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It is known that every unbranched finite covering \(\alpha\colon\widetilde{S}_{g(\alpha)}\rightarrow S\) of a compact Riemann surface \(S\) with genus \(g\geq 2\) induces an isometric embedding \(\Gamma_{\alpha}\) from the Teichmuller space \(T(S)\) to the Teichmuller space \(T(\widetilde{S}_{g(\alpha)})\). Actually, it has been showed that the isometric embedding \(\Gamma_{\alpha}\) can be extended isometrically to the augmented Teichmuller space \(\widehat{T}(S)\) of \(T(S)\). Using this result, we construct a direct limit \(\widehat{T}_{\infty}(S)\) of augmented Teichmuller spaces, where the index runs over all unbranched finite coverings of \(S\). Then, we show that the action of the universal commensurability modular group \(\operatorname{Mod}_{\infty}(S)\) can extend isometrically on \(\widehat{T}_{\infty}(S)\). Furthermore, for any \(X_{\infty}\in T_{\infty}(S)\), its orbit of the action of the universal commensurability modular group \(\operatorname{Mod}_{\infty}(S)\) on \(\widehat{T}_{\infty}(S)\) is dense. Finally, we also construct a direct limit \(\widehat{M}_{\infty}(S)\) of augmented moduli spaces by characteristic towers and show that the subgroup \(\operatorname{Caut}(\pi_{1}(S))\) of \(\operatorname{Mod}_{\infty}(S)\) acts on \(\widehat{T}_{\infty}(S)\) to produce \(\widehat{M}_{\infty}(S)\) as the quotient.

We describe some results on moduli space of logarithmic connections equipped with framings on a $n$-pointed compact Riemann surface., Final version; to appear in Comptes Rendus S\'erie Math\'ematique

We classify compact Riemann surfaces of genus \(g\), where \(g-1\) is a prime \(p\), which have a group of automorphisms of order \(\rho(g-1)\) for some integer \(\rho\ge 1\), and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for \(\rho>6\), and of the first and third authors for \(\rho=\) 3, 4, 5 and 6. As a corollary we classify the orientably regular hypermaps (including maps) of genus \(p+1\), together with the non-orientable regular hypermaps of characteristic \(-p\), with automorphism group of order divisible by the prime \(p\); this extends results of Conder, Siraň and Tucker for maps.

Given a tuple of germs of arbitrary analytic functions at a fixed point, we introduce the polynomial Hermite-Padé -system, which includes the Hermite-Padé polynomials of types I and II. In the generic case we find the weak asymptotics of the polynomials of the Hermite-Padé -system constructed from the tuple of germs of functions that are meromorphic on an -sheeted compact Riemann surface . We show that if for some meromorphic function on , then with the help of the ratios of polynomials of the Hermite-Padé -system we recover the values of on all sheets of the Nuttall partition of , apart from the last sheet. Bibliography: 18 titles.

We perform a consistent reduction of 6d matter-coupled F(4) supergravity on a compact Riemann surface $$ {\Sigma}_{\mathfrak{g}} $$ Σ g of genus $$ \mathfrak{g} $$ g , at the level of the bosonic action. The result is an $$ \mathcal{N} $$ N = 2 gauged supergravity coupled to two vector multiplets and a single hypermultiplet. The four-dimensional model is holographically dual to the 3d superconformal field theories of class ℱ, describing different brane systems in massive type IIA and IIB wrapped on $$ {\Sigma}_{\mathfrak{g}} $$ Σ g . The naive reduction leads to a non-standard 4d mixed duality frame with both electric and magnetic gauge fields, as well as a massive tensor, that can be only described in the embedding tensor formalism. Upon a chain of electromagnetic dualities, we are able to determine the scalar manifolds and electric gaugings that uniquely specify the model in the standard supergravity frame. We then use the result to construct the first examples of static dyonic black holes in AdS6 and perform a microscopic counting of their entropy via the 5d topologically twisted index. Finally, we show the existence of further subtruncations to the massless sector of the 4d theory, such as the Fayet-Iliopoulos gauged T3 model and minimal gauged supergravity. We are in turn able to find new asymptotically AdS4 solutions, providing predictions for the squashed S3 partition functions and the superconformal and refined twisted indices of class ℱ theories.

