Compact Riemann Surface Research Articles

With respect to whom are you critical?

For any compact Riemannian surface S and any point y in S, Qy−1 denotes the set of all points in S for which y is a critical point, and |Qy−1| its cardinality. We proved [2] together with Imre Bárány that |Qy−1|≥1, and that equality for all y∈S characterizes the surfaces homeomorphic to the sphere. Here we show, for any orientable surface S and any point y∈S, the following two main results. There exists an open and dense set of Riemannian metrics g on S for which y is critical with respect to an odd number of points in S, and this is sharp. If S is the torus then |Qy−1|≤5, and if S has genus g≥2 then |Qy−1|≤8g−5. Properties involving points at globally maximal distance on S are eventually presented.

Isometric Embeddings of Subsets of Boundaries of Teichmüller Spaces of Compact Hyperbolic Riemann Surfaces

It is known that every finitely unbranched holomorphic covering π:$$\pi :\tilde{S} \to S$$ of a compact Riemann surface S with genus g ≥ 2 induces an isometric embedding $${{\rm{\Phi}}_{\rm{\pi}}}:Teich\left(S \right) \to Teich\left({\widetilde{S}} \right)$$. By the mutual relations between Strebel rays in Teich(S) and their embeddings in $$Teich\left({\widetilde{S}} \right)$$, we show that the 1st-strata space of the augmented Teichmuller space $$\widehat{Teich}\left(S \right)$$ can be embedded in the augmented Teichmuller space $$\widehat{Teich}\left({\widetilde{S}} \right)$$ isometrically. Furthermore, we show that Φπ induces an isometric embedding from the set Teich(S) B (∞) consisting of Busemann points in the horofunction boundary of Teich(S) into $$Teich{\left({\widetilde{S}} \right)_B}\left(\infty \right)$$ with the detour metric.

A geodesic-preserving map of a Euclidean cone disk is affine

In this paper, we solve a rigidity problem on geodesic-preserving maps of the unit disk D into itself. The unit disk D is equipped with a Euclidean cone metric which is induced by the lift of a holomorphic quadratic differential on a compact Riemann surface. The main result is that an injective map of the disk into itself is an affine homeomorphism if it is geodesic-preserving.

Dynamics of holomorphic correspondences on Riemann surfaces

We study the global dynamics of holomorphic correspondences [Formula: see text] on a compact Riemann surface [Formula: see text] in the case, so far not well understood, where [Formula: see text] and [Formula: see text] have the same topological degree. In the absence of a mild and necessary obstruction that we call weak modularity, [Formula: see text] admits two canonical probability measures [Formula: see text] and [Formula: see text] which are invariant by [Formula: see text] and [Formula: see text] respectively. If the critical values of [Formula: see text] (respectively, [Formula: see text]) are not periodic, the backward (respectively, forward) orbit of any point [Formula: see text] equidistributes towards [Formula: see text] (respectively, [Formula: see text]), uniformly in [Formula: see text] and exponentially fast.

On Some Classical andWeighted Estimates for SL4

The purpose of this paper is two-sided. First, we obtain the correct estimate of the error term in the classical prime geodesic theorem for compact symmetric space SL4. As it turns out, the corrected error term depends on the degree of a certain polynomial appearing in the functional equation of the attached zeta function. This is in line with the known result in the case of compact Riemann surface, or more generally, with the corresponding result in the case of compact locally symmetric spaces of real rank one. Second, we derive a weighted form of the theorem. In particular, we prove that the aforementioned error term can be significantly improved when the classical approach is replaced by its higher level analogue.

A solution to Flinn’s conjecture on weakly expansive flows

In L. W. Flinn’s PhD thesis published in 1972, the author conjectured that weakly expansive flows are also expansive flows. In this paper we use the horocycle flow on compact Riemann surfaces of constant negative curvature to show that Flinn’s conjecture is not true.

