Study foliations from complex ball to another via harmonic maps.
problem Rigidity of complex ball quotients.
method Lattice-equivariant harmonic map of small rank.
result Rigidity of complex ball quotients proven.
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48 results for “ball complexes”
Study shows no smooth embeddings of rational homology balls into complex projective plane.
problem Embedding rational homology balls into complex projective plane.
method Elementary arguments to prove non-existence of almost complex embeddings.
result No smooth embeddings of rational homology balls into complex projective plane.
Study non-existence of complex ball quotients in Torelli locus.
problem Non-existence of totally geodesic complex ball quotients in Torelli locus.
method Analytic techniques.
result Analytic techniques used to study non-existence.
The ball in complex 2-space can contain curves of any shape.
problem Embedding complex curves of arbitrary topology in the ball of .
method Proving existence of curves with any given topological type.
result Complete embedded complex curves of any topological type exist in the ball of .
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.
New homology theories defined for ball complexes with products and signatures.
problem Defining products in homology theories for ball complexes.
method Constructing homology theories with L-spectra and defining products.
result Product formulae clarifying the total surgery obstruction.
Quantifies nearly spherical subsets in complex ball geometry.
problem Isoperimetric inequality for nearly spherical domains in Bergman ball.
method Proves a quantitative isoperimetric inequality for nearly spherical subsets of Bergman ball.
result First result on isoperimetric phenomenon in Bergman ball.
This paper classifies ball quotients of the complex projective plane.
problem Understanding the structure of the complex projective plane as a ball quotient.
method Analyzing the branch locus as a line arrangement and smooth normal-crossing curves.
result The orbifold structure of is isomorphic to either the Deligne-Mostow example or a certain degree 9 cover.
Smooth but not symplectic embeddings of rational balls in complex projective plane found.
problem Finding smooth embeddings of rational balls in complex projective plane that are not symplectic.
method Infinite family of rational homology balls, lattice embedding obstruction from Donaldson's diagonalisation theorem.
result No two examples may be embedded disjointly.
Complex domains covering manifolds are biholomorphic to balls.
problem Covering compact manifolds by bounded domains with smooth boundaries.
method Using biholomorphic mappings and properties of boundaries.
result Bounded domains with smooth boundaries covering compact manifolds are biholomorphic to the unit ball.
In this article we prove first of all the nonexistence of holomorphic submersions other than covering maps between compact quotients of complex unit balls, with a proof that works equally well in a more general equivariant setting. For a non-equidimensional surjective holomorphic map between compact ball quotients, our…
In this paper we get an explicit lower bound for the radius of a Bergman ball contained in the Dirichlet fundamental polyhedron of a torsion-free discrete group acting on complex hyperbolic space. Consequently the volume of all complex hyperbolic n-manifolds is bounded below by the volume of this bal…
The paper introduces new boundary operators and proves higher order CR Sobolev trace inequalities for Siegel domain and complex ball.
problem Establishing higher order CR Sobolev trace inequalities for Siegel domain and complex ball.
method Introducing conformally covariant boundary operators, proving extension theorems, and establishing trace inequalities.
result Generalized CR Sobolev trace inequalities for all γ ∈ (0, n+1) \mathbb{N}.
Study rational homology balls using Casson-Gordon invariants to measure complexity.
problem Measuring complexity of rational homology balls.
method Use Casson-Gordon invariants and Levine-Tristram signatures.
result Obtain lower bounds on the number of 1-handles in handle decompositions.
Parsimonious conesmath.GT
Constructs simplified or complexified simplicial complexes.
problem Efficiently simplifying or complexifying complex spaces.
method Embeddings of simplicial complexes into a simplicial ball with bounded degrees and low volume.
result Realizes complicated spaces as parts of a ball/sphere or gives spheres specific metrics.
Regularizes complex functions on manifolds using coordinate balls.
problem Regularizing quasi-plurisubharmonic functions on compact Kaehler manifolds.
method Regularize on local coordinate balls, then glue, focusing on vanishing higher order terms at centers.
result Higher order terms vanish at centers of coordinate balls, forming a delta-net.
The Farey tree helps embed rational balls and lens spaces into complex projective space.
problem Embedding rational homology balls and lens spaces into complex projective space.
method Recursive Kirby calculus argument using the Farey tree.
result Explicit constructions of embeddings of triples of rational homology balls into homotopy .
