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.
Braid groups relate to disk diffeomorphisms with punctures.
problem Understanding the relationship between braid groups and disk diffeomorphisms.
method Using group cohomology to relate diffeomorphisms to braid groups.
result There is no cohomological obstruction to lifting a specific embedding.
New exotic 4D spaces found using knot slicing techniques.
problem Finding non-diffeomorphic exotic 4D spaces.
method Using RBG links to create slice knots with non-diffeomorphic complements.
result Distinguished new exotic 4D spaces using end Floer homology.
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.
Minimal diffeomorphisms extend uniquely with L1 Hopf differential.
problem Extending minimal diffeomorphisms between disks with specific properties.
method Uniqueness of solutions for a Plateau problem in a product of trees.
result Minimal diffeomorphisms extend uniquely with L1 Hopf differential. This paper extends a 3D result to higher dimensions for manifolds with positive curvature.
problem Proving higher-dimensional manifolds with positive curvature operator and strictly convex boundary are diffeomorphic to the Euclidean disk.
method Using the positive curvature operator and strictly convex boundary conditions to deduce the manifold's diffeomorphism to the Euclidean disk.
result Compact n-manifolds with positive curvature operator and strictly convex boundary are diffeomorphic to the standard n-dimensional Euclidean disk.
Homology stable for diffeomorphisms fixing points or disks.
problem Homology stability for diffeomorphism groups.
method Proving homology stability for classifying spaces of diffeomorphism groups.
result Homology stability for diffeomorphism groups and symmetric diffeomorphism groups.
New ribbon disks in 4D space, non-isotopic to each other.
problem Non-isotopic ribbon disks in 4D.
method Using corks to construct diffeomorphic ribbon disks.
result Non-isotopic ribbon disks constructed in 4D.
Symplectic fillings of prequantization bundles are shown to be disk bundles under certain conditions.
problem Characterizing symplectic fillings of prequantization bundles with finite capacities.
method Analysis of symplectic capacities and diffeomorphisms.
result Symplectic fillings of prequantization bundles are diffeomorphic to disk bundles under finite capacity conditions.
The study shows finiteness in homology and homotopy groups of disk automorphisms.
problem Finiteness of homology and homotopy groups for disk automorphisms.
method Homological stability, embedding calculus, arithmeticity of mapping class groups.
result Finitely generated homology and homotopy groups in specified dimensions.
Corks transform complex curves without changing topology.
problem Transforming complex curves without changing their topological properties.
method Using branched covers of holomorphic disks in the 4-ball and exotic factorizations of quasipositive braids.
result Properly embedded, smooth complex curves that are isotopic through homeomorphisms but not diffeomorphisms.
In this paper, we show that the nonexistence of rotationally symmetric harmonic diffeomorphism between the unit disk without the origin and a punctured disc with hyperbolic metric on the target.
Injective construction proves bounded cohomology dimensions.
problem Injectivity of Gambaudo--Ghys construction on bounded cohomology.
method Generalized Gambaudo--Ghys construction on bounded cohomology.
result Injectivity of the construction and infinite-dimensional bounded cohomology.
The paper explores slice disks and their properties using satellite operations and knot Floer homology.
problem Tackles the conjecture about slice disks and their positive Whitehead doubles.
method Uses techniques from knot Floer homology, Seiberg-Witten theory, and Khovanov homology.
result Provides evidence for the conjecture and constructs exotic disks.
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 study classifies manifolds that can be split into two disk bundles.
problem Understanding manifolds that can be decomposed into two disk bundles.
method Established through topological restrictions and rational ellipticity.
result Classification of manifolds up to diffeomorphism in dimensions five and six.
We show that a large class of right-angled Artin groups (in particular, those with planar complementary defining graph) can be embedded quasi-isometrically in pure braid groups and in the group of area preserving diffeomorphisms of the disk fixing the boundary (with respect to the L2-norm metric); this extends resul…
Method classifies solutions to elliptic problems in disk-like domains.
problem Classifying solutions to overdetermined elliptic problems in topological disks.
method Poincare-Hopf index theorem approach.
result Analogue of Hopf's uniqueness theorem for constant mean curvature spheres in general analytic context.