Gravitating vortices and the Einstein–Bogomol’nyi equations

In this work we consider the gravitating vortex equations. These equations couple a metric over a compact Riemann surface with a hermitian metric over a holomorphic line bundle equipped with a fixed global section — the Higgs field —, and have a symplectic interpretation as moment-map equations. As a particular case of the gravitating vortex equations on $${{\mathbb {P}}}^1$$ , we find the Einstein–Bogomol’nyi equations, previously studied in the theory of cosmic strings in physics. We prove two main results in this paper. Our first main result gives a converse to an existence theorem of Yang for the Einstein–Bogomol’nyi equations, establishing in this way a correspondence with Geometric Invariant Theory for these equations. In particular, we prove a conjecture by Yang about the non-existence of cosmic strings on $${{\mathbb {P}}}^1$$ superimposed at a single point. Our second main result is an existence and uniqueness result for the gravitating vortex equations in genus greater than one.

Periods of holomorphic maps on compact Riemann surfaces and product of spheres

ABSTRACT In this article, we consider non-constant holomorphic maps on Riemann surfaces and product of Riemann spheres, and we give conditions on the maps in order that they have arbitrary large prime numbers as periods. We use Lefschetz fixed point theory and in particular we compute the Lefschetz numbers of period m for large m's.

One-Cycle Sweepout Estimates of Essential Surfaces in Closed Riemannian Manifolds

We present new free-curvature one-cycle sweepout estimates in Riemannian geometry, both on surfaces and in higher dimension. More precisely, we derive upper bounds on the length of one-parameter families of one-cycles sweeping out essential surfaces in closed Riemannian manifolds. In particular, we show that there exists a homotopically substantial one-cycle sweepout of the essential sphere in the complex projective space, endowed with an arbitrary Riemannian metric, whose one-cycle length is bounded in terms of the volume (or diameter) of the manifold. This is the first estimate on sweepout volume in higher dimension without curvature assumption. We also give a detailed account of the situation for compact Riemannian surfaces with or without boundary, in relation with questions raised by P.~Buser and L.~Guth.

CM-points and lattice counting on arithmetic compact Riemann surfaces

Let X(D,1)=Γ(D,1)\\H denote the Shimura curve of level N=1 arising from an indefinite quaternion algebra of fixed discriminant D. We study the discrete average of the error term in the hyperbolic circle problem over Heegner points of discriminant d<0 on X(D,1) as d→−∞. We prove that if |d| is sufficiently large compared to the radius r≈log⁡X of the circle, we can improve on the classical O(X2/3)-bound of Selberg. Our result extends the result of Petridis and Risager for the modular surface to arithmetic compact Riemann surfaces.

Group Actions on cyclic covers of the projective line

We use tools from combinatorial group theory in order to study actions of three types on groups acting on a curve, namely the automorphism group of a compact Riemann surface, the mapping class group acting on a surface (which now is allowed to have some points removed) and the absolute Galois group $\mathrm{Gal}(\bar{\Q}/\Q)$ in the case of cyclic covers of the projective line.

A Dirichlet Problem in Noncommutative Potential Theory

We prove the solvability of a Dirichlet problem for flat hermitian metrics on Hilbert bundles over compact Riemann surfaces with boundary. We also prove a factorization result for flat hermitian metrics on doubly connected domains.

The pseudo-real genus of a group

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 genusψ(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 genusψ⁎(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).

An inequality for the Selberg zeta-function, associated to the compact Riemann surface

Abstract We consider the absolute values of the Selberg zeta-function, associated to the compact Riemann surface, at places symmetric with respect to the line ℛ(s) = 1/2. We prove an inequality for the Selberg zeta-function, extending the result of R. Garunkštis and A. Grigutis.

A note on large automorphism groups of compact Riemann surfaces

Belolipetsky and Jones classified those compact Riemann surfaces of genus g admitting a large group of automorphisms of order λ(g−1), for each λ>6, under the assumption that g−1 is a prime number. In this article we study the remaining large cases; namely, we classify Riemann surfaces admitting 5(g−1) and 6(g−1) automorphisms, with g−1 a prime number. As a consequence, we obtain the classification of Riemann surfaces admitting a group of automorphisms of order 3(g−1), with g−1 a prime number. We also provide isogeny decompositions of their Jacobian varieties.

