Compact Riemann Surface Research Articles

Lifting a symmetry of a Riemann surface

Let X be a compact Riemann surface of genus $$g \ge 2$$ that possesses a fixed point free group H of automorphisms and let $$Y=X/H$$ denote the orbit space of X under the action of H. Assume Y possesses a symmetry $$\sigma ,$$ that is, an anticonformal involution. We give conditions that determine when $$\sigma $$ lifts to an anticonformal automorphism of the surface X. The study splits naturally into three cases according to the different topological types that $$\sigma $$ may possess. We apply the criterion to abelian groups and also to particular presentations of other types of groups.

Hurwitz's regular map 37 of genus 7: A polyhedral realization

A Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84(g − 1) automorphisms, where g is the genus of the surface. The Hurwitz surface of least genus is the Klein quartic of genus 3. A polyhedral realization without self-intersections of Klein’s quartic of genus 3 was found by E. Schulte and J. M. Wills in 1985. For the next possible genus of a Hurwitz surface, i.e., for the genus 7 case with 72 vertices, we provide a polyhedral realization without self-intersections. We also show a topological representation for which we have a corresponding model.

Bubbling analysis for approximate Lorentzian harmonic maps from Riemann surfaces

For a sequence of approximate harmonic maps (u_n,v_n) (meaning that they satisfy the harmonic system up to controlled error terms) from a compact Riemann surface with smooth boundary to a standard static Lorentzian manifold with bounded energy, we prove that identities for the Lorentzian energy hold during the blow-up process. In particular, in the special case where the Lorentzian target metric is of the form g_N -beta dt^2 for some Riemannian metric g_N and some positive function beta on N, we prove that such identities also hold for the positive energy (obtained by changing the sign of the negative part of the Lorentzian energy) and there is no neck between the limit map and the bubbles. As an application, we complete the blow-up picture of singularities for a harmonic map flow into a standard static Lorentzian manifold. We prove that the energy identities of the flow hold at both finite and infinite singular times. Moreover, the no neck property of the flow at infinite singular time is true.

Monodromy of the SL(n) and GL(n) Hitchin fibrations

We compute the monodromy of the Hitchin fibration for the moduli space of $L$-twisted $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$-Higgs bundles for any $n$, on a compact Riemann surface of genus $g>1$. We require the line bundle $L$ to either be the canonical bundle or satisfy $deg(L) > 2g-2$. The monodromy group is generated by Picard-Lefschetz transformations associated to vanishing cycles of singular spectral curves. We construct such vanishing cycles explicitly and use this to show that the $SL(n,\mathbb{C})$ monodromy group is a {\em skew-symmetric vanishing lattice} in the sense of Janssen. Using the classification of vanishing lattices over $\mathbb{Z}$, we completely determine the structure of the monodromy groups of the $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$ Hitchin fibrations. As an application we determine the image of the restriction map from the cohomology of the moduli space of Higgs bundles to the cohomology of a non-singular fibre of the Hitchin fibration.

Bounds of the Group of Automorphisms of Compact Riemann Surface With Reference To The Point Group of Sulphur Molecule

Bounds of the Group of Automorphisms of Compact Riemann Surface With Reference To The Point Group of Sulphur Molecule

Blowup behavior of harmonic maps with finite index

In this paper, we study the blow-up phenomena on the alpha _k-harmonic map sequences with bounded uniformly alpha _k-energy, denoted by {u_{alpha _k}: alpha _k>1 quad text{ and } quad alpha _ksearrow 1}, from a compact Riemann surface into a compact Riemannian manifold. If the Ricci curvature of the target manifold has a positive lower bound and the indices of the alpha _k-harmonic map sequence with respect to the corresponding alpha _k-energy are bounded, then we can conclude that, if the blow-up phenomena occurs in the convergence of {u_{alpha _k}} as alpha _ksearrow 1, the limiting necks of the convergence of the sequence consist of finite length geodesics, hence the energy identity holds true. For a harmonic map sequence u_k:(Sigma ,h_k)rightarrow N, where the conformal class defined by h_k diverges, we also prove some similar results.

