The paper studies fundamental domains in H^3 and their associated polyhedra.
problem Understanding the relationship between polyhedra and groups associated with fundamental domains in H^3.
method Analyzes torsion-free groups and edge classes of abstract polyhedra, proving results about group properties and edge classes.
result Classifies fundamental domains on the cube with torsion-free groups and provides insights into polyhedra and groups.
The paper defines fundamental domains for Fuchsian groups in hyperbolic space.
problem Identifying fundamental domains for Fuchsian groups.
method Provides a condition for sets to be fundamental domains via hyperbolic space.
result A necessary and sufficient condition for fundamental domains in mPSL(2,R). Fundamental gap vanishes for convex domains in hyperbolic space.
problem Behavior of fundamental gap in convex domains in hyperbolic space.
method Proof for Laplace operator with Dirichlet boundary conditions.
result Fundamental gap can be arbitrarily small for domains of any diameter.
Research explores hyperbolic space groups and their fundamental domains.
problem Investigating fundamental domains of space groups in hyperbolic spaces.
method Analyzing symmetries of fundamental polyhedra and considering edge conditions.
result Identifies edge conditions for simplicial fundamental domains of Family F12.
Study shows fundamental gap of horoconvex domains in hyperbolic space has no positive lower bound.
problem Understanding the fundamental gap of horoconvex domains in hyperbolic space.
method Analysis of fundamental gap of geodesic balls as radius goes to infinity.
result Product of fundamental gap and square of diameter has no positive lower bound for horoconvex domains.
New convex domains in hyperbolic space can have lower fundamental gap than constant potentials.
problem Finding convex domains with lower fundamental gap than constant potentials.
method Constructing specific convex domains and potentials with controlled eigenfunctions.
result Fundamental gap of −Δ+V can be strictly smaller than −Δ for convex domains. Proves a fundamental gap lower bound for horoconvex domains in hyperbolic space.
problem Proving a fundamental gap lower bound for horoconvex domains in hyperbolic space.
method Reduces the problem to a radial-height problem, compares Dirichlet forms with angular operators, and uses Green estimates.
result Establishes a polynomial \(D^{-3}\) scale fundamental gap lower bound.
Researchers create a fundamental domain for all Deligne-Mostow lattices in PU(2,1).
problem Building a fundamental domain for all Deligne-Mostow lattices in PU(2,1).
method Using Thurston's approach, the team constructed a fundamental domain for lattices with 2-fold symmetry, completing the list of commensurability classes.
result Fundamental domains were created for all commensurability classes of Deligne-Mostow lattices in PU(2,1).
Crooked planes are piecewise linear surfaces that were introduced by Drumm in the early 1990s to construct fundamental domains for properly discontinuous actions of free groups on Minkowski 3-space. In a previous paper, we introduced analogues of these surfaces, called AdS crooked planes, in the 3-dimensional anti-de S…
Researchers find gap between eigenvalues for convex domains in hyperbolic plane.
problem Estimating the gap between the first and second Dirichlet eigenvalues for convex domains in hyperbolic space.
method Computed fundamental gap for a family of convex domains in H2. result For some convex domains, the gap λ2−λ1<D23π2, where D is the diameter. Negative curvature restricts the gap between the first and second eigenvalues of convex domains.
problem The fundamental gap of convex domains is limited by negative curvature.
method Adapted from Bourni et. al. (2022) for Riemannian manifolds with negative sectional curvature.
result The product of the fundamental gap and the square of the diameter can be arbitrarily small in domains with negative curvature.
New fundamental domain for Hilbert-Blumenthal cusp shapes.
problem Understanding the geometry of Hilbert-Blumenthal surfaces at cusps.
method Constructing a fundamental domain using Dirichlet domains and deformations of lattices.
result Explicitly describes the cusp cross section's Sol 3-manifold structure and Anosov diffeomorphism.
We show the existence of isometric (or Ford) fundamental regions for a large class of subgroups of the isometry group of any rank one Riemannian symmetric space of noncompact type. The proof does not use the classification of symmetric spaces. All hitherto known existence results of isometric fundamental regions and do…
The paper defines cocycles for positive Anosov representations and constructs affine actions with bounded fundamental domains.
problem Positive Anosov representations into SO(2n,2n−1). method Definition of cocycles and construction of affine actions with fundamental domains.
result Quotient manifolds are homeomorphic to handlebodies.
