Compact Riemann Surface Research Articles

Plemelj\u2013Sokhotski isomorphism for quasicircles in Riemann surfaces and the Schiffer operators

Let R be a compact Riemann surface and Gamma be a Jordan curve separating R into connected components Sigma _1 and Sigma _2. We consider Calderón–Zygmund type operators T(Sigma _1,Sigma _k) taking the space of L^2 anti-holomorphic one-forms on Sigma _1 to the space of L^2 holomorphic one-forms on Sigma _k for k=1,2, which we call the Schiffer operators. We extend results of Max Schiffer and others, which were confined to analytic Jordan curves Gamma , to general quasicircles, and prove new identities for adjoints of the Schiffer operators. Furthermore, let V be the space of anti-holomorphic one-forms which are orthogonal to L^2 anti-holomorphic one-forms on R with respect to the inner product on Sigma _1. We show that the restriction of the Schiffer operator T(Sigma _1,Sigma _2) to V is an isomorphism onto the set of exact holomorphic one-forms on Sigma _2. Using the relation between this Schiffer operator and a Cauchy-type integral involving Green’s function, we also derive a jump decomposition (on arbitrary Riemann surfaces) for quasicircles and initial data which are boundary values of Dirichlet-bounded harmonic functions and satisfy the classical algebraic constraints. In particular we show that the jump operator is an isomorphism on the subspace determined by these constraints.

Nikishin systems on star-like sets: Ratio asymptotics of the associated multiple orthogonal polynomials, II

In this paper we continue the investigations initiated in López-García and López Lagomasino (2018) on ratio asymptotics of multiple orthogonal polynomials and functions of the second kind associated with Nikishin systems on star-like sets. We describe in detail the limiting functions found in López-García and López Lagomasino (2018), expressing them in terms of certain conformal mappings defined on a compact Riemann surface of genus zero. We also express the limiting values of the recurrence coefficients, which are shown to be strictly positive, in terms of certain values of the conformal mappings. As a consequence, the limits depend exclusively on the location of the intervals determined by the supports of the measures that generate the Nikishin system.

Metric distortion in the geometric Schottky problem

The classical Schottky problem is concerned with characterization of Jacobian varieties of compact Riemann surfaces among all abelian varieties, or the identification of the Jacobian locus \(J(\mathcal{M}_g)\) in the moduli space \(\mathcal{A}_g\) of principally polarized abelian varieties as an algebraic subvariety. By viewing \(\mathcal{A}_g\) as a noncompact metric space coming from its structure as a locally symmetric space and \(J(\mathcal{M}_g)\) as a metric subspace, we compare the subspace metric d and the induced length metric l on \(J(\mathcal{M}_g)\). Consequently, we clarify the nature of the metric distortion of the subspace \(J(\mathcal{M}_g)\) and hence settle a problem posed by Farb (2006) on the metric distortion of \(J(\mathcal{M}_g)\) inside \(\mathcal{A}_g\) in a certain sense (see Theorem 1.5 and Corollary 1.6).

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].

Rational Singularities, Quiver Moment Maps, and Representations of Surface Groups

Abstract We prove using jet schemes that the zero loci of the moment maps for the quivers with one vertex and at least two loops have rational singularities. This implies that the spaces of representations of the fundamental group of a compact Riemann surface of genus at least two have rational singularities. This has consequences for the numbers of irreducible representations of the special linear groups over the integers and over the $p$-adic integers.

Local existence of a fourth-order dispersive curve flow on locally Hermitian symmetric spaces and its application

A fourth-order nonlinear dispersive partial differential equation arises in the field of mathematical physics, the solution of which is a curve flow on the two-dimensional unit sphere. In recent ten years, a geometric generalization of the sphere-valued physical model has been considered and the solvability of the initial value problem has been investigated. In particular, in the author's previous work, time-local existence and uniqueness result of the solution was established under the assumption that the solution is a closed curve flow on a compact Riemann surface with constant curvature. In the present paper, we propose a new geometric generalization of the sphere-valued physical model. As a main result, we show time-local existence of a solution to the initial value problem under the assumption that the solution is a closed curve flow on a compact locally Hermitian symmetric space. The proof is based on the geometric energy method combined with a gauge transformation to overcome the difficulty of the so-called loss of derivatives. Interestingly, the results can be applied to construct a generalized bi-Schrödinger flow proposed by Ding and Wang. The assumption on the manifold plays a crucial role both to enjoy a good solvable structure of the initial value problem and to reduce the generalized bi-Schrödinger flow equation to the one considered in the present paper.

