Study geodesics on Siegel-Jacobi ball with balanced metric.
problem Understanding geodesics on the Siegel-Jacobi ball.
method Determined Christoffel symbols, studied geodesic equations, calculated covariant derivatives.
result Explicit formulas for geodesics on Siegel-Jacobi ball.
We determine the matrix of the balanced metric of the Siegel-Jacobi ball and its inverse. We calculate the scalar curvature, the Ricci form and the Laplace-Beltrami operator of this manifold. We discuss several geometric aspects related with Berezin quantization on the Siegel-Jacobi ball.
We determine the homogeneous Kähler diffeomorphism FC which expresses the Kähler two-form on the Siegel-Jacobi ball $\mc{D}^J_n=\C^n\times \mc{D}_n$ as the sum of the Kähler two-form on $\C^n$ and the one on the Siegel ball $\mc{D}_n$. The classical motion and quantum evolution on $\mc{D}^J_n$ determined by a hermiti…
We show that the Wei-Norman method applied to describe the evolution on the Siegel-Jacobi disk D1J=D1×C1, where D1 denotes the Siegel disk, determined by a hermitian Hamiltonian linear in the generators of the Jacobi group G1J and Berezin's scheme using coherent …
We find the homogenous Kähler isomorphism FC which expresses the Kähler two-form on the Siegel-Jacobi domain D1J=C×D1 as the sum of the Kähler two-form on C and the one on the Siegel ball D1. The classical motion and quantum evolution on D1J…
Analyzes Berry phases and connection matrices on Siegel-Jacobi spaces.
problem Understanding Berry phases and connection matrices on Siegel-Jacobi spaces.
method Examines the Siegel-Jacobi disk and upper half-plane, calculates connection matrices and covariant derivatives.
result Calculates the connection matrix and covariant derivatives on the extended Siegel-Jacobi upper half-plane.
The paper explores geometry and arithmetic of the Siegel-Jacobi space.
problem None explicitly stated; focuses on theory.
method Theoretical discussion of geometry and arithmetic aspects.
result Theoretical insights into the geometry and arithmetic of the Siegel-Jacobi space.
The paper outlines key geometry issues in Siegel-Jacobi space.
problem Basic problems in the geometry of the Siegel-Jacobi space.
method Proposes basic problems.
result Outlines key geometry issues in Siegel-Jacobi space.
Inverse metric matrices on Siegel-Jacobi spaces are calculated for Berezin quantization.
problem Calculating inverse metric matrices on Siegel-Jacobi spaces.
method Inversion of metric matrices on XnJ and ildeXnJ. result Explicit calculations of inverse metric matrices for n=2. 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 …
We underline some differences between the geometric aspect of Berezin's approach to quantization on homogeneous Kähler manifolds and Bergman's construction for bounded domains in Cn. We construct explicitly the Bergman representative coordinates for the Siegel-Jacobi disk D1J, which is a parti…
Geodesics on extended Siegel-Jacobi upper half-plane determined.
problem Determining geodesics on a complex geometric space.
method Equating parameters in geodesic equations on the extended Siegel-Jacobi upper half-plane.
result Geodesic equations on Siegel-Jacobi, Siegel, and Heisenberg spaces.
Extended Siegel-Jacobi upper half-plane geometry studied with invariant metrics.
problem Characterizing the geometry of the extended Siegel-Jacobi upper half-plane.
method Parameterized using S-coordinates and expressed in terms of invariant metrics.
result Extended Siegel-Jacobi upper half-plane is a reductive, non-symmetric manifold.
The coherent state representation of the Jacobi group G1J is indexed with two parameters, μ(=ℏ1), describing the part coming from the Heisenberg group, and k, characterizing the positive discrete series representation of SU(1,1). The Ricci form, the scalar curvature and the geodesics of th…
Extended metric defined on Siegel-Jacobi space using invariant forms.
problem Defining a metric on the extended Siegel-Jacobi upper half space.
method Matrix embedding, pre-Iwasawa decomposition, invariant forms, sum of squares of forms.
result Invariant metric on the extended Siegel-Jacobi upper half space is derived.