We prove that any one-relator group $G$ is the fundamental group of a compact Sasakian manifold if and only if $G$ is either finite cyclic or isomorphic to the fundamental group of a compact Riemann surface of genus g > 0 with at most one orbifold point of order $n \geq 1$. We also classify all groups of deficiency at least two that are also the fundamental group of some compact Sasakian manifold.

For a complex simple simply connected Lie group $G$, and a compact Riemann surface $C$, we consider two sorts of families of flat $G$-connections over $C$. Each family is determined by a point ${\mathbf u}$ of the base of Hitchin's integrable system for $(G,C)$. One family $\nabla_{\hbar,{\mathbf u}}$ consists of $G$-opers, and depends on $\hbar \in {\mathbb C}^\times$. The other family $\nabla_{R,\zeta,{\mathbf u}}$ is built from solutions of Hitchin's equations, and depends on $\zeta \in {\mathbb C}^\times, R \in {\mathbb R}^+$. We show that in the scaling limit $R \to 0$, $\zeta = \hbar R$, we have $\nabla_{R,\zeta,{\mathbf u}} \to \nabla_{\hbar,{\mathbf u}}$. This establishes and generalizes a conjecture formulated by Gaiotto.

We study a natural functional on the space of holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface, generalizing the energy of equivariant harmonic maps corresponding to twistor lines. We give a link to a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by Hitchin. Moreover, we prove that for a certain class of real holomorphic sections of the Deligne-Hitchin moduli space, the functional is basically given by the Willmore energy of corresponding (equivariant) conformal map to the 3-sphere. As an application we use the functional to distinguish new components of real holomorphic sections of the Deligne-Hitchin moduli space from the space of twistor lines., 33 pages

Our basic objects will be compact, even-dimensional, locally symmetric Riemannian manifolds with strictly negative sectional curvature. The goal of the present paper is to investigate the prime geodesic theorems that are associated with this class of spaces. First, following classical Randol&rsquo, s appraoch in the compact Riemann surface case, we improve the error term in the corresponding result. Second, we reduce the exponent in the newly acquired remainder by using the Gallagher&ndash, Koyama techniques. In particular, we improve DeGeorge&rsquo, s bound Ox&eta, 2&rho, &minus, &rho, n &le, &eta, &lt, up to Ox2&rho, logx&minus, 1, and reduce the exponent 2&rho, n replacing it by 2&rho, 4n+14n2+1 outside a set of finite logarithmic measure. As usual, n denotes the dimension of the underlying locally symmetric space, and &rho, is the half-sum of the positive roots. The obtained prime geodesic theorem coincides with the best known results proved for compact Riemann surfaces, hyperbolic three-manifolds, and real hyperbolic manifolds with cusps.

In this note we prove that the Selberg zeta-function associated to a compact Riemann surface is pseudo-prime and right-prime in the sense of a decomposition., Comment: To appear in Hokkaido Mathematical Journal

We introduce the notion of twisted gravitating vortex on a compact Riemann surface. If the genus of the Riemann surface is greater than 1 and the twisting forms have suitable signs, we prove an existence and uniqueness result for suitable range of the coupling constant generalizing the result of arXiv:1510.03810v2 in the non twisted setting. It is proved via solving a continuity path deforming the coupling constant from 0 for which the system decouples as twisted K\"ahler-Einstein metric and twisted vortices. Moreover, specializing to a family of twisting forms smoothing delta distribution terms, we prove the existence of singular gravitating vortices whose K\"ahler metric has conical singularities and Hermitian metric has parabolic singularities. In the Bogomol'nyi phase, we establish an existence result for singular Einstein-Bogomol'nyi equations, which represents cosmic strings with singularities., Comment: 1 figure

We classify compact Riemann surfaces of genus $g$, where $g-1$ is a prime $p$, which have a group of automorphisms of order $\rho(g-1)$ for some integer $\rho\ge 1$, and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for $\rho>6$, and of the first and third authors for $\rho=3, 4, 5$ and $6$. As a corollary we classify the orientably regular hypermaps (including maps) of genus $p+1$, together with the non-orientable regular hypermaps of characteristic $-p$, with automorphism group of order divisible by the prime $p$; this extends results of Conder, \v Sir\'a\v n and Tucker for maps., Comment: 29 pages, 5 figures

