Compact Riemann Surface Research Articles

Meromorphic quadratic differentials and measured foliations on a Riemann surface

We describe the space of measured foliations induced on a compact Riemann surface by meromorphic quadratic differentials. We prove that any such foliation is realized by a unique such differential q if we prescribe, in addition, the principal parts of \(\sqrt{q}\) at the poles. This generalizes a theorem of Hubbard and Masur for holomorphic quadratic differentials. The proof analyzes infinite-energy harmonic maps from the Riemann surface to \(\mathbb {R}\)-trees of infinite co-diameter, with prescribed behavior at the poles.

Stability of the parabolic Poincaré bundle

Given a compact Riemann surface [Formula: see text] and a moduli space [Formula: see text] of parabolic stable bundles on it of fixed determinant of complete parabolic flags, we prove that the Poincaré parabolic bundle on [Formula: see text] is parabolic stable with respect to a natural polarization on [Formula: see text].

On Representation Zeta Functions for Special Linear Groups

We prove that the numbers of irreducible n-dimensional complex continuous representations of the special linear groups over p-adic integers grow slower than the square of n. We deduce that the abscissas of convergence of the representation zeta functions of the special linear groups over the ring of integers are bounded above by 2. In order to show these results we prove also that if G is a connected, simply connected, semi-simple algebraic group defined over the field of rational numbers, then the G-representation variety of the fundamental group of a compact Riemann surface of genus n has rational singularities if and only if the G-character variety has rational singularities.

Branes through finite group actions

Mid-dimensional $(A,B,A)$ and $(B,B,B)$-branes in the moduli space of flat $G_{\mathbb C}$-connections appearing from finite group actions on compact Riemann surfaces are studied. The geometry and topology of these spaces is then described via the corresponding Higgs bundles and Hitchin fibrations.

On Determinants of Laplacians on Compact Riemann Surfaces Equipped with Pullbacks of Conical Metrics by Meromorphic Functions

Let $\mathsf m$ be any conical (or smooth) metric of finite volume on the Riemann sphere $\Bbb CP^1$. On a compact Riemann surface $X$ of genus $g$ consider a meromorphic funciton $f: X\to {\Bbb C}P^1$ such that all poles and critical points of $f$ are simple and no critical value of $f$ coincides with a conical singularity of $\mathsf m$ or $\{\infty\}$. The pullback $f^*\mathsf m$ of $\mathsf m$ under $f$ has conical singularities of angles $4\pi$ at the critical points of $f$ and other conical singularities that are the preimages of those of $\mathsf m$. We study the $\zeta$-regularized determinant $\operatorname{Det}' \Delta_F$ of the (Friedrichs extension of) Laplace-Beltrami operator on $(X,f^*\mathsf m)$ as a functional on the moduli space of pairs $(X, f)$ and obtain an explicit formula for $\operatorname{Det}' \Delta_F$.

Moduli spaces of meromorphic functions and determinant of the Laplacian

The Hurwitz space is the moduli space of pairs $(X,f)$ where $X$ is a compact Riemann surface and $f$ is a meromorphic function on $X$. We study the Laplace operator $\Delta^{|df|^2}$ of the flat singular Riemannian manifold $(X,|df|^2)$. We define a regularized determinant for $\Delta^{|df|^2}$ and study it as a functional on the Hurwitz space. We prove that this functional is related to a system of PDE which admits explicit integration. This leads to an explicit expression for the determinant of the Laplace operator in terms of the basic objects on the underlying Riemann surface (the prime form, theta-functions, the canonical meromorphic bidifferential) and the divisor of the meromorphic differential $df$. The proof has several parts that can be of independent interest. As an important intermediate result we prove a decomposition formula of the type of Burghelea-Friedlander-Kappeler for the determinant of the Laplace operator on flat surfaces with conical singularities and Euclidean or conical ends. We introduce and study the $S$-matrix, $S(\lambda)$, of a surface with conical singularities as a function of the spectral parameter $\lambda$ and relate its behavior at $\lambda=0$ with the Schiffer projective connection on the Riemann surface $X$. We also prove variational formulas for eigenvalues of the Laplace operator of a compact surface with conical singularities when the latter move.

A generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities

In this paper, using the method of blow-up analysis, we establish a generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularity. Precisely, let (Σ,D) be such a surface with divisor $$D=\Sigma_{i=1}^m\beta_{i}p_{i}$$ , where βi > −1 and pi ∈ Σ for i = 1, …, m, and g be a metric representing D. Denote b0 = 4π(1 + min1⩽i⩽mβi). Suppose ψ : Σ → ℝ is a continuous function with ∫Σψdvg ≠ 0 and define $$\lambda _1^{**} (\sum ,g) = \mathop {\inf }\limits_{u \in H^1 (\sum ,g),\smallint _\sum \psi udv_g = 0,\smallint _\sum u^2 dv_g = 1} \int_\sum {\left| {\nabla _g u} \right|^2 dv_g .}$$ Then for any $$\alpha\in[0,\lambda_1^{**}(\Sigma, g))$$ , we have $$\mathop {\sup }\limits_{u \in H^1 (\sum ,g),\smallint _\sum \psi u = 0,\smallint _\sum \left| {\nabla _g u} \right|^2 dv_g - \alpha \smallint _\sum u^2 dv_g \leqslant 1} \int_\sum {e^{b_0 u^2 } dv_g < + \infty .}$$ When b > b0, the integrals $$\int_\sum {e^{bu^2 } dv_g }$$ are still finite, but the supremum is infinity. Moreover, we prove that the extremal function for the above inequality exists.

On Koyama’s refinement of the prime geodesic theorem

We give a new proof of the best presently known error term in the prime geodesic theorem for compact Riemann surfaces, without the assumption of excluding a set of finite logarithmic measure. Stronger implications of the Gallagher-Koyama approach are derived yielding to a further reduction of the error term outside a set of finite logarithmic measure.

Livsic-type determinantal representations and hyperbolicity

Hyperbolic homogeneous polynomials with real coefficients, i.e., hyperbolic real projective hypersurfaces, and their determinantal representations, play a key role in the emerging field of convex algebraic geometry. In this paper we consider a natural notion of hyperbolicity for a real subvariety X⊂Pd of an arbitrary codimension ℓ with respect to a real ℓ−1-dimensional linear subspace V⊂Pd and study its basic properties. We also consider a class of determinantal representations that we call Livsic-type and a nice subclass of these that we call very reasonable. Much like in the case of hypersurfaces (ℓ=1), the existence of a definite Hermitian very reasonable Livsic-type determinantal representation implies hyperbolicity. We show that every curve admits a very reasonable Livsic-type determinantal representation. Our basic tools are Cauchy kernels for line bundles and the notion of the Bezoutian for two meromorphic functions on a compact Riemann surface that we introduce. We then proceed to show that every real curve in Pd hyperbolic with respect to some real d−2-dimensional linear subspace admits a definite Hermitian, or even definite real symmetric, very reasonable Livsic-type determinantal representation.

Geometry of the 1-skeleta of singular nerves of moduli spaces of Riemann surfaces

Cornalba (Ann Mat Pura Appl 149:135–151, 1987) classified the components of the singular locus of the moduli space of compact Riemann surfaces of genus $$g \ge 2$$ . Here we consider the problem of describing the intersections of these components by examining certain nerves of the cover of singular locus that the Cornalba components provide. We give a description of the 1-skeleton of such nerves which significantly extends the results of our earlier paper written together with A. Weaver where we considered a coarser cover of the singular locus. We compare the results of our earlier work with those of the present one in terms of certain natural simplicial covering maps between them.

Counting saddle connections in flat surfaces with poles of higher order

Flat surfaces that correspond to k-differentials on compact Riemann surfaces are of finite area provided there is no pole of order k higher. We denote by flat surfaces with poles of higher order those surfaces with flat structures defined by a k-differential with at least one pole of order at least k. Flat surfaces with poles of higher order have different geometrical and dynamical properties than usual flat surfaces of finite area. In particular, they can have a finite number of saddle connections. We give lower and upper bounds for the number of saddle connections and related quantities. In the case $$k=1 or 2$$ , we provide a combinatorial characterization of the strata for which there can be an infinite number of saddle connections.

An invitation to 2D TQFT and quantization of Hitchin spectral curves

This article consists of two parts. In Part 1, we present a formulation of two-dimensional topological quantum field theories in terms of a functor from a category of Ribbon graphs to the endofuntor category of a monoidal category. The key point is that the category of ribbon graphs produces all Frobenius objects. Necessary backgrounds from Frobenius algebras, topological quantum field theories, and cohomological field theories are reviewed. A result on Frobenius algebra twisted topological recursion is included at the end of Part 1. In Part 2, we explain a geometric theory of quantum curves. The focus is placed on the process of quantization as a passage from families of Hitchin spectral curves to families of opers. To make the presentation simpler, we unfold the story using SL_2(\mathbb{C})-opers and rank 2 Higgs bundles defined on a compact Riemann surface $C$ of genus greater than $1$. In this case, quantum curves, opers, and projective structures in $C$ all become the same notion. Background materials on projective coordinate systems, Higgs bundles, opers, and non-Abelian Hodge correspondence are explained.

