Compact Hyperbolic Riemann Surface Research Articles

Higgs bundles in the Hitchin section over non‐compact hyperbolic surfaces

AbstractLet be an arbitrary non‐compact hyperbolic Riemann surface, that is, not the complex plane or punctured plane. Let be , or . We show that, on a Higgs bundle in the ‐Hitchin section over , there always exists a harmonic metric satisfying certain nice properties. Moreover, when the Higgs field has bounded spectral curve with respect to the complete hyperbolic metric on , such a harmonic metric is unique. As a result, we obtain a new proof of the existence and uniqueness of a harmonic metric for Higgs bundles in the ‐Hitchin section over a compact hyperbolic Riemann surface. Moreover, we show an existence result of harmonic metrics for a more general family of Higgs bundles that admit a full holomorphic filtration.

Liouville Action for Harmonic Diffeomorphisms

In this paper, we introduce a Liouville action for a harmonic diffeomorphism from a compact Riemann surface to a compact hyperbolic Riemann surface of genus $g\ge 2$. We derive the variational formula of this Liouville action for harmonic diffeomorphisms when the source Riemann surfaces vary with a fixed target Riemann surface.

Energy distribution of harmonic 1-forms and Jacobians of Riemann surfaces with a short closed geodesic

We study the energy distribution of harmonic 1-forms on a compact hyperbolic Riemann surface S where a short closed geodesic is pinched. If the geodesic separates the surface into two parts, then the Jacobian variety of S develops into a variety that splits. If the geodesic is nonseparating then the Jacobian degenerates. The aim of this work is to get insight into this process and give estimates in terms of geometric data of both the initial surface S and the final surface, such as its injectivity radius and the lengths of geodesics that form a homology basis. The Jacobians in this paper are represented by Gram period matrices. As an invariant we introduce new families of symplectic matrices that compensate for the lack of full dimensional Gram-period matrices in the noncompact case.

Isometric Embeddings of Subsets of Boundaries of Teichmüller Spaces of Compact Hyperbolic Riemann Surfaces

It is known that every finitely unbranched holomorphic covering π:$$\pi :\tilde{S} \to S$$ of a compact Riemann surface S with genus g ≥ 2 induces an isometric embedding $${{\rm{\Phi}}_{\rm{\pi}}}:Teich\left(S \right) \to Teich\left({\widetilde{S}} \right)$$. By the mutual relations between Strebel rays in Teich(S) and their embeddings in $$Teich\left({\widetilde{S}} \right)$$, we show that the 1st-strata space of the augmented Teichmuller space $$\widehat{Teich}\left(S \right)$$ can be embedded in the augmented Teichmuller space $$\widehat{Teich}\left({\widetilde{S}} \right)$$ isometrically. Furthermore, we show that Φπ induces an isometric embedding from the set Teich(S) B (∞) consisting of Busemann points in the horofunction boundary of Teich(S) into $$Teich{\left({\widetilde{S}} \right)_B}\left(\infty \right)$$ with the detour metric.

Bergman kernel on Riemann surfaces and Kähler metric on symmetric products

Let [Formula: see text] be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer [Formula: see text], we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle [Formula: see text], where [Formula: see text] is the holomorphic cotangent bundle of [Formula: see text]. Our first main result estimates the corresponding Bergman metric on [Formula: see text] in terms of the Poincaré metric. We then consider a certain natural embedding of the symmetric product of [Formula: see text] into a Grassmannian parametrizing subspaces of fixed dimension of the space of all global holomorphic sections of [Formula: see text]. The Fubini–Study metric on the Grassmannian restricts to a Kähler metric on the symmetric product of [Formula: see text]. The volume form for this restricted metric on the symmetric product is estimated in terms of the Bergman kernel of [Formula: see text] and the volume form for the orbifold Kähler form on the symmetric product given by the Poincaré metric on [Formula: see text].

Kähler Geometry on Hurwitz Spaces

The classical Hurwitz space \mathcal{H}^{n,b} is a fine moduli space for simple branched coverings of the Riemann sphere \mathbb{P}^1 by compact hyperbolic Riemann surfaces. In the article we study a generalized Weil-Petersson metric on the Hurwitz space, which was introduced in [R. Axelsson et al., Manuscr. Math. 147, No. 1–2, 63–79 (2015; Zbl 1319.32012)]. For this purpose, Horikawa's deformation theory of holomorphic maps is refined in the presence of hermitian metrics in order to single out distinguished representatives. Our main result is a curvature formula for a subbundle of the tangent bundle on the Hurwitz space obtained as a direct image. This covers the case of the curvature of the fibers of the natural map \mathcal{H}^{n,b} \to \mathcal{M}_g .

