Hyperbolic Manifolds Research Articles

Tubes in complex hyperbolic manifolds

We prove a tubular neighborhood theorem for an embedded complex geodesic in a complex hyperbolic 2-manifold where the width of the tube depends only on the Euler characteristic χ \chi of the embedded complex geodesic. We give an explicit estimate for this width. We supply two applications of the tubular neighborhood theorem. The first is a lower volume bound for such manifolds. The second is an upper bound on the first eigenvalue of the Laplacian in terms of the geometry of the manifold. Finally, we prove a geometric combination theorem for two C \mathbb {C} -Fuchsian subgroups of PU ⁡ ( 2 , 1 ) \operatorname {PU}(2,1) . Using this combination theorem, we show that the optimal width size of a tube about an embedded complex geodesic is asymptotically bounded between 1 | χ | \frac {1}{|\chi |} and 1 | χ | \frac {1}{\sqrt {|\chi |}} .

On possible limit functions on a Fatou component in non-autonomous iteration

Abstract The possibilities for limit functions on a Fatou component for the iteration of a single polynomial or rational function are well understood and quite restricted. In non-autonomous iteration, where one considers compositions of arbitrary polynomials with suitably bounded degrees and coefficients, one should observe a far greater range of behavior. We show this is indeed the case and we exhibit a bounded sequence of quadratic polynomials which has a bounded Fatou component on which one obtains as limit functions every member of the classical Schlicht family of normalized univalent functions on the unit disc. The proof is based on quasiconformal surgery and the use of high iterates of a quadratic polynomial with a Siegel disc which closely approximate the identity on compact subsets. Careful bookkeeping using the hyperbolic metric is required to control the errors in approximating the desired limit functions and ensure that these errors ultimately tend to zero.

SKT and Gauduchon hyperbolic compact complex manifolds

SKT and Gauduchon hyperbolic compact complex manifolds

Conformally invariant random fields, Liouville quantum gravity measures, and random Paneitz operators on Riemannian manifolds of even dimension

AbstractFor large classes of even‐dimensional Riemannian manifolds , we construct and analyze conformally invariant random fields. These centered Gaussian fields , called co‐polyharmonic Gaussian fields, are characterized by their covariance kernels k which exhibit a precise logarithmic divergence: . They share a fundamental quasi‐invariance property under conformal transformations. In terms of the co‐polyharmonic Gaussian field , we define the Liouville Quantum Gravity measure, a random measure on , heuristically given as and rigorously obtained as almost sure weak limit of the right‐hand side with replaced by suitable regular approximations . In terms on the Liouville Quantum Gravity measure, we define the Liouville Brownian motion on and the random GJMS operators. Finally, we present an approach to a conformal field theory in arbitrary even dimension with an ansatz based on Branson's ‐curvature: we give a rigorous meaning to the Polyakov–Liouville measure and we derive the corresponding conformal anomaly. The set of admissible manifolds is conformally invariant. It includes all compact 2‐dimensional Riemannian manifolds, all compact non‐negatively curved Einstein manifolds of even dimension, and large classes of compact hyperbolic manifolds of even dimension. However, not every compact even‐dimensional Riemannian manifold is admissible. Our results concerning the logarithmic divergence of the kernel rely on new sharp estimates for heat kernels and higher order Green kernels on arbitrary closed manifolds.

Strip deformations of decorated hyperbolic polygons

AbstractIn this paper, we study the hyperbolic and parabolic strip deformations of ideal (possibly once‐punctured) hyperbolic polygons whose vertices are decorated with horoballs. We prove that the interiors of their arc complexes parametrise the open convex set of all uniformly lengthening infinitesimal deformations of the decorated hyperbolic metrics on these surfaces, motivated by the work of Danciger–Guéritaud–Kassel. We also give a version of this result for the undecorated ideal polygons and once‐punctured ideal polygons.

Constant Gaussian curvature foliations and Schläfli formulae of hyperbolic 3-manifolds

Constant Gaussian curvature foliations and Schläfli formulae of hyperbolic 3-manifolds

Rigidity for inscribed radius estimate of asymptotically hyperbolic Einstein manifold

The inscribed radius of a compact manifold with boundary is bounded above if its Ricci curvature and mean curvature are bounded from below. The rigidity result implies that the upper bound can be achieved only in space form. In this paper, we generalize this result to asymptotically hyperbolic (AH) Einstein manifold. We get an upper bound of the relative volume of AH manifold and if we combine it with the recent work of Wang and Zhou, then the rigidity is obtained.

