Narrow Results

We give a lower bound estimate of the sum of the square norm of the sections of the pluricanonical bundles over a Riemann surface of genus greater than 2 and Gauss curvature -1. Such an estimate must depend on the injective radius of the Riemann surface. However, using this estimate, we give a uniform estimate of the corona problem on Riemann surface. Here "uniform" means that the estimate depends only on the genus of Rieman surface, not on the injective radius.

We give relations between the Gauss map and the mean curvature vector field of a surface in the Euclidean unit n-sphere Sⁿ. These relations are necessary and sufficient conditions for the existence of a surface in Sⁿ with prescribed Gauss map and mean curvature vector field. We show that such surfaces can be expressed explicitly by using given data.

We find the upper bound for the total number of ovals of k (5 ≤ k ≤ 8) non-conjugate symmetries of a Riemann surface of genus g, without an additional assumption concerning their separabilities, filling this way the gap existing in known literature of the subject. We also prove that our bound is attained for infinitely many values of g. Finally, as a byproduct of our considerations, we obtain the algebraic structure of the group generated by the extremal configurations of the symmetries, provided it is a 2-group.

A Riemann surface X of genus g > 2 is elliptic-hyperelliptic if it admits a conformal involution h such that the orbit space X/〈h〉 has genus one. This elliptic-hyperelliptic involution h is unique for g > 5 [1]. In a previous article [3], we established the non-connectedness of the subspace of real elliptic-hyperelliptic algebraic curves in the moduli space of Riemann surfaces of genus g, when g is even and > 5. In this paper we improve this result and give a complete answer to the connectedness problem of the space of real elliptic-hyperelliptic surfaces of genus > 5: we show that is connected if g is odd and has exactly two connected components if g is even; in both cases the closure of in the compactified moduli space is connected.

Let X be a compact connected Riemann surface of genus at least two. Let r be a prime number and ξ → X a holomorphic line bundle such that r is not a divisor of degree ξ. Let denote the moduli space of stable vector bundles over X of rank r and determinant ξ. By Γ we will denote the group of line bundles L over X such that L^⊗r is trivial. This group Γ acts on by the rule (E, L) ↦ E ⊗ L. We compute the Chen–Ruan cohomology of the corresponding orbifold.

For a prime p, the gauge group of a principal U(p)-bundle over a compact, orientable Riemann surface is decomposed up to homotopy as a product of spaces, each of which is commonly known. This is used to deduce explicit computations of the homotopy groups of the moduli space of stable vector bundles through a range, answering a question of Daskalopoulos and Uhlenbeck.

We study a natural map from representations of a free (respectively, free abelian) group of rank g in GL_r(ℂ), to holomorphic vector bundles of degree zero over a compact Riemann surface X of genus g (respectively, complex torus X of dimension g). This map defines what is called a Schottky functor. Our main result is that this functor induces an equivalence between the category of unipotent representations of Schottky groups and the category of unipotent vector bundles on X. We also show that, over a complex torus, any vector or principal bundle with a flat holomorphic connection is Schottky.

Let $X$ be a connected open Riemann surface. Let $Y$ be an Oka domain in the smooth locus of an analytic subvariety of ${ℂ}^n$, $n\ge 1$, such that the convex hull of $Y$ is all of ${ℂ}^n$. Let ${\mathcal{𝒪}}_{\ast }(X,Y)$ be the space of nondegenerate holomorphic maps $X\to Y$. Take a holomorphic 1-form $𝜃$ on $X$, not identically zero, and let $\pi :{\mathcal{𝒪}}_{\ast }(X,Y)\to H^1(X,{ℂ}^n)$ send a map $g$ to the cohomology class of $g𝜃$. Our main theorem states that $\pi$ is a Serre fibration. This result subsumes the 1971 theorem of Kusunoki and Sainouchi that both the periods and the divisor of a holomorphic form on $X$ can be prescribed arbitrarily. It also subsumes two parametric h-principles in minimal surface theory proved by Forstnerič and Lárusson in 2016.

In this work, we prove that the hyperelliptic branch locus of orientable Klein surfaces of genus $g$ with one boundary component is connected and in the case of non-orientable Klein surfaces it has $\frac{g+1}{2}$ components, if $g$ is odd, and $\frac{g+2}{2}$ components for even $g$. We notice that, for non-orientable Klein surfaces with two boundary components, the hyperelliptic branch loci are connected for all genera.

A compact Riemann surface $X$ of genus $g$ is called pseudo-real if it admits an anticonformal automorphism but no anticonformal involution. In this paper, we study pseudo-real $(q,n)$-gonal Riemann surfaces of genera greater or equal to two; these surfaces have anticonformal automorphisms of prime order $n$ such that the quotient spaces have genus $q$.