Construction of hyperbolic Riemann surfaces with large systoles

Let S be a compact hyperbolic Riemann surface of genus \({g \geq 2}\). We call a systole a shortest simple closed geodesic in S and denote by \({{\rm sys}(S)}\) its length. Let \({{\rm msys}(g)}\) be the maximal value that \({{\rm sys}(\cdot)}\) can attain among the compact Riemann surfaces of genus g. We call a (globally) maximal surface Smax a compact Riemann surface of genus g whose systole has length \({{\rm msys}(g)}\). In Section 2 we use cutting and pasting techniques to construct compact hyperbolic Riemann surfaces with large systoles from maximal surfaces. This enables us to prove several inequalities relating \({{\rm msys}(\cdot)}\) of different genera. In Section 3 we derive similar intersystolic inequalities for non-compact hyperbolic Riemann surfaces with cusps.

Bounds for Green’s functions on noncompact hyperbolic Riemann orbisurfaces of finite volume

In Jorgenson and Kramer (Compos Math 142:679–700, 2006) derived bounds for the canonical Green’s function and the hyperbolic Green’s function defined on a compact hyperbolic Riemann surface. In this article, we extend these bounds to noncompact hyperbolic Riemann orbisurfaces of finite volume and of genus greater than zero, which can be realized as a quotient space of the action of a Fuchsian subgroup of first kind on the hyperbolic upper half-plane.

A relation of two metrics on a noncompact hyperbolic Riemann orbisurface

Jorgenson and Kramer (Compos Math 142:679–700, 2006) proved a certain key identity which relates the two natural metrics, namely the hyperbolic metric and the canonical metric defined on a compact hyperbolic Riemann surface. In this article, we extend this identity to noncompact hyperbolic Riemann orbisurfaces of finite volume, which can be realized as a quotient space of the action of a Fuchsian subgroup of first kind on the hyperbolic upper half plane. Our result can be seen as an extension of the key identity to elliptic fixed points and cusps at the level of currents.

A uniform bound for geodesic periods of eigenfunctions on hyperbolic surfaces

Abstract We consider periods along closed geodesics and along geodesic circles for eigenfunctions of the Laplace–Beltrami operator on a compact hyperbolic Riemann surface. We obtain uniform bounds for such periods as the corresponding eigenvalue tends to infinity. We use methods from the theory of automorphic functions and, in particular, the uniqueness of the corresponding invariant functionals on irreducible unitary representations of PGL2(ℝ).

Measure-theoretic rigidity for Mumford curves

AbstractOne can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincaré disc that is absolutely continuous for Lebesgue measure if and only if the surfaces are isomorphic. In this paper, we find the corresponding statement for Mumford curves, a non-Archimedean analogue of Riemann surfaces. In this case, the mere absolute continuity of the boundary map (for Schottky uniformization and the corresponding Patterson–Sullivan measure) only implies isomorphism of the special fibers of the Mumford curves, and the absolute continuity needs to be enhanced by a finite list of conditions on the harmonic measures on the boundary (certain non-Archimedean distributions constructed by Schneider and Teitelbaum) to guarantee an isomorphism of the Mumford curves. The proof combines a generalization of a rigidity theorem for trees due to Coornaert, the existence of a boundary map by a method of Floyd, with a classical theorem of Babbage, Enriques and Petri on equations for the canonical embedding of a curve.

Minimal Lagrangian surfaces in $${\mathbb {CH}^2}$$ and representations of surface groups into SU(2, 1)

We use an elliptic differential equation of Ţiţeica (or Toda) type to construct a minimal Lagrangian surface in $${\mathbb {CH}^2}$$ from the data of a compact hyperbolic Riemann surface and a cubic holomorphic differential. The minimal Lagrangian surface is equivariant for an SU(2, 1) representation of the fundamental group. We use this data to construct a diffeomorphism between a neighbourhood of the zero section in a holomorphic vector bundle over Teichmuller space (whose fibres parameterise cubic holomorphic differentials) and a neighborhood of the $${\mathbb {R}}$$ -Fuchsian representations in the SU(2, 1) representation space. We show that all the representations in this neighbourhood are complex-hyperbolic quasi-Fuchsian by constructing for each a fundamental domain using an SU(2, 1) frame for the minimal Lagrangian immersion: the Maurer–Cartan equation for this frame is the Ţiţeica-type equation. A very similar equation to ours governs minimal surfaces in hyperbolic 3-space, and our paper can be interpreted as an analog of the theory of minimal surfaces in quasi-Fuchsian manifolds, as first studied by Uhlenbeck.

