Hyperbolic Surfaces Research Articles

Rigidity Theorems by Capacities and Kernels

Abstract For any open hyperbolic Riemann surface $X$, the Bergman kernel $K$, the logarithmic capacity $c_{\beta }$, and the analytic capacity $c_{B}$ satisfy the inequality chain $\pi K \geq c^2_{\beta } \geq c^2_B$. Moreover, equality holds at a single point between any two of the three quantities if and only if $X$ is biholomorphic to a disk possibly less a relatively closed polar set. We extend the inequality chain by showing that $c_{B}^2 \geq \pi v^{-1}(X)$ on planar domains, where $v(\cdot )$ is the Euclidean volume, and characterize the extremal cases when equality holds at one point. Similar rigidity theorems concerning the Szegö kernel, the higher-order Bergman kernels, and the sublevel sets of the Green’s function are also developed. Additionally, we explore rigidity phenomena related to the multi-dimensional Suita conjecture.

First-return maps of Birkhoff sections of the geodesic flow

This paper compares different pseudo-Anosov maps coming from different Birkhoff sections of a given flow. More precisely, given a hyperbolic surface and a collection of periodic geodesics on it, we study those Birkhoff sections for the geodesic flow on the unit bundle to the surface bounded by the collection. We show that there is a canonical identification of all those surfaces, and that the first-return maps induced by the flow can all be expressed as a composition of negative Dehn twists along the same family of curves: only the order depends on the choice of a particular surface.

Moiré-driven reconstitution on electromagnetic energy transfer

The past decade has witnessed an explosion in the field of twistronics concepts, opening the door to numerous extraordinary physical phenomena such as weak ferroelectric charge transfer, superconductivity, topologically protected states, and so on. Here, inspired by the twistronics concepts, we take a moiré three-body system consisting of three metasurfaces supporting highly confined hyperbolic surface plasmons as an example, introducing an analogous concept into the study of near-field electromagnetic energy transfer within three-body system. We demonstrate the possible existence of the moiré transistor effect induced by the twisted hyperbolic repeater in the polariton coupling system (source and drain parallel to each other), where the near-field electromagnetic energy transfer exchanged between the source and the drain can be reversibly switched, amplified, and modulated by action on moiré patterns. Moreover, we point out that for the symmetry-breaking situations (source and drain not parallel to each other), there is also a magic angle in this system and demonstrate that the twisted hyperbolic repeater can act as a moiré compensation effect at this magic angle, mitigating to a certain extent the electromagnetic energy attenuation caused by the malposition between source and drain. Our findings may open new opportunities for the development of near-field thermal energy transport, enabling new perspectives concerning thermal management at the nanoscale, near-field energy conversion, and infrared spectroscopy.

Currents on cusped hyperbolic surfaces and denseness property

The space \operatorname{GC}(\Sigma) of geodesic currents on a hyperbolic surface \Sigma can be considered as a completion of the set of weighted closed geodesics on \Sigma when \Sigma is compact, since the set of rational geodesic currents on \Sigma , which correspond to weighted closed geodesics, is a dense subset of \operatorname{GC}(\Sigma) . We prove that even when \Sigma is a cusped hyperbolic surface with finite area, \operatorname{GC}(\Sigma) has the denseness property of rational geodesic currents, which correspond not only to weighted closed geodesics on \Sigma but also to weighted geodesics connecting two cusps. In addition, we present an example in which a sequence of weighted closed geodesics converges to a geodesic connecting two cusps, which is an obstruction for the intersection number to extend continuously to \operatorname{GC}(\Sigma) . To construct the example, we use the notion of subset currents. Finally, we prove that the space of subset currents on a cusped hyperbolic surface has the denseness property of rational subset currents.

Geodesic Triangles in H² with Short Sides

We prove that if M is an orientable hyperbolic surface without boundary (possibly compact, possibly with infinitely generated fundamental group) and γ is a closed geodesic in M then any side of any triangle formed by distinct lifts of γ in H2 is shorter than γ.