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.
In this paper we prove a stability theorem for block diffeomorphisms of 2d-dimensional manifolds that are connected sums of S^d x S^d. Combining this with a recent theorem of S. Galatius and O. Randal-Williams and Morlet's lemma of disjunction, we determine the homology of the classifying space of their diffeomorphism …
Derivative map for disk diffeomorphisms induces nontrivial homotopy groups.
problem Proving nontriviality of homomorphisms induced by derivative maps.
method Combining recent results on homotopy spheres, plumbing approach, and explicit constructions.
result Non-zero homomorphism between specific homotopy groups.
The paper proves the existence of constant mean curvature disks with capillary boundary conditions.
problem Existence of constant mean curvature disks with specific boundary conditions.
method Extending Struwe's result to a broader range of boundary angles.
result Existence of constant mean curvature disks with index at most 1.
In this note, we consider generalizations of the asymptotic Hopf invariant, or helicity, for Hamiltonian systems with one-and-a-half degrees of freedom and symplectic diffeomorphisms of a two-disk to itself.
Algebraic structure of the group of pseudo-isotopy classes of diffeomorphisms of the trivial disk bundle over the standard sphere which restrict to the identity map on the boundary is determined.
Curvature conditions distinguish Euclidean space and disks in contractible manifolds.
problem Distinguish Euclidean space and disks among contractible manifolds.
method Investigate curvature conditions on open and compact contractible manifolds with boundary.
result Stronger curvature conditions can distinguish disks from Euclidean spaces.
We show that some riemannian manifolds diffeomorphic to the sphere have the property that the cut loci of general points are smoothly embedded closed disks of codimension one. Ellipsoids with distinct axes are typical examples of such manifolds.
Study on flux homomorphism and its extension in symplectic group of a disk.
problem Understanding the flux homomorphism and its extension in symplectic group.
method Defined and analyzed the flux homomorphism and its extension, determined the Euler class, and investigated its relation to group 2-cocycle and Calabi invariant.
result Determined the Euler class of the flux extension and investigated its relation to group 2-cocycle and Calabi invariant.
The study explores how surface diffeomorphisms of knots relate to their topological properties.
problem Understanding how properties of surface diffeomorphisms of knots relate to their topological properties.
method Examining both braid and fibered knot perspectives to explore the relationship between surface diffeomorphisms and knot properties.
result Properties of surface diffeomorphisms may relate to four-dimensional topological properties of knots, such as the slice genus.
New method proves existence of constant mean curvature disks on convex surfaces.
problem Proving existence of constant mean curvature disks on convex surfaces.
method Sacks-Uhlenbeck type perturbation instead of heat flow.
result Existence for all H∈(0,H0) when Σ is convex and has mean curvature bounded below by H0. We provide a complete set of moves relating any two Lefschetz fibrations over the disk having as their total space the same 4-dimensional 2-handlebody up to 2-equivalence. As a consequence, we also obtain moves relating diffeomorphic 3-dimensional open books, providing a different approach to an analogous previous resu…
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.
The paper shows how certain circle families in S1imesD3 relate to sphere families in S2imesD2 and induces nontrivial barbell diffeomorphisms.
problem Understanding the relationship between circle and sphere families in specific 3-manifolds.
method Analyzing the fundamental groups and ambient extensions of circle and sphere families.
result Induces nontrivial barbell diffeomorphisms of S1imesS2imesI. Classifies smooth manifolds homotopy equivalent to sphere products
problem Classifying smooth manifolds homotopy equivalent to sphere products
method Using normal-invariant map and explicit families of manifolds
result Classifies smooth manifolds up to almost diffeomorphism
The paper compares isotopic and diffeomorphic links in lens spaces.
problem Comparing isotopic and diffeomorphic links in lens spaces.
method Provides a set of moves on disk, band, and grid diagrams to connect diffeomorphic links.
result There are up to four isotopy equivalent links in each diffeo equivalence class.
