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Wang and Ye [Adv. Math. 230 (2012), pp. 294–320] prove a Hardy–Moser–Trudinger inequality in dimension two which improves both the classical Moser–Trudinger inequality and the classical Hardy inequality in the unit disc. In this paper, we generalize their result both to the higher dimensional unit ball and to the singular weighted cases as well. More precisely, we prove that \[ \sup _{\substack {u\in W^{1,n}_0(\mathbb {B}^n),\\ \int _{\mathbb {B}^n} |\nabla u|^n dx -\left (\frac {2(n-1)}n\right)^n \int _{\mathbb {B}^n} \frac {|u|^n}{(1-|x|^2)^n} dx \leq 1}}\int _{\mathbb {B}^n} e^{(1-\frac \beta n)\alpha _n |u|^{\frac n{n-1}}} |x|^{-\beta } dx \leq C \] for any \beta \in [0,n), n\geq 3 where \alpha _n = n \omega _{n-1}^{\frac 1{n-1}} and \omega _{n-1} is the surface area of the n-1 dimensional unit sphere. The proof of Wang and Ye is based on the blow-up analysis method which seems not work in the higher dimensions. In this paper, we propose a new approach based on the method of transplantation of Green's functions to prove our inequality. As a consequence, we obtain a singular Moser–Trudinger inequality in the hyperbolic spaces which confirms affirmatively a conjecture by Mancini, Sandeep and Tintarev [Adv. Nonlinear Anal., 2 (2013), pp. 309–324]. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish the higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane using the Helgason-Fourier analysis on symmetric spaces. A crucial part of our work is to establish appropriate factorization theorems on these spaces, which can be of independent interest. To this end, we need to identify and introduce the "quaternionic Geller operators" and the "octonionic Geller operators", which have been absent on these spaces. Combining the factorization theorems and the Geller type operators with the Helgason-Fourier analysis on symmetric spaces, some precise estimates for the heat and the Bessel-Green-Riesz kernels, and the Kunze-Stein phenomenon for connected real simple groups of real rank one with finite center, we succeed to establish the higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane. The kernel estimates required to prove these inequalities are also sufficient to establish the Adams and Hardy-Adams inequalities on these spaces. This paper, together with our earlier works on real and complex hyperbolic spaces, completes our study of the factorization theorems, higher order Poincaré-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on all rank one symmetric spaces of noncompact type. [ABSTRACT FROM AUTHOR]

We intend to continue our previous papers on dense ball packing hyperbolic space H³ by equal balls, but here with centres belonging to different orbits of the fundamental group Cw(2z, 3 ≤ z ∈ N, odd number), of our new series of tube or cobweb manifolds Cw = H³/Cw with z-rotational symmetry. As we know, Cw is a fixed-point-free isometry group, acting on H³ discontinuously with appropriate tricky fundamental domain Cw, so that every point has a ball-like neighbourhood in the usual factor-topology. Our every Cw(2z) is minimal, i.e. does not cover regularly a smaller manifold. It can be derived by its general symmetry group W(u; v;w = u) that is a complete Coxeter orthoscheme reflection group, extended by the half-turn h (0 ↔ 3, 1 ↔ 2) of the complete orthoscheme A0A1A2A3 ∼ b0b1b2b3 (see Figure 1). The vertices A0 and A3 are outer points of the (Beltrami-Cayley-Klein) B-C-K model of H³, as 1/u+1/v ≤ 1/2 is required, 3 ≤ u = w, v for the above orthoscheme parameters. For the above simple manifold-construction we specify u = v = w = 2z. Then the polar planes a0 and a3 of A0 and A3, respectively, make complete with reflections a0 and a3 the Coxeter reflection group, where the other reflections are denoted by b0, b¹, b², b³ in the sides of the orthoscheme b0b1b2b3. The situation is described first in Figure 1 of the half trunc-orthoscheme W and its usual extended Coxeter diagram, moreover, by the scalar product matrix (bij) = (⟨bi, bj⟩) in formula (1) and its inverse (Ajk) = (⟨Aj, Ak⟩) in (3). These will describe the hyperbolic angle and distance metric of the half trunc-orthoscheme W, then its ball packings, densities, then those of the manifolds Cw(2z). As first results we concentrate only on particular constructions by computer for probable material model realizations, atoms or molecules by equal balls, for general W(u; v;w = u) as well, summarized at the end of our paper. [ABSTRACT FROM AUTHOR]

