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.
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.
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.
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.
A three-manifold equipped with a Heegaard diagram can be used to set up a Floer homology theory whose differential counts pseudo-holomorphic disks in the g-fold symmetric product of the Heegaard surface. This leads to a topological invariant for three-manifolds, Heegaard Floer homology, which is functorial under cobo…
Constructs a spectrum for knot Floer homology without holomorphic geometry.
problem Computing knot Floer homology without using holomorphic geometry.
method Combinatorial definition and inductive construction of models for moduli spaces.
result Conjectures that the filtered homotopy type of the spectrum is an invariant of the knot.
We develop a global twistor correspondence for pseudo-Riemannian conformal structures of signature (++--) with self-dual Weyl curvature. Near the conformal class of the standard indefinite product metric on S^2 x S^2, there is an infinite-dimensional moduli space of such conformal structures, and each of these has the …
Paper proves disks are holomorphic in Teichmüller spaces.
problem Understanding holomorphic disks in Teichmüller spaces.
method Analyzes totally-geodesic isometries and disk-rigid domains.
result Every isometry is holomorphic or anti-holomorphic.
Holomorphic isometries from Poincaré disk to complex symmetric domains found.
problem Finding holomorphic isometries from the Poincaré disk to complex symmetric domains.
method Examined holomorphic isometries from the Poincaré disk into the product of the unit disk and complex unit n-ball, then extended to irreducible bounded symmetric domains of rank at least 2.
result Provided new examples of holomorphic isometries from the Poincaré disk into irreducible bounded symmetric domains of rank at least 2.
The paper proves a disk's energy minimizer is holomorphic and calculates its Morse index.
problem Analyzing the Morse index of a non-holomorphic disk in pseudoconvex domains.
method Proof of holomorphic minimizers and Morse index calculation.
result Non-holomorphic critical disks have a Morse index of at least n-1.
Holomorphic disks on compact Lagrangian surfaces are shown to exist.
problem Existence of holomorphic disks on compact Lagrangian surfaces.
method Analyzes compact Lagrangian surfaces in Kähler manifolds with specific curvature properties.
result Existence of holomorphic disks whose boundary is freely homotopic to a given class in the fundamental group.
Study shows nontrivial links as cross-sections of unknotted holomorphic disks.
problem Resolving parts of Kirby's list problem about nontrivial links.
method Using unknotted holomorphic disks in the four-ball to produce cross-sections.
result Many nontrivial links arise as cross-sections of unknotted holomorphic disks.
The regular type of a real hyper-surface M in an (almost) complex manifold at some point p is the maximal contact order at p of M with germs of non singular (pseudo) holomorphic disks. The main purpose of this paper is to give two intrinsic characterizations the type: one in terms of Lie brackets of a complex tangent v…
Holomorphic cylinders converge to disks joined by flow lines.
problem Convergence of perturbed holomorphic cylinders.
method Exponential estimates and flow line computation.
result Holomorphic cylinders converge to two disks joined by a flow line.
Holomorphic immersions blocked in 9D real hypersurface with specific signature.
problem Existence of holomorphic immersions of bi-disks into a specific 9D real hypersurface.
method Application of Cartan's method to analyze the existence of immersions.
result Holomorphic immersions are obstructed by specific conditions on the hypersurface.
In this paper, we discuss a rigidity property for holomorphic disks in Teichmüller space. In fact, we give a refinement of Tanigawa's rigidity theorem. We will also treat the rigidity property of holomorphic disks for complex manifolds. We observe the rigidity property is valid for bounded strictly pseudoconvex domains…
Study the asymptotics of holomorphic maps from polydisk to disk.
problem Understanding the asymptotics of holomorphic maps from polydisk to disk.
method Analyzing the conjugation action of a unipotent subgroup of PSL2(R).
result Results in the asymptotics of the translation flow on holomorphic maps.
The paper proves Liouville theorems for holomorphic maps on pseudo-Hermitian manifolds.
problem Proving Liouville theorems for holomorphic maps on pseudo-Hermitian manifolds.
method Analyzing maps between different classes of pseudo-Hermitian manifolds, using curvature assumptions.
result Holomorphic maps are constant under certain curvature conditions.
We show that conformally compact, globally hyperbolic, Lorentzian Einstein-Weyl 3-manifolds are in natural one-to-one correspondence with orientation-reversing diffeomorphisms of the 2-sphere. The proof hinges on a holomorphic-disk analog of Hitchin's mini-twistor correspondence.
The paper establishes Schwarz type lemmas for pseudo-Hermitian manifolds.
problem Understanding the geometry and mappings of pseudo-Hermitian manifolds.
method Using sub-Laplacian or Hessian type Bochner formulas and comparison theorems.
result Established Schwarz type results for \emph{CR} maps and transversally holomorphic maps.
Killing fields on compact pseudo-Kähler manifolds are holomorphic.
problem Characterizing Killing fields on compact pseudo-Kähler manifolds.
method Detailed explanation and argumentation of why a previous claim was incomplete.
result Killing fields on compact pseudo-Kähler manifolds are holomorphic.
