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.
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.
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.
Optimal Poincaré constant estimates on manifolds with ends.
problem Estimating the Poincaré constant on manifolds with ends.
method Heat kernel estimates extended to manifolds with ends, focusing on central balls.
result The Poincaré constant is determined by the second largest end.
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.
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.
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.
We first study holomorphic isometries from the Poincaré disk into the product of the unit disk and the complex unit n-ball for n≥2. On the other hand, we observe that there exists a holomorphic isometry from the product of the unit disk and the complex unit n-ball into any irreducible bounded symmetric domain …
The paper establishes lower bounds for the relative volume of Poincaré-Einstein manifolds.
problem Lower bounds for the relative volume of Poincaré-Einstein manifolds.
method Fractional Yamabe constants of the boundary provide lower bounds for the relative volume.
result Explicit lower bounds for the relative volume of Poincaré-Einstein manifolds are derived.
Poincar{é} and Sobolev inequalities for differential forms on Heisenberg balls, involving Rumin's differentials, are given. Furthermore, a global homotopy of Rumin's complex which improves differentiability of Rumin forms is provided on any bounded geometry contact manifold.
The study improves Bochner inequality on Finsler manifolds to derive important inequalities.
problem Improving Bochner inequality on Finsler manifolds to derive new inequalities.
method Using improved Bochner inequality and its integrated form, the study derives a sharp Poincaré-Lichnerowicz inequality, a new proof for logarithmic Sobolev inequality, and an estimate of geodesic ball volumes.
result Derivation of new inequalities and estimates on Finsler manifolds.
In this note we prove the existence of infinitely many positive conformal classes on S7 which cannot be the conformal infinity of a Poincaré-Einstein metric on the ball B8. We also prove a sharp inequality between the Yamabe invariant of the conformal infinity and the Yamabe invariant of the interior (after a sui…
The paper classifies energy-minimizing sets in specific domains.
problem Classifying volume-constraint local energy-minimizing sets.
method Proved a Poincaré-type inequality for stable sets.
result Relative boundary of energy-minimizing sets is smooth.
Study of Poincare-Lovelock metrics on conformally compact manifolds.
problem Understanding Poincare-Lovelock metrics in conformally compact geometry.
method Fefferman-Graham expansion and Lovelock equation analysis.
result Existence of fillings for conformal classes near round sphere.
Sharp Poincaré inequality proved for specific metric spaces.
problem Proving a sharp Poincaré inequality for certain metric measure spaces.
method Identifying model densities and using localization arguments, without assuming geodesic convexity.
result Best possible Poincaré constant as a function of parameters.
Study automorphic forms on bounded domains, proving spanning results and estimating norms.
problem Understanding automorphic forms on bounded symmetric domains and their norms.
method Proving spanning results for vector-valued Poincaré series and analyzing holomorphic automorphic forms.
result Found different asymptotic behaviors of norms for certain submanifolds.
This paper classifies ball quotients of the complex projective plane.
problem Understanding the structure of the complex projective plane as a ball quotient.
method Analyzing the branch locus as a line arrangement and smooth normal-crossing curves.
result The orbifold structure of (P2,D) is isomorphic to either the Deligne-Mostow example or a certain degree 9 cover. No radial balanced metrics found on Kepler manifold unit ball with mild boundary conditions.
problem Finding radial balanced metrics on the unit ball of the Kepler manifold.
method Analyzing boundary behavior and weights of metrics.
result Explicit weights for radial metrics satisfying balanced condition identified.
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.
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 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.
Poincaré VAEs improve hierarchical data representation.
problem Hierarchical data structures are difficult to represent in Euclidean latent spaces.
method Introducing Poincaré ball model of hyperbolic geometry as a latent space for VAEs.
result Better generalization and hierarchical structure recovery in hyperbolic space.
In this paper we study global Poincare inequalities on balls in a large class of sub-Riemannian manifolds satisfying the generalized curvature dimension inequality introduced by F.Baudoin and N.Garofalo. As a corollary, we prove the uniqueness of solutions for the subelliptic heat equation. Our results apply in particu…
Study improves Poincaré-Sobolev inequalities for differential forms.
problem Improving Sobolev space embeddings for differential forms.
method Utilizes Lq,p-cohomology and bi-Lipschitz images to estimate embedding norms. result Estimates for embedding norms in Euclidean balls and their images.
The paper introduces boundary operators for Poincaré-Einstein manifolds and proves higher order trace inequalities.
problem Proving higher order trace inequalities on Poincaré-Einstein manifolds.
method Introducing conformally covariant boundary operators and using them to set up higher order Dirichlet problems.
result Sharp higher order Sobolev trace inequalities on the ball are obtained.
The paper re-evaluates eigenvalue estimates and rigidity of Poincare-Einstein metrics.
problem Eigenvalue estimates and rigidity of Poincare-Einstein metrics on spin manifolds.
method Revisits eigenvalue estimates of the Dirac operator and proves rigidity results under weaker conditions.
result Poincaré-Einstein metrics are rigid under specific eigenvalue conditions.
The paper proves properties for random graphs based on geometric submanifolds.
problem Establishing measure-metric properties of random geometric graphs.
method Analyzing ε-neighborhood graphs with specific conditions on submanifold and distribution. result Volume doubling and local Poincaré inequalities hold for random geometric graphs with high probability.
We prove the existence of a C1,1 conformally compact Einstein metric on the ball that has asymptotic sectional curvature decay to −1 plus terms of order e−2r where r is the distance from any fixed compact set. This metric has no C2 conformal compactification.
