Study on holomorphic discs in bundles over compact surfaces, proving Fredholm regularity under certain conditions.
problem Analyzing holomorphic discs with boundary on surfaces in vector bundles over compact manifolds.
method Proves Fredholm regularity for sections with a single complex point under specific conditions.
result Holomorphic discs are Fredholm regular under certain conditions, including neutral Kähler and symplectic actions.
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.
Classifies 5D stationary black hole domains using plumbing and disc bundles.
problem Classifying the topology of 5D stationary black hole domains.
method Using plumbing constructions and disc bundles over the 2-sphere.
result The domain of outer communication is homeomorphic to S4 or a connected sum of S2imesS2's or CP2. Study of rotation angles in a rotating disc model.
problem Understanding geometric phase in rotating systems.
method Analyzes a simple kinematic model of rotating discs.
result Explicit form of geometric phase Δg found using Baumkuchen lemma. Defines a distance function on a manifold using symplectic embeddings and recovers the metric.
problem Recovering a Riemannian metric from symplectic embeddings in cotangent bundles.
method Defines a distance-like function ρW using symplectic embeddings and recovers the metric when W is the unit disc-cotangent bundle. result The distance function ρW recovers the Riemannian metric when W is the unit disc-cotangent bundle. Applying the techniques developed in [AGG], we construct new real hyperbolic manifolds whose underlying topology is that of a disc bundle over a closed orientable surface. By the Gromov-Lawson-Thurston conjecture [GLT], such bundles M→S should satisfy the inequality ∣eM/χS∣⩽1, where eM stands for the E…
The paper extends the uniqueness of complete harmonic metrics to subharmonic weights and proves their existence on the unit disc.
problem Finding complete harmonic metrics on Riemann surfaces for subharmonic weights.
method Extending Li-Mochizuki's theorem to subharmonic weights and proving existence on the unit disc.
result Complete harmonic metrics exist on the unit disc for subharmonic weights.
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.
We study complex hyperbolic disc bundles over closed orientable surfaces that arise from discrete and faithful representations H_n->PU(2,1), where H_n is the fundamental group of the orbifold S^2(2,...,2) and thus contains a surface group as a subgroup of index 2 or 4. The results obtained provide the first complex hyp…
This article is based on the methods developed in [AGG]. We construct a complex hyperbolic structure on a trivial disc bundle over a closed orientable surface Σ (of genus 2) thus solving a long standing problem in complex hyperbolic geometry (see [Gol1, p. 583] and [Sch, p. 14]). This example answers also [Eli, Open …
We establish existence and regularity results for normal Coulomb frames in the normal bundle of two-dimensional surfaces of disc-type embedded in Euclidean spaces of higher dimensions.
We show the vanishing of the log-term in the Fefferman expansion of the Bergman kernel of the disk bundle over a compact simply-connected homogeneous Kaehler--Einstein manifold of classical type.
New 2D complex hyperbolic structures found on sphere orbibundles.
problem Locally rigid complex hyperbolic structures on sphere orbibundles.
method Constructing families of complex hyperbolic structures on disc orbibundles.
result Examples of non-locally rigid complex hyperbolic structures.
New complex structure on hyperbolic disc within hyperkaehler space.
problem Complex structure on hyperbolic disc within hyperkaehler space.
method Mostow decomposition and complex adjoint orbit analysis.
result Complex structure on hyperbolic disc differs from natural embedding.
The paper proves bounded cohomology properties of Euclidean space groups.
problem Understanding bounded cohomology of transformation groups of Euclidean spaces and discs.
method Analyzes groups of homeomorphisms and diffeomorphisms of Euclidean spaces and discs, proving boundedly acyclic properties.
result Groups of orientation-preserving homeomorphisms and diffeomorphisms of Rn are boundedly acyclic, with implications for characteristic classes of bundles. We show that if K: P \to R is an autonomous Hamiltonian on a symplectic manifold (P,Ω) which attains 0 as a Morse-Bott nondegenerate minimum along a symplectic submanifold M, and if c_1(TP)|_M vanishes in real cohomology, then the Hamiltonian flow of K has contractible periodic orbits with bounded period on all suffici…
Maximal discs in Anti-de Sitter space linked to Teichmüller space.
problem Characterizing maximal discs in Anti-de Sitter space.
method Introduced and studied maximal discs, identified their parametrization space, and used the Mess map to relate them to Teichmüller space.
result Maximal discs of Weil-Petersson class in Anti-de Sitter space are bijectively parametrized by certain submanifolds of Teichmüller space.
We prove that if the unit codisc bundle of a closed Riemannian manifold embeds symplectically into a symplectic cylinder of radius one then the length of the shortest nontrivial closed geodesic is at most half the area of the unit disc.
