Study rolling of a ball over a sphere in n-dimensional space.
problem Rolling of a Chaplygin ball over a fixed sphere in R^n.
method Modifications of L+R and LR systems on Lie groups with invariant measure.
result Hamiltonization of the reduced system for a special inertia operator.
We relate a Chaplygin type system to a Cartan decomposition of a real semi-simple Lie group. The resulting system is described in terms of the structure theory associated to the Cartan decomposition. It is shown to possess a preserved measure and when internal symmetries are present these are factored out via a process…
Unified approach to rolling ball dynamics on spheres proves integrability.
problem Integrability of rolling ball dynamics on spheres.
method Unified Chaplygin multiplier method and Maupertuis principle.
result Complete integrability for specific inertia operators and radii ratios.
Study of gyroscopic Chaplygin systems and magnetic flows on spheres.
problem Integrability and Hamiltonization of magnetic geodesic flows on spheres.
method Analysis of gyroscopic Chaplygin systems with magnetic forces, Hamiltonization, invariant measure existence.
result Integrable magnetic geodesic flows on spheres Sn−1 for n>3. Paper proves trajectories of Chaplygin systems are reparametrized geodesics.
problem Understanding trajectories of Chaplygin systems.
method Constructive proof using modified Riemannian metrics.
result Reparametrized geodesics of Chaplygin systems.
Study on relativistic nonholonomic mechanics with time-dependent constraints.
problem Formulating classical time-dependent nonholonomic mechanics.
method Invariant formulation using moving frames and Chaplygin systems.
result Hamiltonization of time-dependent constraints achieved.
The paper studies a natural n-dimensional generalization of the classical nonholonomic Chaplygin sphere problem. We prove that for a specific choice of the inertia operator, the restriction of the generalized problem onto zero value of the SO(n-1)-momentum mapping becomes an integrable Hamiltonian system after an app…
Hidden symmetries of the Goryachev-Chaplygin and Kovalevskaya gyrostats spacetimes, as well as the Brdička-Eardley-Nappi-Witten pp-waves are studied. We find out that these spacetimes possess higher rank Stäckel-Killing tensors and that in the case of the pp-wave spacetimes the symmetry group of the Stäckel-Killing ten…
This paper simplifies complex nonholonomic systems using momentum map reduction.
problem Reducing complex nonholonomic systems with symmetries.
method Using nonholonomic momentum bundle map and gauge transformation.
result Reduced manifolds are Chaplygin-type leaves with an almost symplectic form.
The aim of this paper is to describe a class of conservative systems on S2 possessing an integral cubic in momenta. We prove that this class of systems consists off the case of Goryachev-Chaplygin, the one-parameter family of systems which has been found by the author in the previous paper (dg-ga/9711005) and a new …
We consider coupled nonholonomic LR systems on the product of Lie groups. As examples, we study n-dimensional variants of the spherical support system and the rubber Chaplygin sphere. For a special choice of the inertia operator, it is proved that the rubber Chaplygin sphere, after reduction and a time reparametrizat…
We study relations between vakonomically and nonholonomically constrained Lagrangian dynamics for the same set of linear constraints. The basic idea is to compare both situations at the level of variational principles, not equations of motion as has been done so far. The method seems to be quite powerful and effective.…
Paper reduces nonholonomic systems with symmetries.
problem Nonholonomic systems with symmetries and conserved quantities.
method Two-step reduction procedure: first results in Chaplygin system, second in almost symplectic structure.
result Almost symplectic manifolds coincide with reduced nonholonomic brackets.
Geodesic extensions for systems with nonholonomic constraints.
problem Extending equations of motion for systems with nonholonomic constraints.
method Constructing extensions to second-order ODEs, investigating geodesic conditions.
result Conditions for nonholonomic trajectories to be geodesics of a Riemannian metric.
This paper studies the construction of geometric integrators for nonholonomic systems. We derive the nonholonomic discrete Euler-Lagrange equations in a setting which permits to deduce geometric integrators for continuous nonholonomic systems (reduced or not). The formalism is given in terms of Lie groupoids, specifyin…
Projective geodesic extensions for nonholonomic systems are derived under conformal transformations.
problem Deriving conditions for projective geodesic extensions in nonholonomic mechanics.
method Analyzing necessary and sufficient conditions for existence under conformal modifications.
result Conditions for existence of projective geodesic extensions in nonholonomic systems under conformal transformations.
It has been proved that on 2-dimensional orientable compact manifolds of genus g>1 there is no integrable geodesic flow with an integral polynomial in momenta. There is a conjecture that all integrable geodesic flows on T2 possess an integral quadratic in momenta. All geodesic flows on S2 and T2 possessing i…
We consider static spacetimes whose spatial part admits foliations with the extrinsic curvature tensor K_{ab}=0. There are two complementary cases when the gradient of the lapse function points 1) to the direction of foliation or 2) orthogonally to it. Case 1) gives generalization of metrics like Bertotti-Robinson or N…
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. 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.
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.
Sharp inequality outside ball proved using Neumann method.
problem Anisotropic isoperimetric inequality for domains outside an Euclidean ball.
method Applied ABP method to Neumann boundary value problem.
result Proved sharp anisotropic isoperimetric inequality.
Classifies surgeries on torus knots and cables that bound rational homology balls.
problem Which surgeries on torus knots and cables bound rational homology balls?
method Classification based on integral surgeries and rational numbers q/p for cables.
result Set of rational numbers q/p for cables of a given knot K is bounded.
New illumination bodies defined for ball-convex shapes, proving convexity and establishing surface area measures.
problem Characterizing properties of ball-convex shapes.
method Introducing illumination bodies and weighted illumination bodies, proving convexity, and establishing surface area measures.
result Illumination bodies are convex and provide surface area measures for ball-convex shapes.
The paper characterizes unit balls among Stein spaces with specific groups using Bergman-Einstein metrics.
problem Characterizing unit balls among Stein spaces with specific groups.
method Study of Bergman metric on finite ball quotients and its Kähler-Einstein property.
result The Bergman-Einstein metric exists only on the unit ball itself for finite ball quotients with trivial groups.