Study of Riemannian geometry on quaternionic unit ball linked to Sp(1,1) group.
problem Understanding the geometry induced by slice Riemannian metric.
method Developed Lie theoretic study, computed isometry group, compared with quaternionic Poincaré geometry.
result Isometry group of slice Riemannian metric is built from symmetries of Sp(1,1) group.
We study the linearization of the Dirichlet-to-Neumann map for Poincaré-Einstein metrics in even dimensions on an arbitrary compact manifold with boundary. By fixing a suitable gauge, we make the linearized Einstein equation elliptic. In this gauge the linearization of the Dirichlet-to-Neumann map appears as the scatte…
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…
Representation learning has become an invaluable approach for learning from symbolic data such as text and graphs. However, while complex symbolic datasets often exhibit a latent hierarchical structure, state-of-the-art methods typically learn embeddings in Euclidean vector spaces, which do not account for this propert…
Perelman's proof confirmed, new method uses 4D topology.
problem Confirming the classical Poincaré conjecture.
method 4D topology, spun torus-knots, ribbonness, disk-chord system, Bing's result.
result Homotopy 3-sphere is diffeomorphic to the 3-sphere.
For each k > 0 we find an explicit function f_k such that the topology of S inside the ball B(p,r) is `bounded' by f_k(r) for every complete Riemannian surface (compact or noncompact) with K\geq -k^2, every point p on the surface, and every r. Using this result, we obtain a characterization (simple to check in practica…
For geometrically finite hyperbolic manifolds Γ\Hn+1, we prove the meromorphic extension of the resolvent of Laplacian, Poincaré series, Einsenstein series and scattering operator to the whole complex plane. We also deduce the asymptotics of lattice points of Γ in large balls of Hn+1 in terms of t…
New adaptive optimization methods for Riemannian manifolds improve training of complex models.
problem Adapting popular adaptive optimization methods to Riemannian manifolds.
method Generalized Adam, Adagrad, and Amsgrad to product Riemannian manifolds.
result Improved convergence and lower train loss on complex embedding tasks.
The paper establishes inequalities for minimal graphs on manifolds with nonnegative Ricci curvature.
problem Establishing inequalities for minimal graphs on manifolds with nonnegative Ricci curvature.
method Combining Cheeger-Colding theory and geometric measure theory to derive Sobolev and Neumann-Poincaré inequalities.
result Gradient estimates and Liouville theorem for minimal graphs over manifolds with nonnegative Ricci curvature.
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. Efficiently learns high-quality hierarchical embeddings in hyperbolic space.
problem Discovering hierarchical relationships from large-scale similarity scores.
method Used the Lorentz model of hyperbolic geometry to learn embeddings efficiently.
result The proposed approach yields high-quality embeddings that improve over Poincaré embeddings, especially in low dimensions.
Poincaré-Hopf theorem extended to Filippov vector fields on 2D manifolds.
problem Extending Poincaré-Hopf theorem to Filippov vector fields.
method Introducing new index definition for Filippov vector fields, including singularities.
result Established a variant of Hairy Ball Theorem for Filippov vector fields.
A formula calculates the Euler class of foliations using dual graphs.
problem Calculating the Euler class of foliations using cooriented branched surfaces.
method Using dual graphs of cooriented branched surfaces to define a simplicial 1-cycle representing the Poincaré dual of the Euler class.
result The formula generalizes previous results and classifies realizable homology classes.
The paper extends Thurston's method to new variants of Mather-Thurston theorem.
problem Proving new variants of Mather-Thurston theorem for PL homeomorphisms and contactomorphisms.
method Generalizing Thurston's technique to prove new variants of Mather-Thurston theorem for PL homeomorphisms and contactomorphisms.
result The paper answers questions posed by Gelfand-Fuks and Greenberg on PL foliations and Rybicki on contactomorphisms.
Explicit formulas for fractional GJMS operators on hyperbolic spaces and inequalities proved.
problem Proving explicit formulas and inequalities for fractional operators on hyperbolic spaces.
method Scattering theory on hyperbolic space, Helgason-Fourier analysis, and special function analysis.
result Sharp constants in fractional Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities coincide with Euclidean space constants.
Paper refines Alesker-Bernig-Schuster theorem, proving Hodge-Riemann relations for Euclidean balls.
problem Understanding translation-invariant valuations and their geometric implications.
method Explicit construction of highest weight vectors and analysis of natural operations on these vectors.
result Proof of Hodge-Riemann relations for Euclidean balls, extending geometric inequalities.
The study explores Hesse manifolds and their symmetries in multifield cosmological models.
problem Understanding symmetries in multifield cosmological models.
method Analyzes Hesse functions and their properties on Riemannian manifolds.
result Complete Hesse manifolds are characterized by their index and are hyperbolic.
Given (M, g0) we consider the problem -ε^2Delta_{g0+h}u + u = (u+)^{p-1} with (ε, h) \in (0, ε0) \times Bρ. Here Bρ is a ball centered at 0 with radius ρ in the Banach space of all Ck symmetric covariant 2-tensors on M. Using the Poincaré polynomial of M, we give an estimate on the number of nonconstant solutions with …
APo-VAE generates text in hyperbolic space for better hierarchical representation.
problem Lack of hierarchical structure in Euclidean embeddings for natural language.
method Adversarial Poincare Variational Autoencoder (APo-VAE) in hyperbolic latent space.
result APo-VAE outperforms Euclidean VAEs in capturing latent language hierarchies.
This work is a short, self-contained introduction to subriemannian geometry with special emphasis on Chow's Theorem. As an application, a regularity result for the Poincaré Lemma is presented. At the beginning, the definitions of a subriemannian geometry, horizontal vector fields and horizontal curves are given. Then t…