Geometrically, the first Betti number of orbits is linked to the Kronrod-Reeb graph.
problem Understanding the first Betti number of orbits of smooth functions.
method Established a correspondence between the first Betti number of f-orbits and the number of orbits of S′(f,V) on the Kronrod-Reeb graph. result The first Betti number of f-orbits is equal to the number of orbits of S′(f,V) on the Kronrod-Reeb graph. Study of symmetries of sphere divisions induced by functions with isolated critical points.
problem Understanding symmetries of sphere divisions induced by functions with isolated critical points.
method Analyzing the group of diffeomorphisms that leave invariant a connected component of a level set and its complement.
result The group of such diffeomorphisms is isomorphic to a finite subgroup of SO(3). 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.
Study on Teichmüller theory using Monge-Ampère equation.
problem Properness and diffeomorphism of Teichmüller spaces.
method Analysis of Monge-Ampère equation and Teichmüller space parametrization.
result Proved classical Teichmüller theorem and properness of energy function.
The paper classifies 4-manifolds with shadow complexity zero using combinatorial methods.
problem Classifying acyclic 4-manifolds with shadow complexity zero.
method Combinatorial approach focusing on simple polyhedra and their collapses.
result Any acyclic 4-manifold with shadow complexity zero and boundary is diffeomorphic to a 4-ball.
Paper constructs a cohomology class related to McDuff's secondary class, proving it transgresses to the Euler class of foliated sphere bundles.
problem Finding higher-dimensional analogs of the Calabi invariant and its transgression to the Euler class.
method Constructing a cohomology class of volume-preserving diffeomorphisms and proving transgression to the Euler class of foliated sphere bundles.
result The cohomology class transgresses to the Euler class of foliated sphere bundles.
The homotopy fiber of the inclusion from the long embedding space to the long immersion space is known to be an iterated based loop space (if the codimension is greater than two). In this paper we deloop the homotopy fiber to obtain the topological Stiefel manifold, combining results of Lashof and of Lees. We also give…
We characterize subgroups of the mapping class group that stabilize a Teichmueller disk in terms of ellipses and strips that are immersed in the associated translation surface. In particular, we show that the space of immersed ellipses/strips that meet at least three cone points is naturally a (non-manifold) 2-dimensio…
Perelman's proof confirmed, new method uses 4D topology.
problem Confirming the classical Poincaré conjecture.
method 4D topology, spun torus-knots, ribbonness, disk-chord system, Bing's result.
result Homotopy 3-sphere is diffeomorphic to the 3-sphere.
We study the maximal entropy per unit generator of push-point mapping classes on the punctured disk. Our work is motivated by fluid mixing by rods in a planar domain. If a single rod moves among N-fixed obstacles, the resulting fluid diffeomorphism is in the push-point mapping class associated with the loop in π_1(D^2 …
Study shows infinite Hofer diameter for Lagrangian orbits in cotangent bundles.
problem Determining boundedness of spectral metric on Lagrangian orbit spaces.
method Utilized wrapped Floer cohomology to define spectral invariant and pseudo-metric.
result Proved infinite Hofer diameter for Lagrangian orbits in cotangent bundles.
Generalizes Laudenbach-Poénaru theorem for group actions on 4-manifolds.
problem Extending diffeomorphisms with group actions on specific 4-manifolds.
method Proves a generalization of the Laudenbach-Poénaru theorem for finite group actions on #n(S1imesS2), showing linearly parted actions and equivariant diffeomorphisms. result Finite group actions on #n(S1imesS2) extend to aturaln(S1imesB3), and extensions are equivariantly diffeomorphic. One can define the complexity of a smooth 4-manifold as the minimal sum of the number of disks, strands and crossings in a Kirby diagram. Martelli proved that the number of homeomorphism classes of complexity less than n grows as n2. In this paper we prove that the number of diffeomorphism classes grows at least as …
New 4-manifold accounts for rationally slice knots.
problem Characterize rationally slice knots.
method Show independence of construction of rational homology ball VK. result Single 4-manifold accounts for all known rationally slice knots.