Equations of motion for linear Hamiltonians in the real Jacobi group
problem Equations of motion for linear Hamiltonians in the real Jacobi group
method Using the energy function on the extended Siegel-Jacobi upper half space
result Equations of motion attached to linear Hamiltonians in the generators of the real Jacobi group
This paper bridges Kahler geometry and quantum mechanics in lognormal statistical models.
problem Evolution of spectral curves in Siegel Jacobi space through Schrodinger equation.
method Kahler geometry induced on lognormal statistical manifold, Dombrowski's construction.
result Time-dependent Schrodinger equation with varying energy.
In this article, we investigate differential operators on the Siegel-Jacobi space that are invariant under the natural action of the Jacobi group. These invariant differential operators play an important role in the arithmetic theory of Jacobi forms of higher degree. We present some explicit invariant differential oper…
Let N be the space of Gaussian distribution functions over R, regarded as a 2-dimensional statistical manifold parameterized by the mean μ and the deviation σ. In this paper we show that the tangent bundle of N, endowed with its natural Kähler structure, is the Siegel-Jacobi space…
Abstract: Study Hamiltonian systems on almost cosymplectic manifolds, extending contact Hamiltonian systems.
problem Extend Hamiltonian systems to almost cosymplectic manifolds.
method Determine Hamiltonian vector field on odd-dimensional almost cosymplectic manifolds.
result Extend equations of motion to generalized transitive almost cosymplectic structures.
Abstract: Study of metrics on line bundles over complex varieties.
problem Chern-Weil and Hilbert-Samuel formulae for singular metrics.
method Theory of b-divisors and multiplier ideal volume function.
result Generalization of a result for elliptic curves to higher degrees.
Researchers explore geometric dualities in statistical manifolds.
problem Understanding geometric dualities in statistical manifolds.
method Exploring the dualistic geometry of statistical manifolds, focusing on Hessian manifolds.
result Moduli space of univariate normal distributions corresponds to Siegel half-space and Siegel-Jacobi space.
Revisited study of real Jacobi group with invariant metrics and forms.
problem Characterizing the real Jacobi group and its invariant structures.
method Matrix realization, S-coordinates, left-invariant forms and metrics.
result Invariant metrics and forms on the Jacobi group and its quotients.
3-balls in 4-sphere become isotopic in 5-ball.
problem Whether 3-balls in 4-sphere become isotopic in 5-ball.
method Analyzing the embedding of 3-balls in 4-sphere and 5-ball.
result Affirmative answer to Gay, Hughes, Kim, and Miller's question.
Study on ball widths and minimal submanifolds in space forms.
problem Understanding widths of balls and minimal submanifolds.
method Analyzing the area of equatorial balls and related bounds for minimal submanifolds.
result Lower bounds for the area of free boundary minimal submanifolds.
This paper describes a method to construct standard 4-balls from homotopy 4-balls in C2.
problem The problem is whether every homotopy 4-ball in S4 is standard. method The approach is to use Stein surfaces and pseudoconvex domains to construct a diffeomorphic domain that is the union of three pseudoconvex domains, ensuring it is a standard 4-ball.
result The construction method ensures that the domain is a standard 4-ball, providing a compelling reimbedding construction for homotopy 4-balls in C2. Diameters of ball intersections decrease as centers move apart.
problem Behavior of intersections of moving balls in Riemannian manifolds.
method Continuous decrease of intersection diameter as centers move apart.
result Diameter of intersections decreases continuously.
Two minimal hypersurfaces in a ball intersect in any half-ball.
problem Intersection properties of minimal hypersurfaces in a ball.
method Analyzing the intersection of two minimal hypersurfaces in a unit Euclidean ball.
result Intersection point in any half-ball, strong Frankel property.