Some counterexamples in surface homology

We present four counterexamples in surface homology. The first example shows that even if the loops inducing a homology basis intersect each other at most once, they still may separate the surface into two parts. The other three examples show some difficulties in working with minimal homology bases. Introducing hyperbolic ribbon graphs we modify the examples so as to have a hyperbolic metric.

Volume and Euler classes in bounded cohomology of transformation groups

Abstract Let $M$ be an oriented smooth manifold and $\operatorname{Homeo}\!(M,\omega )$ the group of measure preserving homeomorphisms of $M$ , where $\omega$ is a finite measure induced by a volume form. In this paper, we define volume and Euler classes in bounded cohomology of an infinite dimensional transformation group $\operatorname{Homeo}_0\!(M,\omega )$ and $\operatorname{Homeo}_+\!(M,\omega )$ , respectively, and in several cases prove their non-triviality. More precisely, we define: • Volume classes in $\operatorname{H}_b^n(\operatorname{Homeo}_0\!(M,\omega ))$ , where $M$ is a hyperbolic manifold of dimension $n$ . • Euler classes in $\operatorname{H}_b^2(\operatorname{Homeo}_+(S,\omega ))$ , where $S$ is an oriented closed hyperbolic surface. We show that Euler classes have positive norms for any closed hyperbolic surface and volume classes have positive norms for all hyperbolic surfaces and certain hyperbolic $3$ -manifolds; hence, they are non-trivial.

Expansiveness, generators and Lyapunov exponents for random bundle transformations

We generalize Fathi's results by showing that a compact metrizable space admits an fiber expansive homeomorphism if and only if it has a compatible hyperbolic metric. Moreover, we prove that a compact metrizable space admits an fiber expansive homeomorphism if and only if it has a generator in detail. Furthermore, we show that a fiber expansive homeomorphism has finite fiber topological entropy. Finally, we show that fiber Lyapunov exponents for a fiber expansive system are different from zero, indicating that the system presents a chaotic system. Meanwhile, we also prove that negative fiber Lyapunov exponents for compact invariant sets of a dynamical system imply that the compact set is a fiber attractor.

Hyperbolic prototype rectification for few-shot 3D point cloud classification

Few-shot point cloud classification is a challenging task in 3D computer vision and has received widespread attention from researchers. Most of the works on deep learning models rely heavily on Euclidean spatial metrics. However, point cloud objects often have complex non-Euclidean geometric structures, with underlying inter/intra-class hierarchical structures, which are difficult to capture by current Euclidean-based deep learning models. Moreover, due to the lack of training samples, many few-shot learning methods often suffer from the overfitting problem. Given the Hyperbolic metric of non-Euclidean geometry offering hierarchical structural prior, as we assume to be able to assist FSL task, we propose Hyperbolic Prototype Rectification (HPR) for few-shot point cloud classification, without requiring extra learnable parameter. Firstly, point clouds are embedded into hyperbolic space to better describe hierarchical similarity relationships in data. Secondly, the HPR utilizes hyperbolic spatial and distributional information to enhance the feature representation and improve the generalization capability, with more appropriate hyperbolic prototypes. The few-shot classification experiments and further ablation studies conducted on widely used point cloud datasets demonstrate the effectiveness of our method. On the real-world ScanObjectNN(-PB) datasets, the average classification accuracy outperforms the SOTA method by 2.08%(0.66%), respectively, indicating that the proposed HPR has great generalization capability and strong robustness against perturbed data. Our code is available at: https://github.com/Jonathan-UCAS/HPR.

Rigidity of flat holonomies

Abstract We prove that the existence of one horosphere in the universal cover of a closed Riemannian manifold of dimension $n \geq 3$ with strongly $1/4$ -pinched or relatively $1/2$ -pinched sectional curvature, on which the stable holonomy along one horosphere coincides with the Riemannian parallel transport, implies that the manifold is homothetic to a real hyperbolic manifold.

