Hyperbolic Surfaces Research Articles

Positive definite functions on a regular domain

Abstract We define positive and strictly positive definite functions on a domain and study these functions on a list of regular domains. The list includes the unit ball, conic surface, hyperbolic surface, solid hyperboloid and the simplex. Each of these domains is embedded in a quadrant or a union of quadrants of the unit sphere by a distance-preserving map, from which characterizations of positive definite and strictly positive definite functions are derived for these regular domains.

Unstable minimal surfaces in $\mathbb{R}^{n}$ and in products of hyperbolic surfaces

We prove that every unstable equivariant minimal surface in \mathbb{R}^{n} produces a maximal representation of a surface group into \prod_{i=1}^{n}\mathrm{PSL}(2,\mathbb{R}) together with an unstable minimal surface in the corresponding product of closed hyperbolic surfaces. To do so, we lift the surface in \mathbb{R}^{n} to a surface in a product of \mathbb{R} -trees, then deform to a surface in a product of closed hyperbolic surfaces. We show that instability in one context implies instability in the other two.

The Moduli Space of Twisted Laplacians and Random Matrix Theory

Abstract Rudnick recently proved that the spectral number variance for the Laplacian of a large compact hyperbolic surface converges, in a certain scaling limit and when averaged with respect to the Weil–Petersson measure on moduli space, to the number variance of the Gaussian Orthogonal Ensemble of random matrix theory. In this article we extend Rudnick’s approach to show convergence to the Gaussian Unitary Ensemble for twisted Laplacians that break time-reversal symmetry, and to the Gaussian Symplectic Ensemble for Dirac operators. This addresses a question of Naud, who obtained analogous results for twisted Laplacians on high degree random covers of a fixed compact surface.

Logarithmic systolic growth for hyperbolic surfaces in every genus

More than thirty years ago, Brooks [J. Reine Angew. Math. 390 (1988), pp. 117–129] and Buser–Sarnak [Invent. Math. 117 (1994), pp. 27–56] constructed sequences of closed hyperbolic surfaces with logarithmic systolic growth in the genus. Recently, Liu and Petri [Random surfaces with large systoles, https://arxiv.org/abs/2312.11428, 2023] showed that such logarithmic systolic lower bound holds for every genus (not merely for genera in some infinite sequence) using random surfaces. In this article, we show a similar result through a more direct approach relying on the original Brooks/Buser–Sarnak surfaces.

No bounded geometry wandering domains for sufficiently regular automorphisms

A question whether sufficiently regular manifold automorphisms may have wandering domains with controlled geometry is answered in the negative for quasiconformal or smooth homeomorphisms of n n -tori, n ≥ 2 n\ge 2 , and hyperbolic surfaces. Besides control on geometry of wandering domains, the assumptions are either analytic, e.g., minimal sets having measure zero or supporting invariant conformal structures, or geometric, such as uniform relative separation of wandering domains.

The action of mapping class groups on de Rham quasimorphisms

We study the action of the mapping class group on the subspace of de Rham classes in the degree-two bounded cohomology of a hyperbolic surface. In particular, we show that the only fixed nontrivial finite-dimensional subspace is the one generated by the Euler class. As a consequence, we get that the action of the mapping class group on the space of de Rham quasimorphisms has no fixed points.

The space of vectored hyperbolic surfaces is path-connected

The space of vectored hyperbolic surfaces is path-connected

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.

Casimir energy of hyperbolic orbifolds with conical singularities

In this article, we obtain the explicit expression of the Casimir energy for compact hyperbolic orbifold surfaces in terms of the geometrical data of the surfaces with the help of zeta-regularization techniques. The orbifolds may have finitely many conical singularities. In computing the contribution to the energy from a conical singularity, we derive an expression of an elliptic orbital integral as an infinite sum of special functions. We prove that this sum converges exponentially fast. Additionally, we show that under a natural assumption known to hold asymptotically on the growth of the lengths of primitive closed geodesics of the (2, 3, 7)-triangle group orbifold, its Casimir energy is positive (repulsive).

