The homology systole of hyperbolic Riemann surfaces

The main goal of this note is to show that the study of closed hyperbolic surfaces with maximum length systole is in fact the study of surfaces with maximum length homological systole. The same result is shown to be true for once-punctured surfaces, and is shown to fail for surfaces with a large number of cusps.     

Self-dual connections,hyperbolic metrics and harmonic mappings on Riemann surfaces

The Sampson-Wolf model of Teichmüller space (using harmonic mappings) is shown to be exactly the same as the more recent Hitchin model (utilizing self-dual connections). Indeed, it is noted how the self-duality equations become the harmonicity equations. An interpretation of the modular group action in this model is mentioned.     

Hyperellipticity and systoles of Klein surfaces

Given a hyperelliptic Klein surface, we construct companion Klein bottles, extending our technique of companion tori already exploited by the authors in the genus 2 case. Bavard??s short loops on such companion surfaces are studied in relation to the original surface so to improve a systolic inequality of Gromov??s. A basic idea is to use length bounds for loops on a companion Klein bottle, and then analyze how curves transplant to the original non-orientable surface. We exploit the real structure on the orientable double cover by applying the coarea inequality to the distance function from the real locus. Of particular interest is the case of Dyck??s surface. We also exploit an optimal systolic bound for the M?bius band, due to Blatter.     

On the Weyl formula for the Laplace operator on hyperbolic Riemann surfaces

An explicit formula which gives an expansion in the zeros of the Selberg function of the second term in the Weyl formula for any strictly hyperbolic group is obtained, and some of its consequences are stated.     

Mixed initial boundary value problem for hyperbolic geometric flow on Riemann surfaces

The hyperbolic geometric flow equations is introduced recently by Kong and Liu motivated by Einstein equation and Hamilton Ricci flow. In this paper, we consider the mixed initial boundary value problem for hyperbolic geometric flow, and prove the global existence of classical solutions. The results show that, for any given initial metric on R² in certain class of metric, one can always choose suitable initial velocity symmetric tensor such that the solutions exist, and the scalar curvature corresponding to the solution metric gij keeps bounded. If the initial velocity tensor does not satisfy the certain conditions, the solutions will blow up at a finite time. Some special explicit solutions to the reduced equation are given.     

Riemann surfaces

Since the classical work of Riemann, Plein, Chobe, and Poincaré, in mathematics the interest in the theory of Riemann surfaces and groups has not abated. The present survey covers papers reviewed in RZhMat during the period 1967–1976 primarily in the sections Algebra. Topology. Geometry. The following topics are considered most completely and thoroughly: the topology of Riemann surfaces and their automorphisms, Fuchsian groups, Teichmüller spaces, and spaces of moduli.Translated from Itogi Nauki i Tekhniki, Algebra, Topologiya, Geometriya, Vol. 16, pp. 191–245, 1978,     

Riemann surfaces with orbifold points

We interpret the previously developed Teichmüller theory of surfaces with marked points on boundary components (bordered surfaces) as the Teichmüller theory of Riemann surfaces with orbifold points of order 2. In the Poincaré uniformization pattern, we describe necessary and sufficient conditions for the group generated by the Fuchsian group of the surface with added inversions to be of the almost hyperbolic Fuchsian type. All the techniques elaborated for the bordered surfaces (quantization, classical and quantum mapping-class group transformations, and Poisson and quantum algebra of geodesic functions) are equally applicable to the surfaces with orbifold points.     

Cluster sets in the large of meromorphic functions on Riemann surfaces

In this note we obtain the relationships in the large between cluster sets, asymptotic values, exceptional values of meromorphic functions on hyperbolic Riemann surfaces. Some of these are generalizations of the boundary theorems given by E. F. Collingwood and M. L. Cartwright.     

BMOA(R,m) and capacity density Bloch spaces on hyperbolic Riemann surfaces

We investigate a number of spaces of functions on Riemann surfaces which are related to Bloch spaces and functions of bounded mean oscillation (BMO). These spaces are defined using properties for the corresponding function spaces on the unit disk in the complex plane, and we show that, in general, different properties lead to different function spaces. We catalogue almost completely the various relationships between these spaces.     

Crystallography and Riemann surfaces

The level set of an elliptic function is a doubly periodic point set in ℂ. To obtain a wider spectrum of point sets, we consider, more generally, a Riemann surface S immersed in ℂ² and its sections (“cuts”) by ℂ More specifically, we consider surfaces S defined in terms of a fundamental surface element obtained as a conformai map of triangular domains in ℂ. The discrete group of isometries of ℂ² generated by reflections in the triangle edges leaves S invariant and generalizes double-periodicity. Our main result concerns the special case of maps of right triangles, with the right angle being a regular point of the map. For this class of maps we show that only seven Riemann surfaces, when cut, form point sets that are discrete in ℂ. Their isometry groups all have a rank 4 lattice subgroup, but only three of the corresponding point sets are doubly periodic in ℂ. The remaining surfaces form quasiperiodic point sets closely related to the vertex sets of quasiperiodic tilings. In fact, vertex sets of familiar tilings are recovered in all cases by applying the construction to a piecewise flat approximation of the corresponding Riemann surface. The geometry of point sets formed by cuts of Riemann surfaces is no less “rigid” than the geometry determined by a tiling, and has the distinct advantage in having a regular behavior with respect to the complex parameter which specifies the cut.