Tunable goniopolarity of graphenelike boron layers in metal diborides

The orientation-dependent conduction polarity, e.g., $p$-type conduction along one direction and $n$-type conduction along another direction, namely ``goniopolarity,'' observed in highly anisotropic materials brought us an alternative concept for novel electronic devices. But goniopolarity has hardly been utilized due to the rarity of goniopolar materials. Here, we associate goniopolarity with the saddle points in electronic band structure and the hyperbolic Fermi surface of materials and highlight the potential of layered metal diboride ($M{\mathrm{B}}_{2}$) materials in achieving this interesting phenomenon using first-principles calculations. We demonstrate that the electron-doped ${\mathrm{MgB}}_{2}$, ${\mathrm{CaB}}_{2}$, and ${\mathrm{SrB}}_{2}$ can exhibit remarkable orientation-dependent conduction polarity with opposite Seebeck coefficients which are detectable in verifying goniopolarity. We attribute the goniopolarity with the ${p}_{x,y}$ orbitals of the boron layers of these materials which constitute anisotropic electronic states along in-plane and out of plane directions. Moreover, their anisotropic Seebeck coefficients can be efficiently enhanced by applying tensile strain. This work offers a promising strategy for design of goniopolar materials and regulation of goniopolarity.

Approximation and Localized Polynomial Frame on Double Hyperbolic and Conic Domains

We study approximation and localized polynomial frames on a bounded double hyperbolic or conic surface and the domain bounded by such a surface and hyperplanes. The main work follows the framework developed recently in Xu (J Funct Anal 281(12):109257, 2021) for homogeneous spaces that are assumed to contain highly localized kernels constructed via a family of orthogonal polynomials. The existence of such kernels will be established with the help of closed-form formulas for the reproducing kernels. The main results provide construction of semi-discrete localized tight frame in weighted $$L^2$$ norm and characterization of best approximation by polynomials on our domains. Several intermediate results, including the Marcinkiewicz–Zygmund inequalities, positive cubature rules, Christoffel functions, and Bernstein inequalities, are shown to hold for doubling weights defined via the intrinsic distance on the domain.

Hyperbolic polyhedral surfaces with regular faces

We study hyperbolic polyhedral surfaces with faces isometric to regular hyperbolic polygons satisfying that the total angles at vertices are at least 2π. The combinatorial information of these surfaces is shown to be identified with that of Euclidean polyhedral surfaces with negative combinatorial curvature everywhere. We prove that there is a gap between areas of non-smooth hyperbolic polyhedral surfaces and the area of smooth hyperbolic surfaces. The numerical result for the gap is obtained for hyperbolic polyhedral surfaces, homeomorphic to the double torus, whose 1-skeletons are cubic graphs.

Observation of Topological p-Orbital Disclination States in Non-Euclidean Acoustic Metamaterials.

Disclinations-topological defects ubiquitously existing in various materials-can reveal the intrinsic band topology of the hosting material through the bulk-disclination correspondence. In low-dimensional materials and nanostructure such as graphene and fullerenes, disclinations yield curved surfaces and emergent non-Euclidean geometries that are crucial in understanding the properties of these materials. However, the bulk-disclination correspondence has never been studied in non-Euclidean geometry, nor in systems with p-orbital physics. Here, by creating p-orbital topological acoustic metamaterials with disclination-induced conic and hyperbolic surfaces, we demonstrate the rich emergent bound states arising from the interplay among the real-space geometry, the bulk band topology, and the p-orbital physics. This phenomenon is confirmed by clear experimental evidence that is consistent with theory and simulations. Our experiment paves the way toward topological phenomena in non-Euclidean geometries and will stimulate interesting research on, e.g., topological phenomena for electrons in nanomaterials with curved surfaces.

Twisted Ruelle zeta function at zero for compact hyperbolic surfaces

Let X be an orientable compact connected hyperbolic surface of genus g. In this paper, we prove that the twisted Selberg and Ruelle zeta functions, associated with an arbitrary finite-dimensional complex representation χ of π1(X) admit a meromorphic continuation to C. Moreover, we study the behavior of the twisted Ruelle zeta function at s=0 and prove that at this point it has a zero of order dim⁡(χ)(2g−2).

Contour Measurement of Object with Arbitrary Surface Using Two-Dimensional Shearography with Source Displacement

A two-dimensional (2D) shearography with source displacement is proposed to measure object contours. Using a dual-shear shearographic setup with two movable laser sources, the full-field slopes along a pair of orthogonal shear directions were obtained. The contour was then obtained by performing 2D integration of the surface slopes. Theoretical derivations and experimental results are presented to demonstrate the performance of the proposed method. The experimental results show that contour of objects with various types of surfaces, such as spherical and hyperbolic paraboloid surfaces, can be effectively measured. The measurement of the contour aids in the precision measurement of strain and the precision location of defects.