A generalized spectral theory for continuous metrics on compact Riemann surfaces

We extend the spectral theory of generalized Laplacians to continuous metrics on compact Riemann surfaces. We define a holomorphic analytic torsion for any continuous metric. As an application of this theory, we partly recover some results of the theory of Bessel functions, for instance, Lommel's theorem on the reality of the zeros of Bessel functions of order exceeding −1.

Exponential Decay for the Asymptotic Geometry of the Hitchin Metric

We consider Hitchin's hyperk\"ahler metric $g_{L^2}$ on the $SU(n)$-Hitchin moduli space moduli space over a compact Riemann surface. We prove that the difference between the metric $g_{L^2}$ and a simpler "semiflat" hyperk\"ahler metric $g_{\mathrm{sf}}$ is exponentially-decaying along generic rays in the Hitchin moduli space, as conjectured by Gaiotto-Moore-Neitzke.

Expansiveness for the geodesic and horocycle flows on compact Riemann surfaces of constant negative curvature

We study expansive properties for the geodesic and horocycle flows on compact Riemann surfaces of constant negative curvature. It is well-known that the geodesic flow is expansive in the sense of Bowen-Walters and the horocycle flow is positive and negative separating in the sense of Gura. In this paper, we give a new proof of the expansiveness in the sense of Bowen-Walters for the geodesic flow and show that the horocycle flow is positive and negative kinematic expensive in the sense of Artigue as well as expansive in the sense of Katok-Hasselblatt but not expensive in the sense of Bowen-Walters. We also point out that the geodesic flow is neither positive nor negative separating.

Morphisms and period matrices

Bounding the number of morphisms between compact Riemann surfaces is a long standing problem coming from complex and algebraic geometry. We show that linear algebra techniques allow to improve the known results when we assume a kind of condition number bound for the period matrix.

Magnetic resonance assessment of the effect of maternal position on fetoplacental blood flow and oxygenation

A compact Riemann surface is called pseudo-real if it admits anti-conformal (orientation-reversing) automorphisms, but no anti-conformal automorphism of order 2, or equivalently, if the surface is reflexible but not definable over the reals. It is known that there exist pseudo-real surfaces of genus g for every integer g≥2, and the number of automorphisms of any such surface is bounded above by 12(g−1).In this paper, we extend the concepts of symmetric genus, strong symmetric genus and symmetric cross-cap genus of a group by defining and investigating two new parameters, as follows: (1) the pseudo-real genus ψ(G) of a finite group G is the smallest genus of those pseudo-real surfaces on which G acts faithfully as a group of automorphisms, some of which might reverse orientation, and (2) the strong pseudo-real genus ψ⁎(G) of G is the smallest genus of those pseudo-real surfaces on which G acts faithfully as a group of automorphisms, some of which do reverse orientation, when there exists such a surface for G.Our main theorem is that for every integer g≥2, there exists a finite group G for which ψ(G)=ψ⁎(G)=g, and hence that the range of each of the functions ψ and ψ⁎ is the set of all integers g≥2. We also give an example of a group G for which ψ⁎(G) is defined but ψ(G)<ψ⁎(G).

A Brief Survey of Higgs Bundles

Considering a compact Riemann surface of genus greater or equal than two, a Higgs bundle is a pair composed of a holomorphic bundle over the Riemann surface, joint with an auxiliar vector field, so-called Higgs field. This theory started around thirty years ago, with Hitchin’s work, when he reduced the self-duality equations from dimension four to dimension two, and so, studied those equations over Riemann surfaces. Hitchin baptized those fields as Higgs fields because in the context of physics and gauge theory, they describe similar particles to those described by the Higgs bosson. Later, Simpson used the name Higgs bundle for a holomorphic bundle together with a Higgs field. Today, Higgs bundles are the subject of research in several areas such as non-abelian Hodge theory, Langlands, mirror symmetry, integrable systems, quantum field theory (QFT), among others. The main purposes here are to introduce these objects, and to present a brief but complete construction of the moduli space of Higgs bundles.

Meromorphic Interpolation on a Compact Riemann Surface

Meromorphic Interpolation on a Compact Riemann Surface

Harmonic maps and wild Teichmüller spaces

We use meromorphic quadratic differentials with higher order poles to parametrize the Teichmüller space of crowned hyperbolic surfaces. Such a surface is obtained on uniformizing a compact Riemann surface with marked points on its boundary components, and has noncompact ends with boundary cusps. This extends Wolf’s parametrization of the Teichmüller space of a closed surface using holomorphic quadratic differentials. Our proof involves showing the existence of a harmonic map from a punctured Riemann surface to a crowned hyperbolic surface, with prescribed principal parts of its Hopf differential which determine the geometry of the map near the punctures.