Graph neural networks (GNNs) are powerful tools for data mining on graph-structured data in various domains, such as social science, finance, and biology. However, most existing GNNs operate in Euclidean space and may fail to preserve the intrinsic network properties, such as self-similarity and hierarchy, that characterize many real-world graphs. Hyperbolic graph neural networks (HGNNs) address this limitation by embedding graphs into hyperbolic space, which can better capture the hierarchical structures of networks. However, HGNNs often involve complex computations in hyperbolic space or its tangent space during training, which may hinder their efficiency. In this paper, we propose hyperbolic attention graph neural networks (HA-GNN), which can leverage both network structure and node features for graph representation learning in an efficient way. Specifically, we design a structural properties attention mechanism that measures the structural connection between nodes based on their hyperbolic embeddings. We also design a node features attention mechanism that quantifies the feature similarity between nodes. We then combine these two attentions to obtain a hyperbolic attention that weights the relevance between all connected nodes. We conduct extensive experiments on five real-world networks and demonstrate that our model consistently and significantly outperforms other state-of-the-art methods. For example, on the Cora network, our model achieves an accuracy of 83.1 (± 0.4) on node classification tasks, which is 1.6% higher than the best baseline method in Euclidean space. [ABSTRACT FROM AUTHOR]

In this paper, we study the following prescribed Gaussian curvature problem: K = f ~ ⁢ (θ) ϕ ⁢ (ρ) α − 2 ⁢ ϕ ⁢ (ρ) 2 + | ∇ ¯ ⁢ ρ | 2 , a generalization of the Alexandrov problem ( α = n + 1 ) in hyperbolic space, where f ~ is a smooth positive function on S n , 휌 is the radial function of the hypersurface, ϕ ⁢ (ρ) = sinh ⁡ ρ and 퐾 is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when α ≥ n + 1 . Our argument provides a parabolic proof in smooth category for the Alexandrov problem in H n + 1 . We also consider the cases 2 < α ≤ n + 1 under the evenness assumption of f ~ and prove the existence of solutions to the above equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability of the volume preserving mean curvature flow of closed hypersurfaces in the hyperbolic space. We prove that an L 2 -almost umbilical hypersurface will be deformed to a totally umbilical hypersurface along the flow. Our result removes the assumption on the mean curvature in the theorems of Huang-Lin-Zhang [Peking J. Math. (2023)] and Leng-Xu-Zhao [Int. J. Math. (2014)]. [ABSTRACT FROM AUTHOR]

We consider the question of whether the high-energy eigenfunctions of certain Schrödinger operators on the d-dimensional hyperbolic space of constant curvature −κ2 are flexible enough to approximate an arbitrary solution of the Helmholtz equation Δh + h = 0 on R d , over the natural length scale O ( λ − 1 / 2 ) determined by the eigenvalue λ ≫ 1. This problem is motivated by the fact that, by the asymptotics of the local Weyl law, approximate Laplace eigenfunctions do have this approximation property on any compact Riemannian manifold. In this paper we are specifically interested in the Coulomb and harmonic oscillator operators on the hyperbolic spaces H d (κ). As the dimension of the space of bound states of these operators tends to infinity as κ ↘ 0, one can hope to approximate solutions to the Helmholtz equation by eigenfunctions for some κ > 0 that is not fixed a priori. Our main result shows that this is indeed the case, under suitable hypotheses. We also prove a global approximation theorem with decay for the Helmholtz equation on manifolds that are isometric to the hyperbolic space outside a compact set, and consider an application to the study of the heat equation on H d (κ). Although global approximation and inverse approximation results are heuristically related in that both theorems explore flexibility properties of solutions to elliptic equations on hyperbolic spaces, we will see that the underlying ideas behind these theorems are very different. [ABSTRACT FROM AUTHOR]