The main result of this paper is a construction of fundamental domains for certain group actions on Lorentz manifolds of constant curvature. We consider the simply connected Lie group G~, the universal cover of the group SU(1,1) of orientation-preserving isometries of the hyperbolic plane. The Killing form on the Lie g…
The study shows that the second fundamental form is intrinsic under certain conditions in space forms.
problem Understanding the intrinsic nature of the second fundamental form in space forms.
method Proving the intrinsic nature of the normalized second fundamental form A under specific conditions. result The normalized second fundamental form A is intrinsic if σ2k+1(A)eq0 for some k≥1. The study improves fundamental gap estimates for surfaces with non-constant positive curvature.
problem Estimating the fundamental gap for surfaces with non-constant positive curvature.
method Using a two-point maximum principle, the study establishes log-concavity and fundamental gap estimates.
result Corresponding log-concavity and fundamental gap estimates for surfaces with non-constant positive curvature are derived.
The article proves a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.
problem Proving a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.
method Establishing conformal log-concavity estimates for the first eigenfunction.
result Proves a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.
We review some recent results in the generic rigidity theory of planar frameworks with forced symmetry, giving a uniform treatment to the topic. We also give new combinatorial characterizations of minimally rigid periodic frameworks with fixed-area fundamental domain and fixed-angle fundamental domain.
Method constructs fundamental domains for Picard modular groups.
problem Classify and understand torsion elements in Picard modular groups.
method Systematic construction of coarse fundamental domains.
result Classification of conjugacy classes of torsion elements.
Sharp estimates for heat flow on nonconvex domains.
problem Quantitative estimates for heat flow on nonconvex domains.
method Sharp gradient and transport estimates with novel dependence on time.
result Equivalent characterization of lower bound on second fundamental form.
Algorithm finds Dirichlet domains for hyperbolic surfaces.
problem Computing explicit Dirichlet domains for hyperbolic surfaces.
method Algorithm based on side pairings of fundamental polygons, using geometric-topological data structures.
result Algorithm runs in polynomial time, dependent on surface perimeter and genus.
Study C-Fuchsian subgroups of non-arithmetic lattices.
problem Understand structure and fundamental domains of C-Fuchsian subgroups. method General procedure to analyze structure and show fundamental domains lie on a complex geodesic.
result Fundamental domains of C-Fuchsian subgroups lie on a complex geodesic homeomorphic to the unit disk. Construct Schottky subgroups for maximal representations in symmetric spaces.
problem Maximal representations of surface groups into Sp(2n, R).
method Construction of Schottky type subgroups of automorphism groups.
result Schottky subgroups correspond to maximal representations of surface groups.
In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap (the difference between the first two eigenvalues) conjecture for convex domains in the Euclidean space and conjectured similar results holds for spaces with constant sectional curvature. We prove the conjecture for the sphere. Namely wh…
Lower bounds for quaternionic hyperbolic orbifold volumes found.
problem Finding explicit lower bounds for quaternionic hyperbolic orbifold volumes.
method Using H. C. Wang's radius bound for fundamental domains of semisimple Lie groups.
result Explicit lower bound for quaternionic hyperbolic orbifold volumes depending only on dimension.
In recently published work Maskit constructs a fundamental domain D_g for the Teichmueller modular group of a closed surface S of genus g>1. Maskit's technique is to demand that a certain set of 2g non-dividing geodesics C_{2g} on S satisfies certain shortness criteria. This gives an a priori infinite set of length ine…
Log-concavity of eigenfunctions on curved surfaces is proven, leading to fundamental gap estimates.
problem Proving log-concavity of eigenfunctions on curved surfaces.
method Analyzing the Laplacian eigenfunctions on positively curved surfaces.
result Strong log-concavity of the first eigenfunction on positively curved surfaces.
We will discuss fundamental domains for actions of discrete groups on the 3-dimensional Einstein Universe. These will be bounded by crooked surfaces, which are conformal compactifications of surfaces that arise in the construction of Margulis spacetimes. We will show that there exist pairwise disjoint crooked surfaces …
Paper constructs two series of Lorentz bi-quotients from polyhedra.
problem Creating fundamental domains for Lorentz bi-quotients.
method Explicit construction of polyhedral fundamental domains.
result Two infinite series of Lorentz bi-quotients constructed.
We prove that for every finitely-presented group G there exists a 2-dimensional irreducible complex-projective variety W with the fundamental group G, so that all singularities of W are normal crossings and Whitney umbrellas.
