Improved curvature behavior on hyperbolic surfaces.
problem Understanding curvature behavior on hyperbolic surfaces.
method Analyzing Ricci flows on almost-hyperbolic surfaces.
result Gauss curvature remains close to hyperbolic value for exponentially long time.
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.
New barycenters defined for hyperbolic balls, differing from spheres.
problem Defining barycenters in hyperbolic geometry.
method Introducing conformal and holomorphic barycenters.
result Holomorphic and conformal barycenters differ in hyperbolic balls.
Two geodesic balls maximize the third Neumann eigenvalue in hyperbolic space.
problem Maximizing the third eigenvalue of the Neumann Laplacian in hyperbolic space.
method Using the disjoint union of two geodesic balls to prove maximality.
result The third eigenvalue is maximal for the union of two geodesic balls.
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 G⊂PU(n,1) acting on complex hyperbolic space. Consequently the volume of all complex hyperbolic n-manifolds is bounded below by the volume of this bal…
Constructs minimal annuli with free boundary in hyperbolic 3-space.
problem Finding minimal surfaces with boundary in hyperbolic geometry.
method Constructs families of non-rotational minimal annuli with shared symmetry.
result Bifurcates from hyperbolic catenoids, forming a countable collection.
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.
Study on eigenvalue rate of geodesic balls in asymptotically hyperbolic Einstein manifolds.
problem Rate of decrease of the first Dirichlet eigenvalue of geodesic balls.
method Investigation of eigenvalues in asymptotically hyperbolic Einstein manifolds with nonnegative Yamabe type conformal infinity.
result Two-term asymptotic of eigenvalues is the same as in hyperbolic space for nonnegative Yamabe type conformal infinity.
The study examines 3D combinatorial flow in hyperbolic geometry, proving conditions for ball packings and convergence.
problem Analyzing 3D combinatorial Yamabe flow in hyperbolic geometry.
method Investigates triangulations and ball packings with vanishing combinatorial scalar curvature.
result Conditions for real or virtual ball packings and convergence of the flow.
In this paper, we construct an asymptotically hyperbolic metric with scalar curvature -6 on unit ball D3, which contains multiple horizons.
Geodesic balls are isoperimetric in hyperbolic spaces with certain densities.
problem Proving isoperimetric properties in hyperbolic spaces with specific densities.
method Using geodesic balls and radial, strictly log-convex densities.
result Geodesic balls are isoperimetric in real hyperbolic space HRn. The study constructs free boundary CMC annuli in spherical and hyperbolic balls.
problem Finding free boundary CMC annuli in spherical and hyperbolic balls.
method Constructing free boundary CMC annuli with constant mean curvature H in geodesic balls of S^3 and H^3.
result Embedded free boundary CMC annuli exist for certain mean curvatures in both spaces.
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.
Study proves uniqueness of certain minimal surfaces in spherical and hyperbolic spaces.
problem Proving uniqueness of free boundary minimal annuli in geodesic balls.
method Using Steklov problem frequency and antipodal map invariance.
result Minimal annuli are congruent to a critical rotational annulus.
Study compares eigenvalues on spherically symmetric manifolds to Euclidean balls.
problem Comparing eigenvalues on spherically symmetric manifolds to Euclidean balls.
method Examines Dirichlet Laplace eigenvalues on balls of spherically symmetric manifolds and Euclidean space.
result Eigenvalues on spherically symmetric manifolds are smaller for small radii, but larger for hyperbolic spaces.
Uniform comparison of hyperbolic ball volumes on universal cover.
problem Comparing hyperbolic ball volumes on universal cover.
method Proving a constant δ_n exists such that for any metric g on M, if the volume ratio is less than δ_n, the volume of hyperbolic balls is at least as large as in hyperbolic space.
result Every Riemannian metric g on M with a specific volume ratio satisfies the volume of hyperbolic balls on the universal cover is at least as large as in hyperbolic space.
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.
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.
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.
Estimates for geodesics on hyperbolic tori improve previous bounds.
problem Counting simple closed geodesics on hyperbolic tori.
method McShane-Rivin norm balls and Markoff numbers.
result The number of simple closed geodesics of length exactly L≥2 is at most CX(logL)2. If (M^n, g) is a complete Riemannian manifold with filling radius at least R, then we prove that it contains a ball of radius R and volume at least c(n)R^n. If (M^n, hyp) is a closed hyperbolic manifold and if g is another metric on M with volume at most c(n)Volume(M,hyp), then we prove that the universal cover of (M,g…
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.
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.
We equip the whole tangent space TM to a hyperbolic manifold M (of constant sectional curvature -1) with a natural metric in an intrinsic way, so that the isometries of M extend to isometries of TM by holomorphic continuation. The image to the tangent space to a geodesic is equivalent to a hyperbolic disk. In t…
The paper pinches conditions for minimal surfaces in hyperbolic space and hemisphere.
problem Characterizing minimal surfaces with free boundary in hyperbolic space and hemisphere.
method Pinching condition involving second fundamental form, support function, and potential function.
result Characterization of totally geodesic disk and rotational annulus.
A new framework for hyperbolic neural networks using the Klein model is introduced.
problem Previous works focused on Poincaré and hyperboloid models, neglecting the Klein model.
method Formulation of operations using the Klein model, study of the Klein linear layer, and comparison with Poincaré ball model.
result The Klein HNN performs similarly to the Poincaré ball model, offering a third option.
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.
We show if M is a closed, connected, orientable, hyperbolic 3-manifold with Heegaard genus g then g >= 1/2 cosh(r) where r denotes the radius of any isometrically embedded ball in M. Assuming an unpublished result of Pitts and Rubinstein improves this to g >= 1/2 cosh(r) + 1/2. We also give an upper bound on the volume…
Reformulates Wasserstein autoencoder for hyperbolic latent space.
problem Learning structured latent representations on non-Euclidean manifolds.
method Uses Poincaré ball model of hyperbolic space for latent space structure.
result Competitive results on graph link prediction task.