Call a smooth knot (or smooth link) in the unit sphere in analytic (respectively, smoothly analytic) if it bounds a complex curve (respectively, a smooth complex curve) in the complex ball. Let be a smoothly analytic knot. For a small tubular neighbourhood of we give a sharp lower bound for the 4…
New insights into a complex hyperbolic braid group quotient.
problem Understanding a complex hyperbolic braid group quotient.
method Analyzing the moduli space of 12-tuples in CP1 and identifying loops.
result Identifying loops in the 9-ball quotient corresponding to standard braid generators.
Symplectic embeddings of balls into specific manifolds are studied, with restrictions and obstructions identified.
problem Understanding symplectic embeddings of balls into complex projective spaces, tori, and K3 surfaces.
method Analyzing embeddings with respect to complex structures compatible with the symplectic form and identifying obstructions.
result Symplectic volume is the primary obstruction for the existence of embeddings of balls into certain manifolds.
Results of R. Stanley and M. Masuda completely characterize the h-vectors of simplicial posets whose order complexes are spheres. In this paper we examine the corresponding question in the case where the order complex is a ball. Using the face rings of these posets, we develop a series of new conditions on their h-vect…
Mapper and Ball Mapper tools for complex data analysis.
problem Exploring and visualizing high-dimensional data and scalar functions.
method Combining Mapper and Ball Mapper, adding new features for encoding structure and symmetries.
result A new hybrid algorithm, Mapper on Ball Mapper, for comparing high-dimensional data descriptors.
Researchers create a smooth family of metrics on a ball, including hyperbolic and complex hyperbolic metrics.
problem Constructing Poincaré-Einstein metrics on the ball.
method Gibbons-Hawking-type ansatz of Page and Pope.
result The family of metrics includes the hyperbolic metric and converges to complex hyperbolic at one end.
Classifies 4-manifolds with genus one horizontal handlebody decomposition.
problem Classifying 4-manifolds with specific properties.
method Classification using Euler characteristic and handlebody decomposition.
result Found a large family of rational homology balls that embed into .
Given a closed complex hypersurface and a compact subset , we prove the existence of a pseudoconvex Runge domain in such that and there is a complete proper holomorphic embedding from into the unit ball of . For ,…
The paper proves that complex hypersurfaces in the ball are foliations.
problem Characterizing complete complex hypersurfaces in the ball.
method Analyzing noncritical holomorphic functions and submersions.
result Every complex hypersurface in the ball is part of a nonsingular holomorphic foliation.
Study cohomology of ball quotients and their compactifications.
problem Cohomology of symmetric power of cotangent bundles of ball quotients and their compactifications.
method Hodge theory for complete hermitian manifolds, Green's operator, extension of results.
result Established existence of Hodge decomposition and Green's operator for ball quotients and their compactifications.
Holomorphic foliations found in ball space with unique properties.
problem Finding holomorphic foliations in the ball space.
method Proving existence of nonsingular holomorphic foliations by closed complex hypersurfaces.
result First example of a holomorphic foliation with complete and incomplete leaves.
Researchers prove the arc complexes of decorated hyperbolic polygons are balls.
problem Understanding the structure of decorated hyperbolic polygons.
method Combinatorial approach using pseudo-manifolds and shellability.
result Arc complexes of decorated hyperbolic polygons are closed piecewise linear balls.
Punctured spheres in complex hyperbolic surfaces and bielliptic ball quotient compactificationsmath.GT
The paper explores ball quotient compactifications and their properties.
problem Smooth toroidal compactifications of ball quotients cannot contain properly holomorphically embedded 3-punctured spheres.
method Use totally geodesic punctured spheres to prove ampleness of for .
result First examples of bielliptic ball quotient compactifications are produced.
New symplectic caps and embeddings found in complex projective plane.
problem Embeddings of homology balls in complex projective plane.
method Handlebody construction of symplectic caps and embeddings.
result First examples of symplectic handlebody decompositions of a closed symplectic 4-manifold.
New method constructs non-arithmetic hyperbolic orbifolds from complex arithmetic ball quotients.
problem Creating non-arithmetic lattices in projective orthogonal groups.
method Using anti-holomorphic involutions on complex arithmetic ball quotients, gluing fixed loci along geodesic subspaces.
result Explicit calculation of the volume of constructed non-arithmetic orbifolds.