We compute the Bott-Morse Floer cohomology of the Clifford torus in $\CP^n$ with all possible spin-structures. Each spin structure is known to determine an orientation of the moduli space of holomorphic discs, and we analyze the change of orientation according to the change of spin structure of the Clifford torus. Also…
Study complex hyperbolic structures on a disc orbibundle with 5 cone points.
problem Constructing complex hyperbolic structures on a disc orbibundle with vanishing Euler number.
method Analyzing involutions in PU(2,1) and using bending-connectedness. result A 4-dimensional bending-connected family of complex hyperbolic structures on a disc orbibundle.
In this paper we establish the existence of periodic orbits belonging to any σ-atoroidal free homotopy class for Hamiltonian systems in the twisted disc bundle, provided that the compactly supported time-dependent Hamiltonian function is sufficiently large over the zero section and the magnitude of the weakly exact $…
We describe explicitly the moduli spaces Mgpst(S,E) of polystable holomorphic structures E with detE≅K on a rank 2 vector bundle E with c1(E)=c1(K) and c2(E)=0 for all minimal class VII surfaces S with b2(S)=1 and with respect to all possible Gauduchon metrics g. These surfaces S are …
Plumbing of surfaces embeds in hyperbolic 4-manifolds.
problem Embedding specific geometric structures in hyperbolic manifolds.
method Using plumbings of disc bundles over surfaces.
result Plumbed structures can be embedded in closed hyperbolic 4-manifolds.
An SU(3)- or SU(1,2)-structure on a 6-dimensional manifold N^6 can be defined as a pair of a 2-form omega and a 3-form rho. We prove that any analytic SU(3)- or SU(1,2)-structure on N^6 with d omega^2 =0 can be extended to a parallel Spin(7)- or Spin_0(3,4)-structure Phi that is defined on the trivial disc bundle N^6\t…
The study preserves positive Ricci curvature on connected sums of fibre bundles.
problem Preserving positive Ricci curvature on connected sums of fibre bundles.
method Lifting core metrics along general fibre bundles and applying to specific spaces.
result All classes in the torsion-free oriented bordism ring can be represented by connected manifolds of positive Ricci curvature.
Let X be a compact toric Kähler manifold with −KX nef. Let L⊂X be a regular fiber of the moment map of the Hamiltonian torus action on X. Fukaya-Oh-Ohta-Ono defined open Gromov-Witten (GW) invariants of X as virtual counts of holomorphic discs with Lagrangian boundary condition L. We prove a formula…
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.
In this note, we reveal that our solution of Demailly's strong openness conjecture implies a matrix version of the conjecture; our solutions of two conjectures of Demailly-Kollár and Jonsson-Mustată implies the truth of twisted versions of the strong openness conjecture; our optimal L2 extension implies Berndtsson…
The paper studies minimal surfaces in complex hyperbolic spaces, showing moduli space connected components.
problem Understanding the structure of moduli spaces of minimal surfaces in complex hyperbolic spaces.
method Relating the moduli space to nilpotent cones in Higgs bundles, analyzing limit points of actions.
result Connected components of the moduli space of minimal immersions in CH2 are indexed by the Toledo invariant and the Euler number of the normal bundle. In this paper we investigate the relationships between closed AdS 3-manifolds and Higgs bundles. We have a new way to construct AdS structures that allows us to see many of their properties explicitly, for example we can recover the very recent formula by Tholozan for the volumes. We also find applications to the theor…
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. 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.
First an `irregular Riemann-Hilbert correspondence' is established for meromorphic connections on principal G-bundles over a disc, where G is any connected complex reductive group. Secondly, in the case of poles of order two, isomonodromic deformations of such connections are considered and it is proved that the classi…
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.
Let E be a holomorphic vector bundle. Let θ be a Higgs field, that is a holomorphic section of End(E)⊗ΩX1,0 satisfying θ2=0. Let h be a pluriharmonic metric of the Higgs bundle (E,θ). The tuple (E,θ,h) is called a harmonic bundle. Let X be a complex manifold, and D be a normal crossing divi…
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.
In this article we introduce a definition for the moduli space of equivariant minimal immersions of the Poincaré disc into a non-compact symmetric space, where the equivariance is with respect to representations of the fundamental group of a compact Riemann surface of genus at least two. We then study this moduli space…
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.
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.
Study calculates homotopy groups and derivatives for disc diffeomorphisms.
problem Understanding the homotopy groups of diffeomorphisms of discs.
method Computes rational homotopy groups and uses Weiss' orthogonal calculus.
result Determines optimal rational concordance stable range for high-dimensional discs.