An upper bound for the number of critical points of the systole function on the moduli space of hyperbolic surfaces

We obtain an upper bound for the number of critical points of the systole function on Mg. Besides, we obtain an upper bound for the number of those critical points whose systole is smaller than a constant.

Estimates of Kähler metrics on noncompact finite volume hyperbolic Riemann surfaces, and their symmetric products

Estimates of Kähler metrics on noncompact finite volume hyperbolic Riemann surfaces, and their symmetric products

Multifractal analysis of homological growth rates for hyperbolic surfaces

Abstract We perform a multifractal analysis of homological growth rates of oriented geodesics on hyperbolic surfaces. Our main result provides a formula for the Hausdorff dimension of level sets of prescribed growth rates in terms of a generalized Poincaré exponent of the Fuchsian group. We employ symbolic dynamics developed by Bowen and Series, ergodic theory and thermodynamic formalism to prove the analyticity of the dimension spectrum.

Topological recursion of the Weil–Petersson volumes of hyperbolic surfaces with tight boundaries

Topological recursion of the Weil–Petersson volumes of hyperbolic surfaces with tight boundaries

Geodesic planes in a geometrically finite end and the halo of a measured lamination

Recent works [22,23,3,33] have shed light on the topological behavior of geodesic planes in the convex core of a geometrically finite hyperbolic 3-manifold M of infinite volume. In this paper, we focus on the remaining case of geodesic planes outside the convex core of M, giving a complete classification of their closures in M.In particular, we show that the behavior is different depending on whether exotic roofs exist or not. Here an exotic roof is a geodesic plane contained in an end E of M, which limits on the convex core boundary ∂E, but cannot be separated from the core by a support plane of ∂E.A necessary condition for the existence of exotic roofs is the existence of exotic rays for the bending lamination. Here an exotic ray is a geodesic ray that has a finite intersection number with a measured lamination L but is not asymptotic to any leaf nor eventually disjoint from L. We establish that exotic rays exist if and only if L is not a multicurve. The proof is constructive, and the ideas involved are important in the construction of exotic roofs.We also show that the existence of geodesic rays satisfying a stronger condition than being exotic, phrased only in terms of the hyperbolic surface ∂E and the bending lamination, is sufficient for the existence of exotic roofs. As a result, we show that geometrically finite ends with exotic roofs exist in every genus. Moreover, in genus 1, when the end is homotopic to a punctured torus, a generic one (in the sense of Baire category) contains uncountably many exotic roofs.

A Hyperbolic Contact Surface Winkler Contact Force Model for Spherical Clearance Joints

Abstract Spherical clearance joints are essential for the successful deployment of space structures. When the clearance is small enough, the contacts will be considered conformal contact, which probably leads to inaccuracies in existing contact force models. To address the limitation, this paper proposes a novel hyperbolic contact surface Winkler model. First, a new fundamental formula incorporating a modified variable exponent is presented. Based on the surrogate modeling method, an optimized surrogate function for the variable coefficient is developed. In the optimization process, the finite element and response surface methods (RSMs) are introduced to improve the precision and reliability of the model. Compared with previous models, this paper organizes a detailed discussion and evaluation to validate the accuracy and application of the new proposed model, after which a dynamic example demonstrates the model's effectiveness. The results highlight the model's accuracy and practical efficacy, showing a strong correlation and minimal margin of error, especially when compared to finite element method (FEM) results. This improvement is attributed to the refined variable exponent, which accurately characterizes the relationship between contact force and penetration depth, and the optimized variable coefficient, which fine-tunes the contact force magnitude. Additionally, the model's versatility extends beyond the geometric properties of the contact bodies, offering a broad application scope. As a foundation of precise impact modeling, it is crucial to address the structural dynamic challenges inherent in high-precision space structures.

