arXiv research

Sharp lower bound found for geodesic ball eigenvalues.

A new framework for hyperbolic neural networks using the Klein model is introduced.

Optimal Poincaré constant estimates on manifolds with ends.

We exhibit an explicit one-parameter smooth family of Poincaré-Einstein metrics on the even-dimensional unit ball whose conformal infinities are the Berger spheres. Our construction is based on a Gibbons-Hawking-type ansätz of Page and Pope. The family contains the hyperbolic metric, converges to the complex hyperbolic…

We carry out calculations of Orlicz cohomology for some basic Riemannian manifolds (the real line, the hyperbolic plane, the ball). Relationship between Orlicz cohomology and Poincaré--Sobolev--Orlicz-type inequalities is discussed.

We first study holomorphic isometries from the Poincaré disk into the product of the unit disk and the complex unit $n$-ball for $n\ge 2$. 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 …

Hyperbolic embeddings have recently gained attention in machine learning due to their ability to represent hierarchical data more accurately and succinctly than their Euclidean analogues. However, multi-relational knowledge graphs often exhibit multiple simultaneous hierarchies, which current hyperbolic models do not c…

The paper establishes lower bounds for the relative volume of Poincaré-Einstein manifolds.

This work presents a reformulation of the recently proposed Wasserstein autoencoder framework on a non-Euclidean manifold, the Poincaré ball model of the hyperbolic space. By assuming the latent space to be hyperbolic, we can use its intrinsic hierarchy to impose structure on the learned latent space representations. W…

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.

Enhances neural networks in hyperbolic space for better data structure capture.

The study improves Bochner inequality on Finsler manifolds to derive important inequalities.

Improves few-shot learning for hierarchical data using hyperbolic space.

We prove a sharp Poincaré inequality for subsets $Ω$ of (essentially non-branching) metric measure spaces satisfying the Measure Contraction Property $\textrm{MCP}(K,N)$, whose diameter is bounded above by $D$. This is achieved by identifying the corresponding one-dimensional model densities and a localization argument…

In this note we prove the existence of infinitely many positive conformal classes on $S^7$ which cannot be the conformal infinity of a Poincaré-Einstein metric on the ball $B^8$. 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.

This paper classifies ball quotients of the complex projective plane.

No radial balanced metrics found on Kepler manifold unit ball with mild boundary conditions.

Study Bergman kernels on complex hyperbolic cusps, generalizing previous results.

The variational auto-encoder (VAE) is a popular method for learning a generative model and embeddings of the data. Many real datasets are hierarchically structured. However, traditional VAEs map data in a Euclidean latent space which cannot efficiently embed tree-like structures. Hyperbolic spaces with negative curvatu…

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.

We are concerned with the discovery of hierarchical relationships from large-scale unstructured similarity scores. For this purpose, we study different models of hyperbolic space and find that learning embeddings in the Lorentz model is substantially more efficient than in the Poincaré-ball model. We show that the prop…

The paper introduces boundary operators for Poincaré-Einstein manifolds and proves higher order trace inequalities.

An important tool in the study of conformal geometry, and the AdS/CFT correspondence in physics, is the Fefferman-Graham expansion of conformally compact Einstein metrics. We show that conformally compact metrics satisfying a generalization of the Einstein equation, Poincare-Lovelock metrics, also have Fefferman-Graham…

We prove the existence of a $C^{1,1}$ 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 $C^2$ conformal compactification.

Study of Riemannian geometry on quaternionic unit ball linked to Sp(1,1) group.

We re-visit the eigenvalue estimate of the Dirac operator on spin manifolds with boundary in terms of the first eigenvalues of conformal Laplace operator as well as the conformal mean curvature operator. These problems were studied earlier by Hijazi-Montiel-Zhang and Raulot and we re-prove them under weaker assumption …

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…

The aim of this paper is to establish two fundamental measure-metric properties of particular random geometric graphs. We consider $\varepsilon$-neighborhood graphs whose vertices are drawn independently and identically distributed from a common distribution defined on a regular submanifold of $\mathbb{R}^K$. We show t…

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…

We prove a spanning result for vector-valued Poincaré series on a bounded symmetric domain. We associate a sequence of holomorphic automorphic forms to a submanifold of the domain. When the domain is the unit ball in ${\Bbb{C}}^n$, we provide estimates for the norms of these automorphic forms and we find asymptotics of…

Perelman's proof confirmed, new method uses 4D topology.

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 $Γ\backslash H^{n+1}$, 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 $H^{n+1}$ in terms of t…

The paper establishes inequalities for minimal graphs on manifolds with nonnegative Ricci curvature.

Matrix Factorization (MF) is a common method for generating recommendations, where the proximity of entities like users or items in the embedded space indicates their similarity to one another. Though almost all applications implicitly use a Euclidean embedding space to represent two entity types, recent work has sugge…

APo-VAE generates text in hyperbolic space for better hierarchical representation.

The study explores Hesse manifolds and their symmetries in multifield cosmological models.

Explicit formulas for fractional GJMS operators on hyperbolic spaces and inequalities proved.

Sharp estimate on harmonic maps at conformal points in balls.

Poincaré-Hopf theorem extended to Filippov vector fields on 2D manifolds.

A formula calculates the Euler class of foliations using dual graphs.

The paper extends Thurston's method to new variants of Mather-Thurston theorem.

Let $Σ$ be a $k$-dimensional complete proper minimal submanifold in the Poincaré ball model $B^n$ of hyperbolic geometry. If we consider $Σ$ as a subset of the unit ball $B^n$ in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold $Σ$ and the ideal boundary $\partial_\infty Σ$, say $\…

Paper refines Alesker-Bernig-Schuster theorem, proving Hodge-Riemann relations for Euclidean balls.

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 …