Consider a three dimensional cusped spherical CR manifold M and suppose that the holonomy representation of π1(M) can be deformed in such a way that the peripheral holonomy is generated by a non-parabolic element. We prove that, in this case, there is a spherical CR structure on some Dehn sur…
Study examines heart and football-shaped metrics, verifying geometric structure.
problem Analyzing reducible spherical conical metrics and their geometric properties.
method Examined 1-parameter heart shape and 3-parameter football shape families, verified structure theorem, used explicit metric and geodesic calculations.
result Naturally arise from Abelian differentials of the third kind, offer new evidence for spherical geometry and complex analytic structure interaction.
We classify hyperbolic monopoles with continuous symmetries and construct new examples.
problem Classifying and constructing hyperbolic monopoles with continuous symmetries.
method Developed a Structure Theorem and used representation theory to simplify the problem.
result Found constraints on structure groups and constructed novel spherically symmetric Sp(n) hyperbolic monopoles. Study on spherical CR manifolds with non-trivial Chern classes.
problem Understanding Chern classes in spherical CR manifolds.
method Construction and proof of constraints on Chern classes.
result Topological obstruction to spherical CR structures on contact manifolds.
Study on spherical conical metrics and their reducibility on compact Riemann surfaces.
problem Existence and geometric structure of reducible spherical conical metrics.
method Analysis of monodromy groups and geometric cutting of surfaces.
result Existence of reducible spherical conical metrics with saddle points on the same geodesic.
We apply a spherical CR Dehn surgery theorem in order to obtain infinitely many Dehn surgeries of the Whitehead link complement that carry spherical CR structures. We consider as starting point the spherical CR uniformization of the Whitehead link complement constructed by Parker and Will, using a Ford domain in the co…
We describe a general geometrical construction of spherical CR structures. We construct then spherical CR structures on the complement of the figure eight knot and the Whitehead link. They have discrete holonomies contained in PU(2,1,Z[ω]) and PU(2,1,Z[i]) respectively. These are the same ring of intege…
The present paper is an addendum to "Spherical structures on torus knots and links", arXiv:1008.0312, and concerns more general case of torus knot and link cone-manifolds.
We establish Bochner-type formulas for operators related to CR automorphisms and spherical CR structures. From such formulas, we draw conclusions about rigidity by making assumptions on the Tanaka-Webster curvature and torsion.
Kähler-Ricci flow on spherical Fano manifolds converges to a soliton.
problem Analyzing the behavior of Kähler-Ricci flow on spherical Fano manifolds.
method Gromov-Hausdorff limit and torus degeneration.
result The limit of Kähler-Ricci flow on spherical Fano manifolds is a spherical Fano variety with a Kähler-Ricci soliton.
Suppose M1 and M2 are two closed (compact with no boundary) spherical CR manifolds with positive CR Yamabe constant. In this note, we show that the connected sum of M1 and M2 also admits a spherical CR structure with positive CR Yamabe constant.
The present paper considers two infinite families of cone-manifolds endowed with spherical metric. The singular strata is either the torus knot t(2n+1,2) or the torus link t(2n,2). Domains of existence for a spherical metric are found in terms of cone angles and volume formulæ are presented.
In this paper we develop a method to compute the Burns-Epstein invariant of a spherical CR homology sphere, up to an integer, from its holonomy representation. As application, we give a formula for the Burns-Epstein invariant, modulo an integer, of a spherical CR structure on a Seifert fibered homology sphere in terms …
The goal of this mostly expository paper is to present several candidates for hyperbolic structures on irreducible Artin-Tits groups of spherical type and to elucidate some relations between them. Most constructions are algebraic analogues of previously known hyperbolic structures on Artin braid groups coming from natu…
Locally classifies 4D spherical symmetric Finsler spaces.
problem Classifying 4D spherical symmetric Finsler spaces.
method Local classification of pseudo-Finsler Berwald structures.
result Six classes of non-Riemannian SO(3)-symmetric pseudo-Finsler Berwald functions.
Study establishes monodromy equivalence for Lamé-type equations and constructs cone spherical metrics.
problem Investigating monodromy equivalence and finite-gap structures of Lamé-type equations.
method Analyzing finite-gap structures and constructing cone spherical metrics.
result Established monodromy equivalence between classical and generalized Lamé-type equations, derived finite-gap structures, and constructed cone spherical metrics.