Proves condition for 4-manifolds with sphere boundary to be standard.
problem Determining when acyclic 4-manifolds with sphere boundary are standard.
method Uses Turaev's shadows to provide a sufficient condition for diffeomorphism to the standard 4-ball.
result If a compact, smooth, acyclic 4-manifold with sphere boundary has shadow-complexity at most 2, it is diffeomorphic to the standard 4-ball.
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.
We introduce the non-pure versions of simplicial balls and spheres with minimum number of vertices. These are a special type of non-homogeneous balls and spheres (NH-balls and NH-spheres) satisfying a minimality condition on the number of maximal simplices. The main result is that minimal NH-balls and NH-spheres are pr…
The unit ball is characterized by a Kähler-Einstein potential.
problem Characterizing the unit ball in complex geometry.
method Using a global potential function of the Kähler-Einstein metric.
result A compact Kähler manifold with an ample canonical bundle is the unit ball if it has a specific potential function.
In this paper we prove the existence of rational homology balls smoothly embedded in regular neighborhoods of certain linear chains of smooth -spheres by using techniques from minimal model program for 3-dimensional complex algebraic variety.
Researchers study curvature invariants on complex domains, finding rigidity for unit balls.
problem Determine if obstruction flat boundary in implies biholomorphic equivalence to the unit ball.
method Analyze curvature invariants and their vanishing orders on bounded strictly pseudoconvex domains.
result The unit ball in is rigid with respect to deformations in the class of strictly pseudoconvex domains with obstruction flat boundary.
Study on curves on specific arithmetic quotients of hyperbolic 2-ball.
problem No complex curves of certain genus on these arithmetic quotients.
method Volume estimates and understanding special subvarieties.
result For large discriminants, no complex curves of fixed genus.
We simplify Thurston norm computation for 2-bridge link complements.
problem Understanding the complexity of Thurston norm unit balls in 3-manifolds.
method Utilized Floyd and Hatcher's surface description and integral class minimization.
result Thurston norm unit balls of 2-bridge link complements have at most 8 faces.
The study examines the topology of complements of polytopal skeletons.
problem Characterizing topological properties of polytopal complexes and their skeletons.
method Constructing a long exact sequence relating homologies of skeleton complements and links of faces.
result Characterizations of Cohen-Macaulay and Leray complexes, stacked balls, and neighbourly spheres in terms of skeleton complements.
A spanning tree model for Khovanov homology, Rasmussen's s-invariant and exotic discs in the -ballmath.GT
New spanning tree model connects knot homology, s-invariant, and exotic discs.
problem Understanding exotic discs in the 4-ball for knots.
method Explicitly defined differential in spanning tree complex, described Rasmussen's s-invariant.
result Identified new infinite family of knots bounding exotic discs.
We study the classification of smooth toroidal compactifications of nonuniform ball quotients in the sense of Kodaira and Enriques. Moreover, several results concerning the Riemannian and complex algebraic geometry of these spaces are given. In particular we show that there are compact complex surfaces which admit Riem…
Three new efficient algorithms project vectors onto weighted l1 ball.
problem Sparse system identification and feature selection.
method Projected gradient descent algorithms with linear or highly competitive quadratic worst case complexities.
result Efficient tools for machine learning methods like compress sensing and feature selection.
In 1967, Chillingworth proved that all convex simplicial 3-balls are collapsible. Using the classical notion of tightness, we generalize this to arbitrary manifolds: We show that all tight simplicial 3-manifolds admit some perfect discrete Morse function. We also strengthen Chillingworth's theorem by proving that all c…
Extending smooth functions from sphere to ball without critical points.
problem Extending smooth functions from a sphere to the ball without critical points.
method Using Morse chain complex and restriction of the function to the sphere.
result Necessary condition for extending a function from a sphere to the ball.
Study on volume growth of metric balls on unbounded hypersurfaces in complex space.
problem Understanding volume growth rates of metric balls on unbounded model hypersurfaces.
method Analyzing the Carnot-Carathéodory metric on unbounded model hypersurfaces in .
result Metric balls of radius have volume on the order of or depending on the hypersurface structure.
Paper proves interior regularity estimates for complex Monge-Ampère solutions.
problem Interior regularity of solutions to complex Monge-Ampère equations.
method Proves interior and estimates for solutions with and data.
result Generalizes Bedford-Taylor interior estimate to and .