The trace formula for a point scatterer on a compact hyperbolic surface

An exact trace formula for the perturbation of the Laplacian by a Dirac delta potential on a compact hyperbolic Riemann surface is derived. The formula can be considered an analogue of the Selberg trace formula. The difference of perturbed and unperturbed trace is expressed as an identity term plus a sum over combinations of diffractive orbits which visit the position of the potential.

Geodesic restrictions for the Casimir operator

We consider the hyperbolic Casimir operator C defined on the tangent sphere bundle SY of a compact hyperbolic Riemann surface Y. We prove a non-trivial bound on the L 2 -norm of the restriction of eigenfunctions of C to certain natural hypersurfaces in SY. The result that we obtain goes beyond known (sharp) local bounds of L. Hörmander.

Zeta functions that hear the shape of a Riemann surface

To a compact hyperbolic Riemann surface, we associate a finitely summable spectral triple whose underlying topological space is the limit set of a corresponding Schottky group, and whose “Riemannian” aspect (Hilbert space and Dirac operator) encode the boundary action through its Patterson–Sullivan measure. We prove that the ergodic rigidity theorem for this boundary action implies that the zeta functions of the spectral triple suffice to characterize the (anti-)complex isomorphism class of the corresponding Riemann surface. Thus, you can hear the complex analytic shape of a Riemann surface, by listening to a suitable spectral triple.

An isomorphism theorem for Teichmüller curves

It is proved that for any Fuchsian group Γ such that ℍ/Γ is a hyperbolic Riemann surface, the Teichmuller curve V(Γ) has a unique complex manifold structure so that the natural projection of the Bers fiber space F(Γ) onto V(Γ) is holomorphic with local holomorphic sections. An isomorphism theorem for Teichmuller curves is deduced, which generalizes a classical result that the Teichmuller curve V(Γ) depends only on the type of Γ and not on the orders of the elliptic elements of Γ when ℍ/Γ is a compact hyperbolic Riemann surface.

A Note on Collars of Simple Closed Geodesics

For compact hyperbolic Riemann surfaces, the collar theorem gives a lower bound on the distance between a simple closed geodesic and all other simple closed geodesics that do not intersect the initial geodesic. Here it is shown that there are two possible configurations, and in each configuration there is a natural collar width associated to a simple closed geodesic. If one extends the natural collar of a simple closed geodesic α by e >0, then the extended collar contains an infinity of simple closed geodesics that do not intersect α.

Triangulations and homology of Riemann surfaces

We derive an algorithmic way to pass from a triangulation to a homology basis of a (Riemann) surface. The procedure will work for any surfaces with finite triangulations. We will apply this construction to Riemann surfaces to show that every compact hyperbolic Riemann surface X has a homology basis consisting of curves whose lengths are bounded linearly by the genus g of X and by the homological systole. This work got started by comments presented by Y. Imayoshi in his lecture at the 37th Taniguchi Symposium which took place in Katinkulta near Kajaani, Finland, in 1995.

Limit constructions over Riemann surfaces and their parameter spaces, and the commensurability group actions

To any compact hyperbolic Riemann surface X, we associate a new type of automorphism group — called its commensurability automorphism group, ComAut(X). The members of ComAut(X) arise from closed circuits, starting and ending at X, where the edges represent holomorphic covering maps amongst compact connected Riemann surfaces (and the vertices represent the covering surfaces). This group turns out to be the isotropy subgroup, at the point represented by X (in $ T_\infty $), for the action of the universal commensurability modular group on the universal direct limit of Teichmuller spaces, $ T_\infty $. Now, each point of $ T_\infty $ represents a complex structure on the universal hyperbolic solenoid. We notice that ComAut(X) acts by holomorphic automorphisms on that complex solenoid. Interestingly, this action turns out to be ergodic (with respect to the natural measure on the solenoid) if and only if the Fuchsian group uniformizing X is arithmetic. Furthermore, the action of the commensurability modular group, and of its isotropy subgroups, on some natural vector bundles over $ T_\infty $, are studied by us.

Invariants for a class of equivariant immersions of the universal cover of a compact Riemann surface into a projective space

We consider the space of all holomorphic immersions of the universal cover of a compact hyperbolic Riemann surface into CP n satisfying the condition that at every point of the image, the order of contact of the image with any hyperplane in CP n is at most n−1. A further restriction that is imposed states as follows: there exists a homomorphism of the Galois group for the universal cover into GL(n+1, C) such that the map from the universal cover to CP n is equivariant for the actions of the Galois group. To each such immersion we associate a i-form on the compact Riemann surface for each i∈[3, n+1], and also associate a projective structure on the Riemann surface. The resulting map from the space of all immersions surjects onto the target space. Moreover, this map gives a bijective correspondence between the target space and the space of all equivalence classes of immersions, where the equivalence relation identifies an immersion with any other immersion obtained by composing it with an automorphism of CP n .