Stein and Weinstein structures are described for disk cotangent bundles of surfaces.
problem Characterizing Stein and Weinstein structures on disk cotangent bundles.
method Using Legendrian handlebody diagrams and symplectic/contact mappings.
result The canonical contact structure on the unit cotangent bundle of S is obtained via surgery.
Authors prove a contact structure result using branched covers and overtwisted disks.
problem Proving a contact structure result using branched covers and overtwisted disks.
method Explicitly constructing an overtwisted disk in the p-fold cyclic branched cover of S3. result An overtwisted disk is contained in the complement of the branch locus.
A genus one Lefschetz fibration is described for disk cotangent bundles of surfaces.
problem Describing Lefschetz fibrations on cotangent bundles.
method Explicit construction of genus one Lefschetz fibration.
result Explicit genus one open book decomposition for unit cotangent bundle.
For a genus two Heegaard splitting of a lens space, the primitive disk complex is defined to be the full subcomplex of the disk complex for one of the handlebodies of the splitting spanned by all vertices of primitive disks. In this work, we describe the complete combinatorial structure of the primitive disk complex fo…
Rich geometry on disk with special geodesics and integer spectrum.
problem Developing a unique geometric structure on the disk.
method Introducing a Riemannian structure with specific properties.
result Eigenfunctions are orthogonal polynomials with integer spectrum.
Study of flows with a single singular point on a 2D disk.
problem Classifying flows with a unique singular point on a 2D disk.
method Used a two-colored rooted tree (destingueshed graph) to classify flows and constructed a flow code.
result Found all possible structures of flows with up to 7 separatrices.
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.
We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weights. In particular, it is shown th…
Study connects plumbing baskets to 3-sphere contact structures.
problem Understanding contact structures in 3-sphere.
method Examine flat plumbing baskets and their relation to contact structures.
result Lower bounds and specific numbers for various knots and links.
Disk Embeddings tackle embedding DAGs with exponential growth.
problem Embedding DAGs with exponentially increasing ancestors and descendants.
method Disk Embeddings framework for quasi-metric spaces, including Hyperbolic Disk Embeddings.
result Disk Embeddings outperform existing methods in complex DAGs.
New examples show non-locally-flat PL-disk bounds in rational homology balls but not in integer homology balls.
problem Characterizing knots that bound PL-disks in integer homology balls.
method Involutive Heegaard Floer homology formal properties.
result Found infinitely many manifold-knot pairs (Y, J) where J does not bound a PL-disk in an integer homology ball but does in a rational homology ball.
Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.
problem Characterizing manifolds that are both rational homology spheres and double disk bundles.
method Analyzing the structure of manifolds as unions of disk bundles and using properties of rational homology and cohomology.
result Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.
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.
The paper finds representations of surface groups in SO(4,1) with specific curvature properties.
problem Finding convex-cocompact representations of surface groups with minimal map properties.
method Complex variation of Hodge structures and embedded minimal maps.
result Examples of generalized almost-Fuchsian representations not deformations of Fuchsian representations.
Study extremal trajectories of a rolling disk using geometric control theory.
problem Optimizing the motion of a vertical rolling disk.
method Geometric control theory and symmetries of geometric structures.
result Demonstrated computations in Maple for extremal trajectories.
Given a stabilized Heegaard splitting of a 3-manifold, the primitive disk complex for the splitting is the subcomplex of the disk complex for a handlebody in the splitting spanned by the vertices of the primitive disks. In this work, we study the structure of the primitive disk complex for the genus two Heegaard spli…
Proves existence of strongly overtwisted contact structures on 3-manifolds.
problem Existence of overtwisted contact structures on 3-manifolds.
method Complete h-principle for overtwisted disks and Legendrians.
result Homotopy type of overtwisted disk embeddings is determined.
Thurston's circle packing approximation of the Riemann Mapping (proven to give the Riemann Mapping in the limit by Rodin-Sullivan) is largely based on the theorem that any topological disk with a circle packing metric can be deformed into a circle packing metric in the disk with boundary circles internally tangent to t…
Paper finds conditions for special geometric structures on certain spaces.
problem Existence of specific geometric structures on double disk bundles.
method Derives a sufficient condition involving geometric data from principal orbits.
result Sufficient condition for the existence of cohomogeneity one Einstein metrics.
Study 2D spaces with curvature, finding a graph structure.
problem Understanding the geometry of 2D spaces with curvature constraints.
method Analyzing spaces as unions of disks, identifying singular points.
result Obtained a graph structure of topological singular points.
The paper studies contact categories of disks and their embeddings.
problem Understanding contact structures on disks and their categories.
method Construction of contact categories and their universal covers, embedding into triangulated envelopes.
result The universal cover of the contact category of a disk can be embedded into its triangulated envelope.
Classifies convex disks with Legendrian boundary in overtwisted contact 3-manifolds.
problem Classifying convex disks with Legendrian boundary in overtwisted contact 3-manifolds.
method Contact isotopy classification, h-principle, fundamental groups, contact mapping class group.
result Establishes an h-principle for convex disks with Legendrian boundary in overtwisted contact 3-manifolds.
This paper is the second in a series where we attempt to give a complete description of the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an emb…
Study on a metric for disk automorphisms with maximal modulus.
problem Characterizing the metric on disk automorphisms.
method Explicit formula for the metric induced by maximal modulus.
result Characterized almost regular Finsler structure.