Motion of charged particles in spacetimes with magnetic fields of spherical and hyperbolic symmetry

The motion of charged particles in spacetimes containing a submanifold of constant positive or negative curvature is considered, with the electromagnetic tensor proportional to the volume two-form form of the submanifold. In the positive curvature case, this describes spherically symmetric spacetimes with a magnetic monopole, while in the negative curvature case, it is a hyperbolic spacetime with magnetic field uniform along hyperbolic surfaces. Constants of motion are found by considering Poisson brackets defined on a phase space with gauge-covariant momenta. In the spherically-symmetric case, we find a correspondence between the trajectories on the Poincar\'{e} cone with equatorial geodesics in a conical defect spacetime. In the hyperbolic case, the analogue of the Poincar\'{e} cone is defined as a surface in an auxiliary Minkowski spacetime. Explicit examples are solved for the Minkowski, $\mathrm{AdS}_4\times S^2$, and the hyperbolic AdS-Reissner--Nordstr\"{o}m spacetimes.

Shrinking target equidistribution of horocycles in cusps

Consider a shrinking neighborhood of a cusp of the unit tangent bundle of a noncompact hyperbolic surface of finite area, and let the neighborhood shrink into the cusp at a rate of T^{-1} as T rightarrow infty . We show that a closed horocycle whose length ell goes to infinity or even a segment of that horocycle becomes equidistributed on the shrinking neighborhood when normalized by the rate T^{-1} provided that T/ell rightarrow 0 and, for any delta >0, the segment remains larger than max left{ T^{-1/6},left( T/ell right) ^{1/2}right} left( T/ell right) ^{-delta }. We also have an effective result for a smaller range of rates of growth of T and ell . Finally, a number-theoretic identity involving the Euler totient function follows from our technique.

Correlation of the renormalized Hilbert length for convex projective surfaces

AbstractIn this paper, we focus on dynamical properties of (real) convex projective surfaces. Our main theorem provides an asymptotic formula for the number of free homotopy classes with roughly the same renormalized Hilbert length for two distinct convex real projective structures. The correlation number in this asymptotic formula is characterized in terms of their Manhattan curve. We show that the correlation number is not uniformly bounded away from zero on the space of pairs of hyperbolic surfaces, answering a question of Schwartz and Sharp. In contrast, we provide examples of diverging sequences, defined via cubic rays, along which the correlation number stays larger than a uniform strictly positive constant. In the last section, we extend the correlation theorem to Hitchin representations.

Интерферометр для контроля выпуклых гиперболических поверхностей малого диаметра

We used Fourier transforms to perform a theoretical analysis of how the process of interference pattern formation in a laboratory-scale Fizeau laser interferometer is affected by the error in the working wave front. The interferometer is designed to monitor convex hyperbolic surfaces characterised by small diameters and low aperture angles. We indicate their application scope. The interferometer consists of a laser illuminator, a microlens, a beam splitter, that is, a thin plane-parallel plate with a semitransparent coating on its front surface, and a screen. A thin translucent plate located perpendicular to the line connecting the geometric foci of the hyperbolic surface acts as a separator and a reference simultaneously. The paper provides the main equations describing the wave aberration of the working branch and the position of optical elements in the interferometer design. We present an example of computing aberration in the working branch of the interferometer. This calculation reveals that aberrations in the thin plane-parallel plate can be discarded at low aperture angles. The interferometer forms a high-precision optical surface map, that is, interference band distortion caused by the interferometer error does not exceed 0.1 of the distortion caused by the error inherent to the surface being monitored. The interferometer provides highly accurate measurement of hyperbolic surfaces while featuring an obviously simple design

Length statistics of random multicurves on closed hyperbolic surfaces

A multicurve can be decomposed into components. The ratios of the length of each component to the total length give a hint of the shape of the multicurve. In this paper, we determine the distribution of these ratios of a random multicurve with a given topological type on a closed hyperbolic surface, using the methods of Margulis’ thesis and Mirzakhani’s equidistribution theorem for horospheres. This distribution admits a polynomial density whose coefficients can be expressed explicitly in terms of intersection numbers of psi-classes on the Deligne–Mumford compactification of the moduli space of complex curves, and in particular it does not depend on the hyperbolic metric. This result generalizes prior work of M. Mirzakhani in the case of random pants decompositions. Results very close to ours were obtained independently and simultaneously by F. Arana-Herrera.