It is known that the universal cover of compact Riemann surface is either the projective line, the complex plane or the unit disk. In this article we construct a very explicit family of complex surfaces that gives rise to uncountably many mutually non-biholomorphic universal covers. The slope of these surfaces, which are going to be total spaces of Kodaira fibrations, is also determined, 10 pages

We consider a compact Riemann surface $R$ of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate $R$ into two subsets: a connected Riemann surface $\Sigma$, and the union $\mathcal{O}$ of a finite collection of simply-connected regions. We prove that the Schiffer integral operator mapping the Bergman space of anti-holomorphic one-forms on $\mathcal{O}$ to the Bergman space of holomorphic forms on $\Sigma$ is an isomorphism. We then apply this to prove versions of the Plemelj-Sokhotski isomorphism and jump decomposition for such a configuration. Finally we obtain some approximation theorems for the Bergman space of one-forms and Dirichlet space of holomorphic functions on $\Sigma$ by elements of Bergman space and Dirichlet space on fixed regions in $R$ containing $\Sigma$., Comment: 25 pages

We show a coincidence of index of rigidity of differential equations with irregular singularities on a compact Riemann surface and Euler characteristic of the associated spectral curves which are recently called irregular spectral curves. Also we present a comparison of local invariants, so called Milnor formula which links the Komatsu-Malgrange irregularity of differential equations and Milnor number of the spectral curves., 22 pages. To appear in J. Geom. Phys

The topological data of a finite group G acting conformally on a compact Riemann surface is often encoded using a tuple of non-negative integers ( h ; m 1 , … , m s ) called its signature, where the m i are orders of non-trivial elements in the group. There are two easily verifiable arithmetic conditions on a tuple which are necessary for it to be a signature of some group action. We derive necessary and sufficient conditions on a group for the situation where all but finitely many tuples that satisfy these arithmetic conditions actually occur as the signature for an action of G on some Riemann surface. As a consequence, we show that all non-abelian finite simple groups exhibit this property.

Considering a compact Riemann surface of genus greater 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" beacuse 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 construction of the moduli space of Higgs bundles., 6 pages. Presented as a communication on SIMMAC--2018, UCR

In this paper, we show that if $R$ is a compact Riemann surface and $M=R\setminus\,\bigcup_i D_i$ is a domain in $R$ whose complement is a union of countably many pairwise disjoint smoothly bounded closed discs $D_i$, then $M$ is the complex structure of a complete bounded minimal surface in $\mathbb R^3$. We prove that there is a complete conformal minimal immersion $X:M\to\mathbb R^3$ extending to a continuous map $X:\overline M\to\mathbb R^3$ such that $X(bM)=\bigcup_i X(bD_i)$ is a union of pairwise disjoint Jordan curves. This extends a recent result for bordered Riemann surfaces., Rev. Mat. Iberoam., to appear

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 ${\mathcal M}_X(G)$ parametrizing holomorphic $G$--connections on $X$ and the $G$--character variety $${\mathcal R}(G):= \text{Hom}(\pi_1(X, x_0), G)/\!\!/G\, .$$ While ${\mathcal R}(G)$ is known to be affine, we show that ${\mathcal M}_X(G)$ is not affine. The scheme ${\mathcal R}(G)$ has an algebraic symplectic form constructed by Goldman. We construct an algebraic symplectic form on ${\mathcal M}_X(G)$ with the property that the Riemann--Hilbert correspondence pulls back to 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., Comment: Final version

We investigate a parabolic-elliptic system for maps $(u,v)$ from a compact Riemann surface $M$ into a Lorentzian manifold $N\times{\mathbb{R}}$ with a warped product metric. That system turns the harmonic map type equations into a parabolic system, but keeps the $v$-equation as a nonlinear second order constraint along the flow. We prove a global existence result of the parabolic-elliptic system by assuming either some geometric conditions on the target Lorentzian manifold or small energy of the initial maps. The result implies the existence of a Lorentzian harmonic map in a given homotopy class with fixed boundary data., to appear in J. Math. Pures Appl

The present paper is devoted to the further development of the discrete theory of Riemann surfaces, which was started in the papers by M. Baker and S. Norine and their followers at the beginning of the century. This theory considers finite graphs as analogs of Riemann surfaces and branched coverings of graphs as holomorphic mappings. The genus of a graph is defined as the rank of its fundamental group. The main object of investigation in the paper is automorphism groups of a graph acting freely on the set of half-edges of the graph. These groups are discrete analogs of groups of conformal automorphisms of a Riemann surface. The celebrated Hurwitz theorem (1893) states that the order of the group of conformal automorphisms of a compact Riemann surface of genus g > 1 does not exceed 84(g — 1). Later, K. Oikawa and T. Arakawa refined this bound in the case of groups that fix several finite sets of prescribed cardinalities. This paper provides proofs of discrete versions of the mentioned theorems. In addition, a discrete analog of the E. Bujalance and G. Gromadzki theorem improving one of Arakawa’s results is obtained.

