for Funk and Hilbert discs.
method Finding and describing isometric models and computing Busemann functions.
result Proving asymptotic harmonicity of the Funk disc and showing its dependence on measure.
Study geodesic discs with boundary length bounds, finding their closure in metric space.
problem Geodesic discs with boundary length constraints in metric spaces.
method Investigate closure in Gromov-Hausdorff space, relate to disc retracts.
result Closure of geodesic discs is related to disc retracts in metric spaces.
The rotation angle of a rolling disc is shown to be a geometric phase related to the Hopf fibration.
problem Understanding the geometric nature of rotation angles in kinematic models.
method Using the Hopf fibration and Gauss map, the geometric phase is decomposed into dynamical and geometric components.
result The geometric phase of rotation is described as the holonomy of the Hopf fibration.
Study smooth manifolds using disc-presheaves.
problem Understanding smooth manifolds.
method Using presheaves on a category of discs.
result Disc-presheaves have desirable properties and strong applications.
This note characterizes monohedral tilings of regular polygons with up to three tiles.
problem Characterizing monohedral tilings of regular polygons with up to three tiles.
method Connecting the results for squares and circles to generalize for any regular n-gon. result Characterization of monohedral tilings of any regular n-gon with up to three tiles. Study horizontal discs in fat distributions, proving their existence.
problem Existence of embedded horizontal discs in fat distributions.
method Analyzing nonlinear PDEs and proving local invertibility.
result Existence of germs of embedded horizontal discs.
Study on invariants of complex hyperbolic disc bundles over surfaces, proving a conjecture.
problem Investigating relationships between three invariants of complex hyperbolic disc orbibundles.
method Analyzing Euler characteristic, Euler number, and Toledo invariant of disc orbibundles over 2-orbifolds.
result Proved that -3|τ| = 2e + 2χ holds for certain complex hyperbolic disc orbibundles.
We show that any compact almost-complex manifold of complex dimension m can be pseudo-holomorphically embedded in R^(6m) equipped with a suitable almost-complex structure.
We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to an embedded disc on the surface, as well as the average amount of time it spends in the disc during each visit. This includes the case where the center of the disc is a cone point.
The paper classifies homotopy ribbon discs with specific groups.
problem Classifying homotopy ribbon discs with given fundamental groups.
method Using geometric and algebraic properties of groups, particularly Farrell-Jones conjecture.
result Classification of homotopy ribbon discs for specific knot groups and Baumslag-Solitar groups.