Adaptive passive fault tolerant control of DFIG-based wind turbine using a self-tuning fractional integral sliding mode control

Controlling variable wind speed turbine (VWT) system based on a doubly-fed induction generator (DFIG) is a challenging task. It requires a control law that is both adaptable and robust enough to handle the complex dynamics of the closed control loop system. Sliding mode control (SMC) is a robust control technology that has shown good performance when employed as a passive fault-tolerant control for wind energy systems. To improve the closed control loop of VWT based on DFIG with the aim of improving energy efficiency, even in presence of nonlinearities and a certain range of bounded parametric uncertainties, whether electrically or mechanically, an adaptive passive fault tolerant control (AP-FTC) based on a self-tuning fractional integral sliding mode control law (ST-FISMC) developed from a novel hyperbolic fractional surface is proposed in this paper. ST-FISMC introduces a nonlinear hyperbolic function into the sliding manifold for self-tuning adaptation of control law, while fractional integral of the control law smooths discontinuous sign function to reduce chattering. Additionally, this work introduces an adaptive observer, developed and proved based on a chosen Lyapunov function. This observer is designed to estimate variations in electrical parameters and stator flux, ensuring sensorless decoupling in indirect field- oriented control (SI-FOC) of DFIG. Lyapunov theory is also used to prove stability of states vectors in closed control loop with presence of bounded parameters uncertainties or external disturbances. Simulation results show that the proposed approach offers better performance in capturing optimal wind energy, as well as the ability to regulate active/reactive power and high resilience in presence of occurring parameter uncertainties or external disturbances.

Strichartz estimates for the Schrödinger equation on negatively curved compact manifolds

We obtain improved Strichartz estimates for solutions of the Schrödinger equation on negatively curved compact manifolds which improve the classical universal results of Burq, Gérard and Tzvetkov [11] in this geometry. In the case where the spatial manifold is a hyperbolic surface we are able to obtain no-loss Lt,xqc-estimates on intervals of length log⁡λ⋅λ−1 for initial data whose frequencies are comparable to λ, which, given the role of the Ehrenfest time, is the natural analog of the universal results in [11]. We also obtain improved endpoint Strichartz estimates for manifolds of nonpositive curvature, which cannot hold for spheres.

On procongruence curve complexes and their automorphisms

This paper launches the exploration of the procongruence completions for three varieties of curve complexes attached to hyperbolic surfaces, as well as their automorphisms groups. The discrete counterparts of these objects, especially the curve complex and the so-called pants complex were defined long ago and have been the subject of numerous studies. Introducing some form of completions is natural and indeed necessary to lay the ground for a topological version of the Grothendieck–Teichmüller theory. Here several results of foundational nature are stated and proved, among which reconstruction theorems in the discrete and complete settings, which give a graph theoretic characterization of versions of the curve complex as well as a rigidity theorem for the complete pants complex, in sharp contrast with the case of the (complete) curve complex, whose automorphisms actually define a version of the Grothendieck–Teichmüller group, to be studied elsewhere. We work all along with the procongruence completions — and for good reasons — recalling however that the so-called congruence conjecture predicts that this completion should coincide with the full profinite completion.

Process planning for laser peen forming of complex geometry: An analytical-based inverse study

Laser peen forming (LPF) is a promising process for the flexible manufacturing of complex thin-walled structures. However, process planning for LPF remains challenging due to the deformed geometry relying on the cumulative effect of thousands of laser shocks without dies. The optimization-based planning method is adaptable but easily gets trapped in local optima, leading to insufficient robustness and efficiency. To overcome these limitations, this study introduces an analytical-based inverse study to complement the optimization-based method. Firstly, an inherent and straightforward analytical model between eigen-moment and surface curvature is proposed, enabling a direct inverse determination of the eigen-moment field from the desired geometry shape. Subsequently, a clustering method is employed to achieve aggregation control over the analytically determined eigen-moment field, avoiding extensive optimization iterations. Moreover, a physical-based surface decomposition method is devised to formulate double-side LPF strategies for surfaces with negative or zero Gaussian curvature. To validate the proposed analytical-based process planning method, an elliptic paraboloid-like surface and a hyperbolic paraboloid surface are selected as the objectives. Experimental results show high conformity between the deformed and objective surfaces, validating the proposed method. The analytical-based planning method facilitates the efficient and robust determination of forming strategies, complementing the optimization-based method and providing a comprehensive solution for LPF process planning under small deformations.

Projective embedding of stably degenerating sequences of hyperbolic Riemann surfaces

Projective embedding of stably degenerating sequences of hyperbolic Riemann surfaces