This paper introduces two novel scores for detecting local perturbations in networks. For this, we consider a non-Euclidean representation of networks, namely, their embedding onto the Poincaré disk model of hyperbolic geometry. We numerically evaluate the performances of these scores for the detection and localization of perturbations on homogeneous and heterogeneous network models. To illustrate our approach, we study latent geometric representations of real brain networks to identify and quantify the impact of epilepsy surgery on brain regions. Results suggest that our approach can provide a powerful tool for representing and analyzing changes in brain networks following surgical intervention, marking the first application of geometric network embedding in epilepsy research. [ABSTRACT FROM AUTHOR]

Actions on hyperbolic metric spaces are an important tool for studying groups, and so it is natural, but difficult, to attempt to classify all such actions of a fixed group. In this paper, we build strong connections between hyperbolic geometry and commutative algebra in order to classify the cobounded hyperbolic actions of numerous metabelian groups up to a coarse equivalence. In particular, we turn this classification problem into the problems of classifying ideals in the completions of certain rings and calculating invariant subspaces of matrices. We use this framework to classify the cobounded hyperbolic actions of many abelian‐by‐cyclic groups associated to expanding integer matrices. Each such action is equivalent to an action on a tree or on a Heintze group (a classically studied class of negatively curved Lie groups). Our investigations incorporate number systems, factorization in formal power series rings, completions, and valuations. [ABSTRACT FROM AUTHOR]

In this paper, we prove sharp isoperimetric inequalities for lower order eigenvalues of Neumann Laplacian on bounded domains in both compact and noncompact rank-1 symmetric spaces. Our results generalize the work of Wang and Xia for bounded domains in the hyperbolic space (Xia and Wang in Math Ann 385(1–2):863–879, 2023), and Szegö–Weinberger inequalities in rank-1 symmetric spaces obtained by Aithal and Santhanam (Trans Am Math Soc 348(10):3955–3965, 1996). [ABSTRACT FROM AUTHOR]

In this short paper, we derive a new result on Umehara algebra. As a consequence, we prove that an indefinite complex hyperbolic space and an indefinite complex projective space do not share a common complex submanifold with induced metrics, answering a question raised in Cheng et al. [ABSTRACT FROM AUTHOR]

In the first part of this paper, we consider a partially overdetermined mixed boundary value problem in space forms and generalize the main result in [11] to the case of general domains with partial umbilical boundary in space forms. Precisely, we prove that a partially overdetermined problem in a domain with partial umbilical boundary admits a solution if and only if the rest part of the boundary is also part of an umbilical hypersurface. In the second part of this paper, we prove a Heintze–Karcher–Ros-type inequality for embedded hypersurfaces with free boundary lying on a horosphere or an equidistant hypersurface in the hyperbolic space. As an application, we show an Alexandrov-type theorem for constant mean curvature hypersurfaces with free boundary in these settings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of principal angles between subspaces of the same signature in a (real, complex or quaternionic) pseudo-euclidean space of index 1. We show that these determine the relative position of an important class of pairs of hyperbolic and elliptic subspaces (Theorems 3.9 and 4.10). Also, as an application, we will see that these angles can be used to study the relative position of pairs of an important class of totally geodesic submanifolds of the (real, complex or quaternionic) hyperbolic space (Theorem 6.3). As a consequence we obtain a nice interpretation of these principal angles as geometric invariants of the hyperbolic space. [ABSTRACT FROM AUTHOR]

The problem of the prescribed curvature measure is one of the important problems in differential geometry and nonlinear partial differential equations. In this paper, we consider the prescribed curvature measure problem in the hyperbolic space. We obtain the existence of star‐shaped k‐convex bodies with prescribed (n‐k)‐th curvature measures (k