We study the holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface that are invariant under the natural anti-holomorphic involutions of the moduli space. Their relationships with the harmonic maps are established. As a bi-product, a question of Simpson on such sections, posed in \cite{Si2}, is answered., Comment: Final version; to appear in Communications in Mathematical Physics

Let $\{u_n\}$ be a sequence of maps from a compact Riemann surface $M$ with smooth boundary to a general compact Riemannian manifold $N$ with free boundary on a smooth submanifold $K\subset N$ satisfying \[ \sup_n \ \left(\|\nabla u_n\|_{L^2(M)}+\|\tau(u_n)\|_{L^2(M)}\right)\leq \Lambda, \] where $\tau(u_n)$ is the tension field of the map $u_n$. We show that the energy identity and the no neck property hold during a blow-up process. The assumptions are such that this result also applies to the harmonic map heat flow with free boundary, to prove the energy identity at finite singular time as well as at infinity time. Also, the no neck property holds at infinity time., Comment: to appear in Mathematische Annalen. First version online MPI MIS Preprint: 26/2016, 21. Mar. 2016

Any irreducible Dynkin diagram $\Delta$ is obtained from an irreducible Dynkin diagram $\Delta_h$ of type $\mathrm{ADE}$ by folding via graph automorphisms. For any simple complex Lie group $G$ with Dynkin diagram $\Delta$ and compact Riemann surface $\Sigma$, we give a Lie-theoretic construction of families of quasi-projective Calabi-Yau threefolds together with an action of graph automorphisms over the Hitchin base associated to the pair $(\Sigma, G)$ . These give rise to Calabi-Yau orbifolds over the same base. Their intermediate Jacobian fibration, constructed in terms of equivariant cohomology, is isomorphic to the Hitchin system of the same type away from singular fibers., Comment: 25 pages. Comments welcome!

We often call an extremal Kahler metric with finite singularities on a compact Riemann surface an HCMU (the Hessian of the Curvature of the Metric is Umbilical) metric. In this paper we consider the following question: if we give $N$ points $p_1, \ldots, p_N$ on $S^2$ and $N$ positive real numbers $2\pi \alpha_1, \ldots, 2\pi \alpha_N$ with $\alpha_n \neq 1$, $n = 1, \ldots, N$, what condition can guarantee the existence of a non-CSC HCMU metric which has conical singularities $p_1, \ldots, p_N$ with singular angles $2\pi \alpha_1, \ldots, 2\pi \alpha_N$ respectively. We prove that if there are at least $N-2$ integers in $\alpha_1, \ldots, \alpha_N$ then there exists one non-CSC HCMU metric on $S^2$ satisfying the condition stated above no matter where the given points are.

The period for a compact Riemann surface, defined by the integral of differential 1-forms, is a classical complex analytic invariant, strongly related to the complex structure of the surface. In this paper, we treat another complex analytic invariant called the pointed harmonic volume. As a natural extension of the period defined using Chen's iterated integrals, it captures more detailed information of the complex structure. It is also one of a few explicitly computable examples of complex analytic invariants. We obtain its new value for a certain pointed hyperelliptic curve. An application of the pointed harmonic volume is presented. We explain the relationship between the harmonic volume and first extended Johnson homomorphism on the mapping class group of a pointed oriented closed surface., 15 pages

Existence of Maxwell-Chern-Simons-Higgs (MCSH) vortices in a Hermitian line bundle $\L$ over a general compact Riemann surface $\Sigma$ is proved by a continuation method. The solutions are proved to be smooth both spatially and as functions of the Chern-Simons deformation parameter $\kappa$, and exist for all $|\kappa, Comment: 22 pages, 3 figures