Study on CR structures on 3D Lie groups, focusing on equivalence and closed chains.
problem Characterizing CR structures on 3D Lie groups and their chains.
method Analyzing left-invariant CR structures on 3D Lie groups and their equivalence.
result All chains on G are closed if and only if G is CR equivalent to specific spherical CR structures.
Characterizes representations for complex projective structures with specific branch data.
problem Understanding representations of surface groups as holonomy of complex projective structures.
method Computing holonomies for spherical metrics and affine structures with prescribed conical angles.
result Computed holonomies for spherical metrics and affine structures with specific conical angles.
Uniformizes CR structure on a specific 3-manifold.
problem Prove discrete and faithful representation of a complex hyperbolic triangle group.
method Analyzes subgroup properties and applies to CR structure.
result Even subgroup represents a uniformizable spherical CR structure.
SRCA reduces high-dimensional data to lower dimensions while preserving geometric structures.
problem High-dimensional datasets with underlying geometric structures.
method Spherical Rotation Component Analysis (SRCA) incorporating geometric loss functions.
result SRCA provides a low-rank spherical representation of data with general theoretic guarantees.
New exact spherically symmetric vacuum solutions found in Finsler gravity.
problem Finding exact vacuum solutions in Finsler gravity.
method Spherically symmetric, asymptotically flat Berwald spacetimes solved for Finsler gravity vacuum equation.
result Only one class of spherically symmetric Berwald spacetimes is compatible with asymptotic flatness and a well-defined causal structure.
We study deformation of spherical CR circle bundles over Riemann surfaces of genus > 1. There is a one to one correspondence between such deformation space and the so-called universal Picard variety. Our differential-geometric proof of the structure and dimension of the unramified universal Picard variety has its own…
Study spherical doubly warped spacetimes for stellar collapse and cosmology.
problem Analyzing spherically symmetric spacetimes for stellar collapse and cosmology.
method Obtained results for Weyl and Ricci tensors on general doubly warped spacetimes.
result Friedmann equations deviate from standard FRW cosmology due to electric tensor terms.
A string-net model associates a vector space to a surface in terms of graphs decorated by objects and morphisms of a pivotal fusion category modulo local relations. String-net models are usually considered for spherical fusion categories, and in this case the vector spaces agree with the state spaces of the correspondi…
Study connects contact structures to cone geodesics and contactomorphisms.
problem Understanding contact structures on cone geodesics.
method Review and generalize cone geodesics to contact manifolds, establish correspondence with contactomorphisms.
result Established correspondence between contactomorphisms and cone structures.
New spherical Milnor spaces for diffeological groups with geometric and topological properties.
problem Understanding higher topological structures in diffeological spaces.
method Spherical Milnor construction based on quadratic normalization.
result Provides a natural setting for studying principal bundles with Z2-twists and higher cohomology. Study local features of decorated representation spaces for spherical surfaces.
problem Local structure of moduli space of spherical surfaces with conical points.
method Analysis of decorated representation spaces of fundamental groups in SU(2).
result Smooth locus of decorated representation spaces is dense and connected.
Study spherical twists on K3 surfaces, compute their centers.
problem Understanding autoequivalence groups of K3 surfaces.
method Introduced spherical twists, studied their intersection numbers, and classified subgroups.
result Computed the center of autoequivalence groups of K3 surfaces.
Survey on 4-manifolds with specific curvature properties.
problem Understanding the structure of 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth.
method Analysis of blow-downs and cone-like structures at infinity.
result Manifolds look like cones over spherical space forms at infinity.
It is shown that the qc Yamabe problem has a solution on any compact qc manifold which is non-locally qc equivalent to the standard 3-Sasakian sphere. Namely, it is proved that on a compact non-locally spherical qc manifold there exists a qc conformal qc structure with constant qc scalar curvature
The paper provides examples of geometric transitions in low dimensions.
problem Exploring geometric transitions between different types of structures in low dimensions.
method Explicit examples and computations of transitions from hyperbolic to Euclidean, spherical, and Anti-de Sitter structures.
result Details of elementary computations and techniques are provided to explain geometric transitions.
Bayesian framework for sphere regression using Gaussian fields.
problem Nonparametric regression on the sphere with Gaussian priors.
method Isotropic Gaussian field priors, harmonic structure, exact posterior distributions, optimal spectral truncation, posterior contraction rates.
result Sharp posterior contraction rates for Gaussian priors with polynomially decaying angular power spectra.