Decomposition of Jacobian varieties of curves with dihedral actions via equisymmetric stratification

Given a compact Riemann surface X with an action of a finite group G , the group algebra \mathbb{Q}[G] provides an isogenous decomposition of its Jacobian variety JX , known as the group algebra decomposition of JX . We consider the set of equisymmetric Riemann surfaces \mathcal{M}(2n-1, D_{2n}, \theta) for all n\geq 2 . We study the group algebra decomposition of the Jacobian JX of every curve X\in \mathcal{M}(2n-1, D_{2n},\theta) for all admissible actions, and we provide affine models for them. We use the topological equivalence of actions on the curves to obtain facts regarding its Jacobians. We describe some of the factors of JX as Jacobian (or Prym) varieties of intermediate coverings. Finally, we compute the dimension of the corresponding Shimura domains.

The isometric embedding of the augmented Teichmüller space of a Riemann surface into the augmented Teichmüller space of its covering surface

It is known that every finitely unbranched holomorphic covering $\pi:\widetilde{S}\rightarrow S$ of a compact Riemann surface $S$ with genus $g\geq2$ induces an isometric embedding $\Phi_{\pi} :Teich(S)\rightarrow Teich(\widetilde{S})$. By the mutual relations between Strebel rays in $Teich(S)$ and their embeddings in $Teich(\widetilde{S})$, we show that the augmented Teichmüller space $\widehat{Teich}(S)$ can be isometrically embedded in the augmented Teichmüller space $\widehat{Teich}(\widetilde{S})$.

On the moduli space of holomorphic G-connections on a compact Riemann surface

Let X be a compact connected Riemann surface of genus at least two and G a connected reductive complex affine algebraic group. The Riemann–Hilbert correspondence produces a biholomorphism between the moduli space $${{\mathscr {M}}}_X(G)$$ parametrizing holomorphic G-connections on X and the G-character variety While $${{\mathscr {R}}}(G)$$ is known to be affine, we show that $${{\mathscr {M}}}_X(G)$$ is not affine. The scheme $${{\mathscr {R}}}(G)$$ has an algebraic symplectic form constructed by Goldman. We construct an algebraic symplectic form on $${{\mathscr {M}}}_X(G)$$ with the property that the Riemann–Hilbert correspondence pulls back the Goldman symplectic form to it. Therefore, despite the Riemann–Hilbert correspondence being non-algebraic, the pullback of the Goldman symplectic form by the Riemann–Hilbert correspondence nevertheless continues to be algebraic.

Flat Bundles Over Some Compact Complex Manifolds

We construct examples of flat fiber bundles over the Hopf surface such that the total spaces have no pseudoconvex neighborhood basis, admit a complete Kahler metric, or are hyperconvex but have no nonconstant holomorphic functions. For any compact Riemannian surface of positive genus, we construct a flat $${\mathbb {P}}^1$$ bundle over it and a Stein domain with real analytic boundary in it whose closure does not have pseudoconvex neighborhood basis. For a compact complex manifold with positive first Betti number, we construct a flat bundle over it such that the total space is hyperconvex but admits no nonconstant holomorphic functions.

Vector Bundle of Prym Differentials over Teichm¨uller Spaces of Surfaces with Punctures

In this paper we study multiplicative meromorphic functions and differentials on Riemann surfaces of finite type. We prove an analog of P. Appell’s formula on decomposition of multiplicative functions with poles of arbitrary multiplicity into a sum of elementary Prym integrals. We construct explicit bases for some important quotient spaces and prove a theorem on a fiber isomorphism of vector bundles and n!-sheeted mappings over Teichm¨uller spaces. This theorem gives an important relation between spaces of Prym differentials (Abelian differentials) on a compact Riemann surfaces and on a Riemann surfaces of finite type

Explicit Bounds on Integrals of Eigenfunctions Over Curves in Surfaces of Nonpositive Curvature

Let (M, g) be a compact Riemannian surface with nonpositive sectional curvature and let $$\gamma $$ be a closed geodesic in M. And let $$e_\lambda $$ be an $$L^2$$-normalized eigenfunction of the Laplace–Beltrami operator $$\Delta _g$$ with $$-\Delta _g e_\lambda = \lambda ^2 e_\lambda $$. Sogge et al. (Camb J Math 5(1):123–151, 2017) showed using the Gauss–Bonnet Theorem that $$\int _\gamma e_\lambda \, \mathrm{{d}}s = O((\log \lambda )^{-1/2}),$$ an improvement over the general O(1) bound. We show this integral enjoys the same decay for a wide variety of curves, where M has nonpositive sectional curvature. These are the curves $$\gamma $$ whose geodesic curvature avoids, pointwise, the geodesic curvature of circles of infinite radius tangent to $$\gamma $$.