Uniqueness and Non-Uniqueness Results for Spacetime Extensions

Abstract Given a function $f: A \to{\mathbb{R}}^{n}$ of a certain regularity defined on some open subset $A \subseteq{\mathbb{R}}^{m}$, it is a classical problem of analysis to investigate whether the function can be extended to all of ${\mathbb{R}}^{m}$ in a certain regularity class. If an extension exists and is continuous, then certainly it is uniquely determined on the closure of $A$. A similar problem arises in general relativity for Lorentzian manifolds instead of functions on ${\mathbb{R}}^{m}$. It is well-known, however, that even if the extension of a Lorentzian manifold $(M,g)$ is analytic, various choices are in general possible at the boundary. This paper establishes a uniqueness condition for extensions of globally hyperbolic Lorentzian manifolds $(M,g)$ with a focus on low regularities: any two extensions that are anchored by an inextendible causal curve $\gamma : [-1,0) \to M$ in the sense that $\gamma $ has limit points in both extensions must agree locally around those limit points on the boundary as long as the extensions are at least locally Lipschitz continuous. We also show that this is sharp: anchored extensions that are only Hölder continuous do in general not enjoy this local uniqueness result.

Quasineutral multistability in an epidemiological-like model for defective-helper betacoronavirus infection in cell cultures

It is well known that, during replication, RNA viruses spontaneously generate defective viral genomes (DVGs). DVGs are unable to complete an infectious cycle autonomously and depend on coinfection with a wild-type helper virus (HV) for their replication and/or transmission. The study of the dynamics arising from a HV and its DVGs has been a longstanding question in virology. It has been shown that DVGs can modulate HV replication and, depending on the strength of interference, result in HV extinctions or self-sustained persistent fluctuations. Extensive experimental work has provided mechanistic explanations for DVG generation and compelling evidences of HV-DVGs virus coevolution. Some of these observations have been captured by mathematical models. Here, we develop and investigate an epidemiological-like mathematical model specifically designed to study the dynamics of betacoronavirus in cell culture experiments. The dynamics of the model is governed by several degenerate normally hyperbolic invariant manifolds given by quasineutral planes - i.e., filled by equilibrium points. Three different quasineutral planes have been identified depending on parameters and involving: (i) persistence of HV and DVGs; (ii) persistence of non-infected cells and DVG-infected cells; and (iii) persistence of DVG-infected cells and DVGs. Key parameters involved in these scenarios are the maximum burst size (B), the fraction of DVGs produced during HV replication (β), and the replication advantage of DVGs (δ). More precisely, in the case 0<B<1+β the system displays tristability, where all three scenarios are present. In the case 1+β<B<1+β+δ this tristability persists but attracting scenario (ii) is reduced to a well-defined half-plane. For B>1+β+δ, the scenario (i) becomes globally attractor. Scenarios (ii) and (iii) are compatible with the so-called self-curing since the HV is removed from the population. Sensitivity analyses indicate that model dynamics largely depend on DVGs production rate (β) and their replicative advantage (δ), and on both the infection rates and virus-induced cell deaths. Finally, the model has been fitted to single-passage experimental data using an artificial intelligence methodology based on genetic algorithms and key virological parameters have been estimated.

Semi-symmetric types on hyperbolic Kenmotsu manifolds

In this paper, we give some necessary conditions for an hyperbolic Kenmotsu manifolds M satisfying semi-symmetric, Ricci semi-symmetric, Weyl semi-symmetric and Einstein semi-symmetric. Next, we study the notion of n-Ricci soliton in context of hyperbolic Kenmotsu 3-manifolds M3 and we construct an example to illustrate the existence of n-Ricci soliton. Finally, we present some conditions to M admitting Codazzi type of Ricci tensor and cyclic n-recurrent Ricci tensor.

Discontinuity in RG flows across dimensions: entanglement, anomaly coefficients and geometry

We study the entanglement entropy associated with a holographic RG flow from AdS7 to AdS4 × ℍ3, where ℍ3 is a 3-dimensional hyperbolic manifold with curvature κ. The dual six-dimensional RG flow is disconnected from Lorentz-invariant flows. In this context we address various notions of central charges and identify a monotonic candidate c-function that captures IR aspects of the flow. The UV behavior of the holographic entanglement entropy and, in particular its universal term, display an interesting dependence on the curvature, κ. We then contrast our holographic results with existing field theory computations in six dimensions and find a series of new corrections in curvature to the universal term in the entanglement entropy.