Finite quotients of fibered hyperbolic 3-manifold groups detect taut polynomials.
problem Detecting taut polynomials of fibered faces of Thurston norm balls
method Developing a framework for profinite invariance of twisted multivariable Alexander polynomials
result Proving finite quotients detect taut polynomials
Paper shows approximate Dirichlet domain works as well as exact one for tiling hyperbolic balls.
problem Empirical success of SnapPea's length spectrum algorithm despite using approximate data.
method Showed under certain conditions, approximate Dirichlet domain can perform equivalently to exact one.
result Empirical success of SnapPea's length spectrum algorithm explained.
In this paper, we prove uniform lower bounds on the volume growth of balls in the universal covers of Riemannian surfaces and graphs. More precisely, there exists a constant δ>0 such that if (M,hyp) is a closed hyperbolic surface and h another metric on M with $\area(M,h)\leq δ\area(M,hyp)$ then for every radiu…
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.
Calculations of Orlicz cohomology for simple manifolds and related inequalities.
problem Understanding Orlicz cohomology and inequalities for basic manifolds.
method Calculations of Orlicz cohomology for real line, hyperbolic plane, and ball; discussion of inequalities.
result Relationship between Orlicz cohomology and Poincaré--Sobolev--Orlicz inequalities.
New mappings solve long-standing problems in 3-space.
problem Longstanding problems for bounded locally homeomorphic quasiregular mappings in 3-space.
method Constructed bounded locally homeomorphic quasiregular mappings in the unit 3-ball using non-trivial hyperbolic cobordisms.
result Solved long-standing problems for quasiregular mappings in 3-space.
We derive an explicit lower bound on the radius of a ball embedded in a quaternionic hyperbolic manifold.
Improves few-shot learning for hierarchical data using hyperbolic space.
problem Few-shot class-incremental learning for hierarchical data.
method Contrastive learning in hyperbolic space, Poincaré ball model, hyperbolic contrastive loss, maximum entropy distribution.
result Effective improvement of coarse and fine class accuracies in few-shot conditions.
Study free boundary minimal submanifolds in geodesic balls in hyperbolic and spherical spaces.
problem Characterize free boundary minimal submanifolds in geodesic balls of hyperbolic and spherical spaces.
method Define and analyze functionals related to critical metrics and spectral indices.
result Critical metrics of defined functionals arise from free boundary minimal immersions in geodesic balls of hyperbolic and spherical spaces.
Enhances neural networks in hyperbolic space for better data structure capture.
problem Capturing hierarchical data structures efficiently.
method Unified hyperbolic model for neural network components.
result Superior parameter efficiency and outperformance over Euclidean methods.
Study hyperbolic 2-spheres with cone points, describing spaces for n=3.
problem Characterize the space of hyperbolic 2-spheres with cone points.
method Analyzing the space C(a0,a1,…,an) for n=3 and n=4. result Detailed description of spaces for n=3 and examples for n=4. New tube manifolds model hyperbolic crystallography with dense ball packings.
problem Finding dense ball packings in hyperbolic space by specific tube manifolds.
method Using tube or cobweb manifolds $Cw = \HYP/\BCw$ with z-rotational symmetry, derived from Coxeter orthoscheme reflection groups. result Derived minimal tube manifolds Cw(2z) that are not covered by smaller manifolds, with dense ball packings. Study Bergman kernels on complex hyperbolic cusps, generalizing previous results.
problem Localization of Bergman kernels on Kähler manifolds with complex hyperbolic cusps.
method Revisiting Tian's peak section method, applying to Kähler-Einstein metrics and quotients of complex balls.
result Partial localization result for Poincaré type cusps.
The paper studies nonlocal isoperimetric problems in hyperbolic space and finds unique minimizers for small volumes.
problem Nonlocal isoperimetric problem in hyperbolic space.
method Investigates minimization of a functional with perimeter and a nonlocal term derived from the negative power of distance.
result Geodesic balls are unique minimizers for small volumes in hyperbolic space.
We establish sharp Sobolev inequalities of order four on Euclidean d-balls for d greater than or equal to four. When d=4, our inequality generalizes the classical second order Lebedev-Milin inequality on Euclidean 2-balls. Our method relies on the use of scattering theory on hyperbolic d-balls. As an application, we ch…
Sharp estimate on harmonic maps at conformal points in balls.
problem Estimating harmonic maps at conformal points in balls.
method Sharp estimate on differential norm using Schwarz-Pick lemma.
result Generalizes classical Schwarz-Pick lemma and gives optimal for n≥3. 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 KX+αD for α∈(41,1). result First examples of bielliptic ball quotient compactifications are produced.
MuRP embeds multi-relational graphs in hyperbolic space for better hierarchical representation.
problem Current hyperbolic models struggle with multi-relational knowledge graphs that exhibit multiple hierarchies.
method MuRP embeds multi-relational graph data in the Poincaré ball model of hyperbolic space, learning relation-specific parameters for entity embeddings.
result MuRP embeddings outperform Euclidean counterparts and other methods on link prediction tasks, especially at lower dimensions.
The paper finds a relationship between hyperbolic manifold volumes and cycle areas.
problem Finding a relationship between volumes of hyperbolic manifolds and areas of cycles.
method Using a theorem of Guth and Kronheimer-Mrowka's scalar curvature theorem, the paper establishes a proportional relationship between the volumes of unit balls in the Riemannian universal cover and the area of a specific homology class.
result A lower bound on the area of the Z2--homology class [Σimes∗] on ΣimesS1 is proportional to the hyperbolic area of Σ.