Symplectic geometry of p-adic Teichmüller uniformization for ordinary nilpotent indigenous bundles

The aim of the present paper is to provide a new aspect of the $p$-adic Teichm\"{u}ller theory established by S. Mochizuki. We study the symplectic geometry of the $p$-adic formal stacks $\widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$ (= the moduli classifying $p$-adic formal curves of fixed genus $g>1$) and $\widehat{\mathcal{S}}_{g, \mathbb{Z}_p}$ (= the moduli classifying $p$-adic formal curves of genus $g$ equipped with an indigenous bundle). A major achievement in the (classical) $p$-adic Teichm\"{u}ller theory is the construction of the locus $\widehat{\mathcal{N}}_{g, \mathbb{Z}_p}^{\mathrm{ord}}$ in $\widehat{\mathcal{S}}_{g, \mathbb{Z}_p}$ classifying $p$-adic canonical liftings of ordinary nilpotent indigenous bundles. The formal stack $\widehat{\mathcal{N}}_{g, \mathbb{Z}_p}^{\mathrm{ord}}$ embodies a $p$-adic analogue of uniformization of hyperbolic Riemann surfaces, as well as a hyperbolic analogue of Serre-Tate theory of ordinary abelian varieties. In the present paper, the canonical symplectic structure on the cotangent bundle $T^\vee_{\mathbb{Z}_p} \widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$ of $\widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$ is compared to Goldman's symplectic structure defined on $\widehat{\mathcal{S}}_{g, \mathbb{Z}_p}$ after base-change by the projection $\widehat{\mathcal{N}}_{g, \mathbb{Z}_p}^{\mathrm{ord}} \rightarrow \widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$. We can think of this comparison as a $p$-adic analogue of certain results in the theory of projective structures on Riemann surfaces proved by S. Kawai and other mathematicians.

Around the Combinatorial Unit Ball of Measured Foliations on Bordered Surfaces

Abstract The volume $\mathcal {B}_{\sum }^{\textrm {comb}}({\mathbb {G}})$ of the unit ball—with respect to the combinatorial length function $\ell _{{\mathbb {G}}}$—of the space of measured foliations on a stable bordered surface $\sum $ appears as the prefactor of the polynomial growth of the number of multicurves on $\sum $. We find the range of $s \in {\mathbb {R}}$ for which $(\mathcal {B}_{\sum }^{\textrm {comb}})^{s}$, as a function over the combinatorial moduli spaces, is integrable with respect to the Kontsevich measure. The results depend on the topology of $\sum $, in contrast with the situation for hyperbolic surfaces where [6] recently proved an optimal square integrability.

Molding Broadband Dispersion in Twisted Trilayer Hyperbolic Polaritonic Surfaces.

Recent advancement in twisted layered metasurfaces can be employed to control the nanoscale flow of light, including the exotic hyperbolic-to-elliptic topological transitions in twisted bilayers (tBL). Such topological transitions can only occur to limited frequency ranges, restricted by the intrinsic in-plane dispersion of individual hyperbolic surfaces. Here, we report that, by controlling interlayer evanescent coupling in twisted polaritonic trilayers, moldable topological transitions of light can be achieved in broadband. We reveal that the required minimum open angle of the individual hyperbolic polaritonic surface for open-to-close topological transitions can be significantly lowered compared to that of the twisted bilayer counterpart. This increases the degree of freedom to enhance and control near-field light-matter interactions and energy management. As an example, we demonstrate a knob to manipulate near-field radiative heat transfer (NFRHT). By rotating the relative angles of trilayers, exotic and tunable thermal conductance can be achieved. Our findings enrich the controllability of light at the nanoscale in broadband, bringing twisted optical materials one step closer to practical applications.

Ergodic optimization for some dynamical systems beyond uniform hyperbolicity

ABSTRACT In Ergodic optimization, one wants to find ergodic measures to maximize or minimize the integral of given continuous functions. This has been succefully studied for uniformly hyperbolic systems for generic continuous functions by Bousch and Brémon. In this paper, we show that for several interesting systems beyond uniform hyperbolicity, any generic continuous function has a unique maximizing measure with zero entropy. In some cases, we also know that the maximizing measure has full support. These interesting systems include singular hyperbolic attractors, surface diffeomorphisms and diffeomorphisms away from homoclinic tangencies.We try to give a uniform mechanism for these non-hyperbolic systems.