Many solutions found for a ball boundary problem.
problem Finding solutions for a Dirichlet problem on balls.
method Infinitely many solutions provided.
result Many solutions found for a Dirichlet problem on balls.
New proofs show how to embed certain 3D shapes into a 4D ball.
problem Embedding specific 3D shapes into a 4D ball.
method Embedded surfaces in the 4-ball and branched double covers.
result New proofs of embedding theorems for rational homology balls.
New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.
problem Defining and analyzing surface area measures for ball-convex bodies.
method Introducing Lp relative surface areas, proving invariance and inequalities, and using geometric interpretations. result Established inequalities and a new notion of entropy for ball-convex bodies.
Sharp lower bound found for geodesic ball eigenvalues.
problem Finding the minimum eigenvalue for geodesic balls.
method Applied Li-Schoen's uniform Poincare inequality for non-negative Ricci curvature manifolds.
result Sharp lower bound of the first Dirichlet eigenvalue for geodesic balls.
New Brieskorn spheres found to bound rational homology balls but not integral ones.
problem Identifying Brieskorn spheres that bound rational homology balls but not integral ones.
method Using modifications of Fintushel and Stern's argument and handlebody diagrams.
result Found new families of Brieskorn spheres (Σ(2,4n+1,12n+5) and Σ(3,3n+1,12n+5)) that bound rational homology balls but not integral ones. Paper calculates ball number of links using Lorentz geometry and circle packing.
problem Calculating the minimum number of balls needed to represent a link.
method Lorentz geometry and circle packing theorem applied to ball packings.
result Shows ball(L)≤5cr(L) for any link L. 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.
Extends small-ball method to broader class without uniform small-ball condition.
problem Obtaining high probability lower bounds on quadratic empirical processes.
method Extends small-ball method to allow broader class without uniform small-ball condition, motivated by tournament learning.
result Obtains high probability, almost-isometric lower bound on quadratic empirical process.
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.
Sharp geometric inequalities for free boundary hypersurfaces in balls.
problem Understanding geometric properties of free boundary hypersurfaces in balls.
method Proving a family of sharp geometric inequalities.
result Family of sharp geometric inequalities for free boundary hypersurfaces in balls.
New minimal surfaces found in ball with boundary constraints.
problem Finding minimal surfaces with boundary conditions.
method Equivariant differential geometry approach.
result A family of free boundary minimal surfaces in the unit ball.
New families of Brieskorn spheres bound rational homology balls.
problem Identifying new Brieskorn spheres that bound rational homology balls.
method Using techniques from Akbulut and Larson's work, we present new families of Brieskorn spheres.
result We discover new infinite families of Brieskorn spheres that non-trivially bound rational homology balls.
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.
Minimal surfaces in hyperbolic space have area bounds.
problem Bounding the area of minimal surfaces in geodesic balls of hyperbolic space.
method Proving an area lower bound for minimal submanifolds in a geodesic ball of hyperbolic space.
result The area of minimal surfaces is no less than the area of totally geodesic surfaces.
Manifolds covered by two balls are homeomorphic to spheres.
problem Conditions for a manifold to be homeomorphic to a sphere using metric balls.
method Optimal geometric conditions for a Riemannian manifold covered by two metric balls.
result A Riemannian manifold covered by two metric balls is homeomorphic to a sphere.
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 constructs disks with curved boundaries in a 3D ball.
problem Constructing non-planar free boundary disks in a unit ball.
method Infinite family of non-planar disks with non-positive Gaussian curvature.
result Constructs disks with curved boundaries in a unit ball.
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.
Study geodesics on ball quotients to find nonvanishing sections.
problem Finding nonvanishing holomorphic sections on ball quotients.
method Analyzing sequences of pluricanonical bundles associated to closed geodesics.
result Obtain asymptotics of holomorphic sections on ball quotients.
Fourth-order problem on half-ball with corner behavior.
problem Fourth-order problem with corner behavior on half-ball.
method Conformal mapping to isolate corner effect.
result Gauss-Bonnet formula simplifies to constant term at corner.