Algorithm detects free products in disk mapping class groups.
problem Detecting free products in mapping class groups of punctured disks.
method Algorithm based on Dynnikov coordinates to verify completeness and reveal free product structure.
result Algorithm determines exact structure of free products generated by Dehn twists.
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.
After appropriate normalizations an embedded disk whose second fundamental form has large norm contains a multi-valued graph, provided the L^P norm of the mean curvature is sufficiently small. This generalizes to non-minimal surfaces a well known result of Colding and Minicozzi.
Study properties of contact structures on symplectic disk bundles with concave boundaries.
problem Understanding the geometric properties of contact structures on concave boundaries of symplectic disk bundles.
method Use tools from toric geometry and algebraic torsion measurements from embedded contact homology.
result All such contact manifolds have a global contact toric structure, and can be tight or overtwisted.
Study on planar graphs in Poincare model of hyperbolic geometry.
problem Investigating Morse flows on a 2-disk using planar graphs.
method Using planar graphs and spherical graphs to describe topological structures.
result Listed all planar graphs with at least 3 edges and described those with 4 edges.
We define a moduli space of translation structures on the open topological disk with a basepoint and endow it with a locally-compact metrizable topology. We call this the immersive topology, because it is defined using the concept of immersions: continuous maps between subsets of translation surfaces that respect the b…
In this article we consider nonholonomic deformations of disk solutions in general relativity to generic off-diagonal metrics defining knew classes of exact solutions in 4D and 5D gravity. These solutions possess Lie algebroid symmetries and local anisotropy and define certain generalizations of manifolds with Killing …
The Bergman space and conformally flat 2-disk operads are linked to vertex operator algebras.
problem Understanding invariants of 2D Riemannian manifolds using algebraic structures.
method Introducing a suboperad and showing algebraic structures, using conformally flat factorization homology.
result The Bergman space is identified with the ind-Hilbert space completion of the affine Heisenberg vertex operator algebra.
This paper is the first in a series where we attempt to give a complete description of the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed Riemannian 3-manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure…
Disk surgery on primitive disks of genus-3 Heegaard splittings of 3-sphere yields no primitive disks.
problem Characterizing primitive disks in genus-3 Heegaard splittings of 3-sphere.
method Analyzing the effect of disk surgery on primitive disks in genus-3 Heegaard splittings of 3-sphere.
result Primitive disks in genus-3 Heegaard splittings of 3-sphere are not weakly closed under disk surgery.
Study on minimal disks in metric spaces, focusing on branch set structure.
problem Structure of branch set in minimal disks in metric spaces.
method Analysis of Plateau's problem in metric spaces with quadratic isoperimetric inequality.
result Examples of spaces with large branch sets and planar branch sets.
This paper is the fifth and final in a series on embedded minimal surfaces. Following our earlier papers on disks, we prove here two main structure theorems for non-simply connected embedded minimal surfaces of any given fixed genus. The first of these asserts that any such surface without small necks can be obtained b…
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 …
Slice and ribbon disks with identical exteriors found.
problem Identifying slice and ribbon disks with the same exterior.
method Constructed an infinite family of slice disks proving they are also ribbon disks.
result Found an infinite family of slice and ribbon disks with identical exteriors.
For a smooth immersion f from the punctured disk D\{0} into Rn extendable continuously at the puncture, if its mean curvature is square integrable and the measure of f(D)∩Brk=o(rk) for a sequence rk→0, we show that the Riemannian surface (Dr\{0},g) where g is …
Example shows disk surgeries can alter weak reducibility property.
problem Weak reducibility of compressing disks after surgery.
method Example of weak reducing disks and their surgeries.
result Weak reducibility property is not preserved by disk surgery.
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.
Note on connectedness of primitive disk complex.
problem Whether primitive disk complex is connected for genus > 3 Heegaard splittings.
method Defined and quotiented primitive disk complex to prove connectedness.
result Homotopy primitive disk complex is connected.
This paper classifies planar-Rips complexes and their unit disk graphs up to homotopy.
problem Classifying planar-Rips complexes and their unit disk graphs.
method Simplicial classification, homotopy equivalence, and hereditary properties.
result Classification of planar-Rips complexes and unit disk graphs up to homotopy.
Unique geodesics selected by energy minimization in Teichmüller space.
problem Finding a unique geodesic between points in Teichmüller space.
method Energy minimization of harmonic map rays, extending Thurston boundary.
result Selection of a unique Thurston geodesic through points in Teichmüller space.
Study uses knot Floer homology to distinguish slice disks.
problem Classifying slice disks of knots up to isotopy and diffeomorphism.
method Invariants in knot Floer homology to compute and distinguish slice disks.
result Invariant can distinguish non-isotopic slice disks with diffeomorphic complements.
New knots bound multiple non-isotopic ribbon disks.
problem Finding knots that bound multiple non-isotopic ribbon disks.
method Classification of fibered, homotopy-ribbon disks for generalized square knots.
result Infinitely many knots bound infinitely many pairwise non-isotopic ribbon disks.
We study cobordisms and cobordisms rel boundary of PL locally-flat disk knots $D^{n-2}\into D^n$. Cobordisms of disk knots that do not fix the boundary sphere knots are easily classified by the cobordism properties of these boundaries, and any two even-dimensional disk knots with isotopic boundary knots are cobordant r…
Unbraided wiring diagrams for Stein fillings of lens spaces are described.
problem Constructing Stein fillings of lens spaces with canonical contact structures.
method Algorithm to draw unbraided wiring diagrams equivalent to Lefschetz fibrations.
result Wiring diagrams can be extended to symplectic graphical disks with marked points.