Metrics of Constant Positive Curvature with Conical Singularities, Hurwitz Spaces, and Determinants of Laplacians

Let $f: X\to {\Bbb C}P^1$ be a meromorphic function of degree $N$ with simple poles and simple critical points on a compact Riemann surface $X$ of genus $g$ and let $\mathsf m$ be the standard round metric of curvature $1$ on the Riemann sphere ${\Bbb C}P^1$. Then the pullback $f^*\mathsf m$ of $\mathsf m$ under $f$ is a metric of curvature $1$ with conical singularities of conical angles $4\pi$ at the critical points of $f$. We study the $\zeta$-regularized determinant of the Laplace operator on $X$ corresponding to the metric $f^*\mathsf m$ as a functional on the moduli space of the pairs $(X, f)$ (i.e. on the Hurwitz space $H_{g, N}(1, \dots, 1)$) and derive an explicit formula for the functional.

The full group of automorphisms of non-orientable unbordered Klein surfaces of topological genus 7

To determine the full automorphism group of compact Riemann and Klein surfaces is a hard problem, although some partial results are known. For example, the automorphisms groups of hyperelliptic surfaces, and those of (compact, non-orientable, unbordered) Klein surfaces whose genus is less or equal to 6 are known. In this paper the full automorphism groups of the surfaces of genus 7 are calculated.

On the solution of the Liouville equation

We give a short and rigorous proof of the existence and uniqueness of the solution of the Liouville equation with sources, both elliptic and parabolic, on the sphere and on all higher genus compact Riemann surfaces.

Hermite–Padé approximants for meromorphic functions on a compact Riemann surface

The problem of the limiting distribution of the zeros and the asymptotic behaviour of the Hermite–Padé polynomials of the first kind is considered for a system of germs of meromorphic functions , , on an -sheeted Riemann surface . Nuttall's approach to the solution of this problem, based on a particular `Nuttall' partition of into sheets, is further developed. Bibliography: 36 titles.

One Existence Theorem for non-CSC Extremal Kähler Metrics with Conical Singularities on $S^2$

We often call an extremal Kähler 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 Flow of Gauge Transformations on Riemannian Surface with Boundary

We consider the gauge transformations of a metric $G$-bundle over a compact Riemannian surface with boundary. By employing the heat flow method, the local existence and the long time existence of generalized solution are proved.

Uniformization of one-parametric families of complex tori

We suggest an approximate method to find an elliptic function uniformizing a compact Riemann surface of genus 1 which is given as a ramified covering of the Riemann sphere. The method is based on including the surface into a smooth one-parametric family. We deduce a system of ordinary differential equations for critical points of elliptic functions uniformizing surfaces of the family.

Higgs bundles, the Toledo invariant and the Cayley correspondence

Motivated by the study of the topology of the character variety for a non-compact Lie group of Hermitian type G, we undertake a uniform approach, independent of classification theory of Lie groups, to the study of the moduli space of G-Higgs bundles over a compact Riemann surface. We give an intrinsic definition of the Toledo invariant of a G-Higgs bundle which relies on the Jordan algebra structure of the isotropy representation for groups defining a symmetric space of tube type, and prove a general Milnor–Wood type bound of this invariant when the G-Higgs bundle is semistable. Finally, we prove rigidity results when the Toledo invariant is maximal, establishing in particular a Cayley correspondence when G is of tube type, which reveals new topological invariants only seen in particular cases from the character variety viewpoint.