Stabilization of the Homotopy Groups of the Moduli Spaces of k-Higgs Bundles

El trabajo de Hausel prueba que la estratificación de Bialynicki-Birula del espacio moduli de fibrados de Higgs de rango dos coincide con su estratificación de Shatz. Él usa este hecho para calcular algunos grupos de homotopía del espacio moduli de k-fibrados de Higgs de rango dos. Desafortunadamente, estas dos estratificaciones no coinciden en general. Aquí, el objetivo es presentar una prueba diferente de la estabilización de los grupos de homotopía de Mk(2, d), y generalizarla a Mk(3, d), los espacios moduli de k-fibrados de Higgs de grado d, y rangos dos y tres respectivamente, sobre una superficie de Riemann compacta X, usando los resultados de los trabajos de Hausel y Thaddeus, entre otras herramientas.

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 .

Proper Holomorphic Mappings onto Symmetric Products of a Riemann Surface

We show that the structure of proper holomorphic maps between the n -fold symmetric products, n\geq 2 , of a pair of non-compact Riemann surfaces X and Y , provided these are reasonably nice, is very rigid. Specifically, any such map is determined by a proper holomorphic map of X onto Y . This extends existing results concerning bounded planar domains, and is a non-compact analogue of a phenomenon observed in symmetric products of compact Riemann surfaces. Along the way, we also provide a condition for the complete hyperbolicity of all n -fold symmetric products of a non-compact Riemann surface.

A generalized Trudinger–Moser inequality on a compact Riemannian surface

Let (Σ,g) be a compact Riemannian surface. Let ψ, h be two smooth functions on Σ with ∫Σψdvg≠0 and h≥0, h≢0. In this paper, using a method of blow-up analysis, we prove that the functional (1)Jψ,h(u)=12∫Σ|∇gu|2dvg+8π1∫Σψdvg∫Σψudvg−8πlog∫Σheudvgis bounded from below in W1,2(Σ,g). Moreover, we obtain a sufficient condition under which Jψ,h attains its infimum in W1,2(Σ,g). These results generalize the main results in Ding–Jost–Li–Wang (1997) and Yang–Zhu (2017).

Extremal disc packings in compact hyperbolic surfaces

We study packings of metric discs with respect to the canonical hyperbolic metric of a compact Riemann surface of genus greater than one. We find the maximum radius of a packing as a function of the genus and the number of discs and we investigate some properties of the surfaces that contain an extremal packing.

On (q,n)-gonal pseudo-real Riemann surfaces

A compact Riemann surface [Formula: see text] of genus [Formula: see text] is called pseudo-real if it admits an anticonformal automorphism but no anticonformal involution. In this paper, we study pseudo-real [Formula: see text]-gonal Riemann surfaces of genera greater or equal to two; these surfaces have anticonformal automorphisms of prime order [Formula: see text] such that the quotient spaces have genus [Formula: see text].

Stability for a magnetic Schrödinger operator on a Riemann surface with boundary

We consider a magnetic Schrödinger operator $(\nabla^X)^*\nabla^X+q$ on a compact Riemann surface with boundary and prove a $\log\log$-type stability estimate in terms of Cauchy data for the electric potential and magnetic field under the assumption that they satisfy appropriate a priori bounds. We also give a similar stability result for the holonomy of the connection 1-form $X$.

Existence of solutions to a class of Kazdan-Warner equations on compact Riemannian surface

Let (Σ, g) be a compact Riemannian surface without boundary and λ1(Σ) be the first eigenvalue of the Laplace-Beltrami operator Δ g . Let h be a positive smooth function on Σ. Define a functional $${J_{\alpha ,\beta }}\left( u \right) = \frac{1}{2}\int_\sum {\left( {{{\left| {{\nabla _g}u} \right|}^2} - \alpha {u^2}} \right)} d{v_g} - \beta \log \int_\sum {h{e^u}} d{v_g}$$ on a function space H = {u ∈ W1,2(Σ): ∫Σ udv g = 0}. If α 8π, then for any α ∈ R, there holds infu∈HJα,β(u) = −∞. Moreover, we consider the same problem in the case that α is large, where higher order eigenvalues are involved.