Study characterizes points on projective surfaces using a cubic form.
problem Characterize points on projective surfaces and their impact on Euler characteristic.
method Define local indices for umbilics and godrons, use fundamental cubic form.
result Formulas relating indices to Euler characteristic determine coexistences of points.
Deligne and Mostow constructed a class of lattices in PU(2,1) using monodromy of hypergeometric functions. Later, Thurston reinterpreted them in terms of cone metrics on the sphere. In this spirit we construct a fundamental domain for all lattices with three fold symmetry in Deligne-Mostow list. This is a generalisatio…
Minimal partitions with minimal perimeter found in metric spaces.
problem Finding minimal partitions with minimal perimeter in metric spaces.
method Existence proof and regularity analysis of minimal domains.
result Existence and regularity of minimal partitions in various metric spaces.
In this article we prove a reverse Hölder inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains.
New framework tackles DG under posterior drift, where optimal classifier varies by domain.
problem Generalizing from multiple domains with varying optimal classifiers.
method Decision-theoretic framework for DG under posterior drift.
result Optimal classifier can vary significantly across domains, challenging existing DG approaches.
Researchers extend a gap theorem to convex domains with smaller diameters.
problem Proving a minimum gap between the first two Dirichlet eigenvalues for convex domains.
method Extending a previous result to domains with smaller diameters.
result Extended the gap theorem to convex domains with diameters less than π.
We give a diameter bound for fundamental domains for isometric actions of the fundamental group of a closed hyperbolic surface on a delta-hyperbolic space, where the bound depends on the hyperbolicity constant delta, the genus of the surface, and the injectivity radius of the action, which we assume to be strictly posi…
We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral gap) of a Schrödinger operator with convex potential and Dirichlet boundary data on a convex domain is bounded below by the spectral gap on an interval of the same diameter with zero poten…
Study shows how many domains are needed for generalization, using a new measure called domain shattering dimension.
problem How many domains are needed for domain generalization?
method Introduced a new combinatorial measure called the domain shattering dimension to model domain sample complexity.
result Established a tight quantitative relationship between domain shattering dimension and classic VC dimension.
Part-of-speech (POS) tagging is a fundamental component for performing natural language tasks such as parsing, information extraction, and question answering. When POS taggers are trained in one domain and applied in significantly different domains, their performance can degrade dramatically. We present a methodology f…
Estimates the mass gap for domains with integral Ricci curvature bounds.
problem Estimating the mass gap for domains with specific curvature conditions.
method Proving domains are John domains to estimate the first nonzero Neumann eigenvalue.
result Fundamental gap estimates for domains with integral Ricci curvature bounds.
Unique foliation of AdS3 domains by constant mean curvature surfaces.
problem Foliation of domains of dependence in AdS3. method Proving existence and uniqueness of constant mean curvature foliation.
result Existence and uniqueness of foliation by constant mean curvature surfaces.
Study concavity of solutions to elliptic equations under conformal deformations.
problem Establish concavity estimates for the principle eigenfunction of weighted Schrödinger operators.
method Analyzing the Dirichlet problem for the weighted Schrödinger operator \[-Δu + Vu = λρu\] with conformal connections.
result Partial resolution of Nguyen's conjecture on fundamental gap of horoconvex domains and power convexity estimate for solutions in spherical geometry.
The spectral properties of p-forms on the fundamental domains of regular tesselations of the d-dimensional sphere are discussed. The degeneracies for all ranks, p, are organised into a double Poincare series which is explicitly determined. In the particular case of coexact forms of rank (d-1)/2, for odd d, it is shown …
The study solves open problems in complex geometry by analyzing bounded domains with finite-volume quotients.
problem Analyzing bounded pseudoconvex domains with finite-volume quotients in complex geometry.
method Using semi-simplicity of automorphism groups and applying results to specific settings.
result The automorphism group of certain domains is discrete, and domains with specific properties are biholomorphic to the unit ball.
For a standard Finsler metric F on a manifold M, its domain is the whole tangent bundle TM and its fundamental tensor g is positive-definite. However, in many cases (for example, the well-known Kropina and Matsumoto metrics), these two conditions are relaxed, obtaining then either a pseudo-Finsler metric (with arbitrar…
Method addresses label shift in adversarial domain adaptation.
problem Label shift in behavioral studies.
method DATS (Domain Adversarial nets for Target Shift) framework.
result DATS framework performs well under large label shift.