The occurrence of finite simple permutation groups of fixity 2 as automorphism groups of Riemann surfaces

Motivated by the theory of Riemann surfaces and specifically the significance of Weierstrass points, we study finite simple groups from a permutation action point of view. We classify all possibilities for finite simple groups acting faithfully on a compact Riemann surface of genus at least 2 in such a way that all non-trivial elements have at most two fixed points on each non-regular orbit and at most four fixed points in total.

On the Existence of Non-CSC Extremal Kähler Metrics with Finite Singularities on $$S^2$$

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 problem: suppose \(\alpha _{1},\alpha _{2},\ldots ,\alpha _{N}\) are \(N\ge 4\) nonnegative real numbers with \(\alpha _{j}\ge 2\;(1\le j\le J\le N-3)\) being integers such that $$\begin{aligned} \sum _{j=1}^{J}\alpha _{j}+2-N\ge 0, \end{aligned}$$ given any J points \(p_{1},\ldots ,p_{J}\) on \(S^{2}\setminus \{0,\infty \}\), whether there exists a non-CSC conformal HCMU metric g with singular angles\(2\pi \alpha _{1},\ldots ,2\pi \alpha _{N}\), which belongs to the first class (see Definition 1.1) such that \(p_{1},\ldots ,p_{J}\) are all saddle points of scalar curvature R of g and \(0,\infty \) are extremal point of R. We will give a sufficient condition when R has only one saddle point. As its application, we prove that when the number of the singularities is 4, Obstruction Theorem is also a sufficient condition for the existence of a non-CSC conformal HCMU metric on \(S^{2}\).

𝑉-harmonic morphisms between Riemannian manifolds

A V V -harmonic morphism u : M → N u:M\to N between Riemannian manifolds is a smooth map which pulls back germs of harmonic functions on N N to germs of V V -harmonic functions on M M , where V V is a smooth vector field on M M . In this paper, we give some characterizations and examples of V V -harmonic morphisms. In addition, a dilation estimate and a Liouville-type theorem of V V -harmonic morphisms from noncompact complete manifolds are also established. As applications, we obtain the Liouville-type theorems for V V -harmonic morphisms from complete manifolds of nonnegative Bakry-Émery Ricci curvature, especially complete steady or shrinking Ricci solitons, to manifolds of dimension at least three or compact Riemann surface of genus at least two.

Mixed threefolds isogenous to a product

In this paper, we study threefolds isogenous to a product of mixed type i.e. quotients of a product of three compact Riemann surfaces [Formula: see text] of genus at least two by the action of a finite group [Formula: see text], which is free, but not diagonal. In particular, we are interested in the systematic construction and classification of these varieties. Our main result is the full classification of threefolds isogenous to a product of mixed type with [Formula: see text] in the absolutely faithful case, i.e. any automorphism in [Formula: see text], which restricts to the trivial element in [Formula: see text] for some [Formula: see text], is the identity on the product. Since the holomorphic Euler–Poincaré-characteristic of a smooth threefold of general type with ample canonical class is always negative, these examples lie on the boundary, in the sense of threefold geography. To achieve our result we use techniques from computational group theory. Indeed, we develop a MAGMA algorithm to classify these threefolds for any given value of [Formula: see text] in the absolutely faithful case.

A Proof of the Weierstra\xdf Gap Theorem not Using the Riemann\u2013Roch Formula

Usually, the Weierstraß gap theorem is derived as a straightforward corollary of the Riemann–Roch theorem. Our main objective in this article is to prove the Weierstraß gap theorem by following an alternative approach based on “first principles”, which does not use the Riemann–Roch formula. Having mostly applications in connection with modular functions in mind, we describe our approach for the case when the given compact Riemann surface is associated with the modular curve X_0(N).