For bounded domains of Euclidean spaces with piecewise smooth boundary, the integral of outward unit normal vectors of the boundary is zero. In this paper we consider a similar theorem on Minkowski spaces (Minkowski spaces does not mean finite dimensional Banach spaces but finite dimensional vector spaces with pseudo-inner products). We also consider Gauss' divergence theorem on Minkowski spaces, which implies above. Remark that this theorem implies easily the equation to calculate a kind of centroids of hyperbolic simplices. [ABSTRACT FROM AUTHOR]

The object of the present paper is to characterize two classes of almost Kenmotsu manifolds admitting almost η-Ricci solitons. In this context we have shown that in a (k, μ) and (k, μ)′-almost Kenmotsu manifold admitting an almost η-Ricci soliton the curvature conditions (i) the manifold is Einstein, (ii) the manifold is Ricci symmetric (∇S = 0), (iii) the manifold is Ricci semisymmetric (R · S = 0) and (iv) the manifold is projective Ricci semisymmetric (P · S = 0) are equivalent. Also we have shown that the curvature condition Q · P = 0 in a (k, μ)-almost Kenmotsu manifold admitting an almost η-Ricci soliton holds if and only if the manifold is locally isometric to the hyperbolic space H2n+1(-1) and if a (k, μ)′-almost Kenmotsu manifold admitting an almost η-Ricci soliton satisfies the curvature condition Q · R = 0, then it is locally isometric to the Riemannian product Hn+1(-4) × Rn. [ABSTRACT FROM AUTHOR]

A classical result of Cheng states that the bottom spectrum of complete manifolds of fixed dimension and Ricci curvature lower bound achieves its maximal value on the corresponding hyperbolic space. The paper establishes an analogous result for three-dimensional complete manifolds with scalar curvature lower bound subject to some necessary topological assumptions. The rigidity issue is also addressed and a splitting theorem is obtained for such manifolds with the maximal bottom spectrum. [ABSTRACT FROM AUTHOR]

In this paper, we consider the α -Gauss curvature flow for complete convex graphs over horosphere in the hyperbolic space. We show that for all positive power α > 0 , if the initial hypersurface is smooth, complete non-compact uniformly convex graph over R n and bounded by two horospheres, then the solution of the flow exists for all time. Moreover, the evolution of horospheres act as barriers along the flow. [ABSTRACT FROM AUTHOR]

This paper proposes a novel 3D point cloud segmentation network called EDGCNet. Structurally, the network combines the encoder–decoder structure and graph convolution to improve the processing efficiency and the exploration of point-to-point correlation. In addition, it contains two sub-modules: the dynamic hyperbaric graph convolution module and the dual squeeze-and-attention module. To mitigate the issue of incorrect inference in existing graph convolution networks for multi-category hybrid regions, the dynamic hyperbolic graph convolution module maps the features captured in the Euclidean space to the hyperbolic space to aggregate the neighborhoods and dynamically update the edge weights. Compared to Euclidean geometry, hyperbolic embedding with learnable curvature can deeply fit the geometric topology of point cloud by transforming the manifold shape to enrich feature representation. The dual squeeze-and-attention module recalibrates the original features by modeling the correlation between channels from both global and local perspectives to improve the feature shift caused by the deepening of the convolution. Empirical experiments on three public datasets verify that EDGCNet has an excellent segmentation effect on both artificial shapes and real scenes. Furthermore, ablation studies and confirmatory experiments show that the modules in EDGCNet are contributing, low-complexity and robust. • Our network covers the advantages of encoder–decoder structure and graph convolution network. • We are the first to use dynamic hyperbolic manifolds to represent the correlation of 3D point clouds. • The proposed network EDGCNet can monitor and correct the biased features in time. • We improve Feature Propagation model which always be ignored by others. • Empirical experiments show that EDGCNet has excellent segmentation performance. [ABSTRACT FROM AUTHOR]