Betti multiplets, flows across dimensions and c-extremization

We consider 4d mathcal{N} = 1 SCFTs, topologically twisted on compact constant curvature Riemann surfaces, giving rise to 2d mathcal{N} = (0, 2) SCFTs. The exact R-current of these 2d SCFT extremizes the central charge c2d, similarly to the 4d picture, where the exact R-current maximizes the central charge a4d. There are global currents that do not mix with the R-current in 4d but their mixing becomes non trivial in 2d. In this paper we study the holographic dual of this process by analyzing a 5d mathcal{N} = 2 truncation of T1,1 with one Betti vector multiplet, dual to the baryonic current on the CFT side. The holographic realization of the flow across dimensions connects AdS5 to AdS3 vacua in the supergravity picture. We verify the existence of the flow to AdS3 solutions and we retrieve the field theory results for the mixing of the Betti vector with the graviphoton. Moreover, we extract the central charge from the Brown-Henneaux formula, matching with the results obtained in field theory. We develop a general formalism to obtain the central charge of a 2d SCFT from 5d mathcal{N} = 2 gauged supergravity with a generic number of vector multiplets, showing that its extremization corresponds to an attractor mechanism for the scalars in the supergravity picture.

Holonomy limits of complex projective structures

We study the limits of holonomy representations of complex projective structures on a compact Riemann surface in the Morgan–Shalen compactification of the character variety. We show that the dual R-trees of the quadratic differentials associated to a divergent sequence of projective structures determine the Morgan–Shalen limit points up to a natural folding operation. For quadratic differentials with simple zeros, no folding is possible and the limit of holonomy representations is isometric to the dual tree. We also derive an estimate for the growth rate of the holonomy map in terms of a norm on the space of quadratic differentials.

Asymptotic behaviors of Landau–Lifshitz flows from $$\mathbb {R}^2$$ R 2 to Kähler manifolds

In this paper, we study the asymptotic behaviors of finite energy solutions to the Landau–Lifshitz flows from \(\mathbb {R}^2\) into Kahler manifolds. First, we prove that the solution with initial data below the critical energy converges to a constant map in the energy space as \(t\rightarrow \infty \) for the compact Riemannian surface targets. In particular, when the target is a two dimensional sphere, we prove that the solution to the Landau–Lifshitz–Gilbert equation with initial data having an energy below \(4\pi \) converges to some constant map in the energy space. The proof bases on the method of induction on energy and geometric renormalizations. Second, for general compact Kahler manifolds and initial data of an arbitrary finite energy, we obtain a bubbling theorem analogous to the Struwe’s results on the heat flows.

The Jacobian of a Riemann surface and the geometry of the cut locus of simple closed geodesics

To any compact Riemann surface of genus g one may assign a principally polarized abelian variety of dimension g, the Jacobian of the Riemann surface. The Jacobian is a complex torus, and a Gram matrix of the lattice of a Jacobian is called a period Gram matrix. This paper provides upper and lower bounds for all the entries of the period Gram matrix with respect to a suitable homology basis. These bounds depend on the geometry of the cut locus of non-separating simple closed geodesics. Assuming that the cut loci can be calculated, a theoretical approach is presented followed by an example where the upper bound is sharp. Finally we give practical estimates based on the Fenchel-Nielsen coordinates of surfaces of signature (1,1), or Q-pieces. The methods developed here have been applied to surfaces that contain small non-separating simple closed geodesics in [BMMS].

A vanishing theorem for co-Higgs bundles on the moduli space of bundles

We consider smooth moduli spaces of semistable vector bundles of fixed rank and determinant on a compact Riemann surface $X$ of genus at least $3$. The choice of a Poincar\'e bundle for such a moduli space $M$ induces an isomorphism between $X$ and a component of the moduli space of semistable sheaves over $M$. We prove that $h^0(M, \text{End}({\mathcal E})\otimes TM)= 1$ for a vector bundle $\mathcal E$ on $M$ coming from this component. Furthermore, there are no nonzero integrable co-Higgs fields on $\mathcal E$.

Complex geometry in the entanglement entropy of fermionic chains

The geometry of Riemann surfaces plays a relevant role in the study of entanglement entropy in the ground state of a free fermionic chain. Recently, a new symmetry for the entropy in non critical theories has been discovered. It is based on the Möbius transformations in a compact Riemann surface associated to the Hamiltonian of the system. Here, we argue how to extend it to critical theories supporting our conjectures with numerical tests. We also highlight the intriguing parallelism that exists with conformal symmetry in real space.