On the interplay between boundary conditions and the Lorentzian Wetterich equation

In the framework of the functional renormalization group and of the perturbative, algebraic approach to quantum field theory (pAQFT), in D’Angelo et al. [Ann. Henri Poinc. 25 (2024) 2295–2352] it has been derived a Lorentzian version of a flow equation à la Wetterich, which can be used to study nonlinear, quantum scalar field theories on a globally hyperbolic spacetime. In this work, we show that the realm of validity of this result can be extended to study interacting scalar field theories on globally hyperbolic manifolds with a timelike boundary. By considering the specific examples of half-Minkowski spacetime and of the Poincaré patch of Anti-de Sitter, we show that the form of the Lorentzian Wetterich equation is strongly dependent on the boundary conditions assigned to the underlying field theory. In addition, using a numerical approach, we are able to provide strong evidences that there is a qualitative and not only a quantitative difference in the associated flow and we highlight this feature by considering Dirichlet and Neumann boundary conditions on half-Minkowski spacetime.

From constraints fusion to manifold optimization: A new directional transport manifold metaheuristic algorithm

The ascent of geometry-based models and methodologies, exemplified by geometric deep learning and manifold numerical optimization algorithms, has inaugurated a novel domain across various applications that grapple with geometric data complexities, such as electroencephalogram signals represented by symmetric positive definite matrix manifold, hierarchical data represented by hyperbolic manifold. The imperative fuels this inevitable paradigm shift to encapsulate the intricacies and richness inherent in data, areas where traditional methods prove inadequate. While metaheuristic algorithms are renowned for their versatile adaptability across applications, offering practical solutions within reasonable timeframes. However, the conventional metaheuristic algorithms fail on manifold applications with meaningless solutions. From an extrinsic optimization perspective, we treat manifold optimization problems as general optimization problems with multiple fused constraints that limit the optimization path to the manifold. This study pioneered the proposal and implementation of a metaheuristic manifold optimization, introducing a novel directional transport operator to rectify previously identified issues. Through experimentation across five sets of 25 problems, comparing against five algorithms, including both gradient-free and gradient-dominant counterparts, our proposed algorithm emerges as the optimal performer within the gradient-free category, demonstrating competitiveness even against gradient-dominant algorithms. Furthermore, we applied the proposed algorithm to the robot dynamic manipulation problem, achieving a close-optimal solution that eludes gradient-dominant approaches. This paper delves into the inherent capabilities and establishes the generalization of a metaheuristic algorithm within non-Euclidean functional landscapes. The source code will be available at https://github.com/lingping-fuzzy/metaheuristic-manifold-optimization.

A two parameter family of lightcone-like hyperbolic string vertices

We introduce a two parameter family of string field theory vertices, which we refer to as hyperbolic Kaku vertices. It is defined in terms of hyperbolic metrics on the Riemann surface, but the geometry is allowed to depend on inputs of the states. The vertices are defined for both open and closed strings. In either case, the family contains the hyperbolic vertices. Then we show that the open string lightcone vertex is obtained as the flat limit of the hyperbolic Kaku vertices. The open string Kaku vertices, which interpolate between the Witten vertex and the open string lightcone vertex, is also obtained as a flat limit. We use the same limit on the case of closed strings to define the closed string Kaku vertices: a one parameter family of vertices that interpolates between the polyhedral vertices — which are covariant, but not cubic — and the closed string lightcone vertex — which is cubic, but not Lorentz covariant.

Existence and stability of a quasi-periodic two-dimensional motion of a self-propelled particle

The mechanism of self-propelled particle motion has attracted much interest in mathematical and physical understanding of the locomotion of living organisms. In a top-down approach, simple time-evolution equations are suitable for qualitatively analyzing the transition between the different types of solutions and the influence of the intrinsic symmetry of systems despite failing to quantitatively reproduce the phenomena. We aim to rigorously show the existence of the rotational, oscillatory, and quasi-periodic solutions and determine their stabilities regarding a canonical equation proposed by Koyano et al. (J Chem Phys 143(1):014117, 2015) for a self-propelled particle confined by a parabolic potential. In the proof, the original equation is reduced to a lower dimensional dynamical system by applying Fenichel’s theorem on the persistence of normally hyperbolic invariant manifolds and the averaging method. Furthermore, the averaged system is identified with essentially a one-dimensional equation because the original equation is O(2)-symmetric.