In this paper, we show that the gauge group of a principal $PU(n)$-bundle over a compact Riemann surface decomposes up to homotopy as the product of factors, one of which is a corresponding gauge group for $S^2$ and the others are immediately recognizable spaces. Further, when $n$ is a prime $p$, the gauge group for $S^2$ decomposes as a product of immediately recognizable factors. These gauge groups have strong connections to moduli spaces of stable vector bundles.

We give a geometric construct of a modular functor for any simple Lie-algebra and any level by twisting the constructions in [16, 19] by a certain fractional power of the abelian theory first considered in [13] and further studied in [2].

Partial cross-sections of the $n\alpha$ and $dt$ collisions in the quantum state $J^{\pi }={\frac{3}{2}}^{+}$ near the $dt$-threshold, taken from an available $R$-matrix analysis, are fitted using the semi-analytic multi-channel Jost matrix with proper analytic structure and some adjustable parameters. Then the spectral points are sought as zeros of the Jost matrix determinant (which correspond to the $S$-matrix poles) at complex energies. The correct analytic structure makes it possible to calculate (analytically continue) the fitted Jost matrix on any sheet of the Riemann surface whose topology involves not only the square-root but also the logarithmic branching caused by the Coulomb interaction. As a result of such an analytic continuation, it was found that the previously established ${\frac{3}{2}}^{+}$-resonance (at $\sim 47\mathrm{\text{keV}}$) and its shadow pole (at $\sim 80\mathrm{\text{keV}}$) both are split in overlapping pairs. Apart from studying the properties of a specific nuclear state, it is also proved for a general multi-channel problem that the Coulomb forces change the topology of the Riemann surface as well as destroy the so-called mirror symmetry of the $S$-matrix.

The available $R$-matrix parametrization of experimental data on the excitation functions for the elastic and inelastic $p{}^7\mathrm{\text{Be}}$ scattering at the collision energies up to 3.4MeV is used to generate the corresponding partial-wave cross-sections in the states with $J^{\pi }=0^{+}$. Thus, obtained data are fitted using the semi-analytic two-channel Jost matrix with a proper analytic structure and some adjustable parameters. Then the spectral points are sought as zeros of the Jost matrix determinant (which correspond to the $S$-matrix poles) at complex energies. The correct analytic structure makes it possible to calculate the fitted Jost matrix on any sheet of the Riemann surface whose topology involves not only the square-root but also the logarithmic branching caused by the Coulomb interaction. In this way, two overlapping $0^{+}$ resonances at the excitation energies $\sim 1.79$ and $\sim 1.96$MeV have been found.

In this paper, we consider the elliptic systems

We prove that every open Riemann surface $M$ is the complex structure of a complete surface of constant mean curvature $1$ ($\mathrm{\text{CMC}}$‒1) in the three-dimensional hyperbolic space ${ℍ}^3$. We go further and establish a jet interpolation theorem for complete conformal $\mathrm{\text{CMC}}$‒1 immersions $M\to {ℍ}^3$. As a consequence, we show the existence of complete densely immersed $\mathrm{\text{CMC}}$‒1 surfaces in ${ℍ}^3$ with arbitrary complex structure. We obtain these results as application of a uniform approximation theorem with jet interpolation for holomorphic null curves in ${ℂ}^2\times {ℂ}^{\ast }$ which is also established in this paper.

Let G be a finite group. The strong symmetric genusσ⁰(G) is the minimum genus of any Riemann surface on which G acts preserving orientation. We show that a 2-group G has strong symmetric genus congruent to 3 (mod 4) if and only if G is in one of 14 families of groups. A consequence of this classification is that almost all positive integers that are the genus of a 2-group are congruent to 1 (mod 4).

Let G be a finite group. The strong symmetric genus σ⁰(G) is the minimum genus of any Riemann surface on which G acts faithfully and preserving orientation. Assume that G is non-abelian and generated by two elements, one of which is an involution, and that n is relatively prime to |G|. Our first main result is the determination of the strong symmetric genus of the direct product Z_n ×G in terms of n, |G|, and a parameter associated with the group G. We obtain a variety of genus formulas. Finally, we apply these results to prove that for each integer g ≥ 2, there are at least four groups of strong symmetric genus g.

There are two families of geometric structures associated to a surface, with both structures related to representations of the fundamental group of the surface into SL(2, ℂ). These are projective structures on the surface, and Higgs bundles for a given conformal structure of the surface. This note discusses the links between the two.

The flow of a superfluid film adsorbed on a porous medium can be modeled by a meromorphic differential on a Riemann surface of high genus. In this paper, we define the mixed Hodge metric of meromorphic differentials on a Riemann surface and justify using this metric to approximate the kinetic energy of a superfluid film flowing on a porous surface.