Moduli spaces of holomorphic disks in a complex manifold Z, with boundaries constrained to lie in a maximal totally real submanifold P, have recently been found to underlie a number of geometrically rich twistor correspondences. The purpose of this paper is to develop a general Fredholm regularity criterion for holomor…
Let L⊂J1(M) be a Legendrian submanifold of the 1-jet space of a Riemannian n-manifold M. A correspondence is established between rigid flow trees in M determined by L and boundary punctured rigid pseudo-holomorphic disks in T∗M, with boundary on the projection of L and asymptotic to the doubl…
This paper generalizes a result about bounded differentials to higher-order differentials and studies their geometric implications.
problem The study of bounded holomorphic differentials and their geometric properties.
method Generalization of Wan's result to r-differentials and analysis of induced curvature.
result Equivalences between the boundedness of holomorphic differentials and negative upper bounds of induced curvature on various geometric objects.
A self-dual harmonic 2-form on a 4-dimensional Riemannian manifold is symplectic where it does not vanish. Furthermore, away from the form's zero set, the metric with the 2-form give a compatible almost complex structure and thus pseudo-holomorphic subvarieties. Such a subvariety is said to have finite energy when the …
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…
New examples of knots with infinitely many inequivalent slice disks.
problem Existence of knots with infinitely many pairwise nonisotopic slice disks.
method Classification of fibered, homotopy-ribbon disks and use of satellite operations.
result Examples of fibered, hyperbolic knots bounding infinitely many fibered, ribbon disks that are pairwise nonisotopic modulo local knotting.
We study the holomorphic unitary representations of the Jacobi group based on Siegel-Jacobi domains. Explicit polynomial orthonormal bases of the Fock spaces based on the Siegel-Jacobi disk are obtained. The scalar holomorphic discrete series of the Jacobi group for the Siegel-Jacobi disk is constructed and polynomial …
The paper establishes Schwarz type lemmas for holomorphic maps between pseudo-Hermitian and Hermitian manifolds.
problem Analyzing holomorphic maps between pseudo-Hermitian and Hermitian manifolds.
method Using Bochner formulas and comparison theorems.
result Established Schwarz type lemmas for holomorphic maps.
Compact pseudo-Hermitian spaces have rigid holomorphic isometries.
problem Characterizing the rigidity of pseudo-Hermitian homogeneous spaces.
method Analysis of Tits fibration and automorphism groups of compact spaces.
result Holomorphic isometries of compact pseudo-Hermitian spaces are compact.
New physics models generalize existing brane brick models.
problem Physics models involving special Lagrangian submanifolds with boundary.
method Relates to geometry of coassociative submanifolds and Spin(7) metrics. result Proposes new brane models in type IIA string theory.
For J-holomorphic mappings for a strongly pseudo-convex manifold, we prove elliptic regularity by the argument of boots-strapping.
Souriau studies Gibbs states for symplectic manifolds with group actions.
problem Understanding Gibbs states for symplectic manifolds with symmetries.
method Adaptation of cross product for pseudo-Euclidean spaces, detailed proofs, examples of Gibbs states.
result Presentation of Gibbs states and associated thermodynamic functions for various symplectic manifolds.
Teichmüller disks converge to Thurston boundary points.
problem Understanding convergence of Teichmüller disks to Thurston boundary points.
method Analyzing continuous extension of Teichmüller disks and proving convergence rates.
result Continuous extension of Teichmüller disks and independent convergence rates.
A recent paper (\cite{BJM}) by Biringer, Johnson, and Minsky prove that any pseudo-Anosov whose stable lamination is the limit of disks in a compression body has a power which extends over some non-trivial minimal compression body. This paper presents an alternative proof of their theorem, using techniques of Long and …
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.
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.
Let S be a Riemann surface of type (p,n) with 3p−3+n>0. Let ω be a pseudo-Anosov map of S that is obtained from Dehn twists along two families {A,B} of simple closed geodesics that fill S. Then ω can be realized as an extremal Teichmüller mapping on a surface of type (p,n) which is also denoted by $…
We define a Floer-homology invariant for links in S3, and study its properties.
We define a Floer-homology invariant for knots in an oriented three-manifold, closely related to the holomorphic disk Floer homologies for three-manifolds defined in an earlier paper. We set up basic properties of these invariants, including an Euler characteristic calculation, behaviour under connected sums. Then, we …
The paper resolves a problem about metric inequivalence and characterizes proper holomorphic maps.
problem Metric inequivalence and characterization of proper holomorphic maps.
method Explicit characterization of proper holomorphic maps from a finitely-connected planar domain onto the unit disk.
result Characterization of proper holomorphic maps from a finitely-connected planar domain onto the unit disk.
The paper studies holomorphic curves in a pseudo-Riemannian space and their moduli space.
problem Understanding the moduli space of holomorphic curves in a pseudo-Riemannian space.
method Using Frenet framing and G2′-Higgs bundles, the paper describes the moduli space of equivariant alternating holomorphic curves. result Equivariant alternating holomorphic curves are infinitesimally rigid.
Loxodromic elements are pseudo-Anosov on specific graphs.
problem Characterizing loxodromic elements in specific groups.
method Analyzing subgroups acting on multiarc and curve graphs, and the handlebody group on disk graphs.
result Loxodromic elements are pseudo-Anosov on witness graphs.
Researchers define new algebraic structures for knot Floer homology.
problem Computing knot Floer homology using bordered Floer homology.
method Introduce new bimodules over B(n) for singular crossings, motivated by holomorphic disk counts.
result New algebraic structures for knot Floer homology.
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.
Simplified computation of SFT invariants for Legendrian links.
problem Computing SFT invariants for Legendrian links is combinatorially intractable.
method Left-right-simplification of Legendrian links and analysis of holomorphic maps.
result SFT invariants of Legendrian links are combinatorially computable using disks with ≤ 2 positive punctures.
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…