The paper constructs examples of coupled Dirac-Yang-Mills pairs on Riemannian manifolds.
problem Constructing coupled Dirac-Yang-Mills pairs on Riemannian manifolds.
method Constructing spherically symmetric Dirac-Yang-Mills pairs on Riemannian 3-manifolds with SU(2) structure group.
result The construction yields coupled solutions, including on S^1(r_1) x S^2(r_2) for certain radii.
A new distribution addresses scalability and numerical stability issues of the vMF.
problem Scalability and numerical stability issues in sampling from the von Mises-Fisher (vMF) distribution.
method Proposes the Power Spherical distribution, retaining vMF's properties but addressing its drawbacks.
result Demonstrates the stability of Power Spherical distributions and applies it to a variational auto-encoder.
Novikov equation symmetries, solutions, and pseudo-spherical surfaces studied.
problem Analyzing geometrically integrable Novikov equation properties.
method Lie symmetries, group-invariant solutions, conservation laws, unique continuation, pseudo-spherical surfaces.
result Classification of invariant solutions and existence of analytic metrics for pseudo-spherical surfaces.
New method uses scalar-based models to approximate spherical tensors efficiently.
problem Efficiently approximating spherical tensors with equivariant functions.
method Expressing equivariant functions as the product of a scalar function and a small tensor basis.
result Approximations are fast, simple to implement, and accurate in practical settings.
Paper explores neural network approximations on sphere domains.
problem Approximating functionals on sphere domains using neural networks.
method Encoder-decoder framework with spherical harmonics for infinite-dimensional domain.
result Approximation rates of neural networks with different encoder structures.
Discrete Laplacians defined for spherical and hyperbolic surfaces.
problem Defining discrete Laplacians for non-Euclidean geometries.
method Definitions close to Euclidean, structure-preserving properties proven.
result Connection between discrete and smooth Laplacians in non-Euclidean settings.
We construct a state-sum type invariant of smooth closed oriented 4-manifolds out of a G-crossed braided spherical fusion category (G-BSFC) for G a finite group. The construction can be extended to obtain a (3+1)-dimensional topological quantum field theory (TQFT). The invariant of 4-manifolds generalizes s…
Study integrability of generalized almost complex structures on S^6.
problem Integrability of generalized almost complex structures on the 6-dimensional sphere.
method Local coordinate criteria for integrability with respect to brackets and Courant integrability for strong structures.
result No nontrivial spherical combinations of the canonical structures are integrable with respect to the Levi-Civita connection.
In this paper, we consider generic corank 2 sub-Riemannian structures, and we show that the Spherical Hausdorf measure is always a C^1-smooth volume, which is in fact generically C^2- smooth out of a stratified subset of codimension 7. In particular, for rank 4, it is generically C^2 . This is the continuation of a pre…
We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles ≤π (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles ≤π, possibly with boundary consisting of totally geodesic hyperbo…
Paper introduces spherical knot mosaics for knot and link invariants.
problem Representing knots on a sphere with tiles.
method Tiling a 2-sphere with 11 knot mosaic tiles to define new invariants.
result New knot invariants derived from spherical mosaic tiling.
The so-called Mom-structures on hyperbolic cusped 3-manifolds without boundary were introduced by Gabai, Meyerhoff, and Milley, and used by them to identify the smallest closed hyperbolic manifold. In this work we extend the notion of a Mom-structure to include the case of 3-manifolds with non-empty boundary that does …
Sharp curvature condition implies spherical space form structure.
problem Characterizing manifolds with specific curvature properties.
method Proving diffeomorphism and homeomorphism to spherical space forms using curvature conditions.
result Closed manifolds with $4 rac{1}{2}$-positive curvature operator of the second kind are spherical space forms.
Study geodesics on spherical polyhedra, estimating their number.
problem Counting simple closed geodesics on spherical polyhedra.
method Examined regular spherical octahedra, cubes, and tetrahedra.
result Estimated the number of simple closed geodesics on spherical polyhedra.
We obtain a branched spherical CR structure on the complement of the figure eight knot with a given holonomy representation (called rho_2). There are essentially two boundary unipotent representations from the complement of the figure eight knot into PU(2,1), we call them rho_1 and rho_2. We make explicit some fundamen…
The abstract proves spherical surface decompositions with conical singularities.
problem Decomposing surfaces with spherical metrics and conical singularities.
method Geometric triangulations and irreducible components of standard shapes.
result Spherical polygons, including half-spherical concave polygons, can be arbitrarily complicated.