Weierstrass factorizations in compact Riemann surfaces

Letν′ be the complementary in a compact Riemann surface ν of a point (or a finite set). In this paper are characterized the subfields, of the field of meromorphic functions inν′, containing sufficient functions to verify a factorization property, similar to that of the classical Weierstrass theorem. It is also seen that the field generated by the Baker functions is not of this type, and the problem is solved of determining the divisors, inν′, of the holomorphic functions admiting Weierstrass factorizations with Baker functions as factors. As an application, a theorem is obtained characterizing the infinite products, of meromorphic functions in ν with bounded degree, which converge normally inν′. The first-named author is partially supported by PB91-0188.     

Polyanalytic forms on compact Riemann surfaces

A sheaf of differentials on a compact Riemann surface supplied with a projective structure is said to be n-analytic if, in local projective coordinates, sections of the sheaf satisfy the differential equation { } For the projective structure induced by a covering mapping from the disk, an explicit characterization of the space of cross sections and of the space of first cohomologies of an n-analytic sheaf is given in terms of known spaces of sections of certain holomorphic sheaves. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 15–25. Translated by S. V. Kislyakov.     

Approximation by ridge function fields over compact sets

We study the approximation of a continuous function field over a compact set T by a continuous field of ridge approximants over T, named ridge function fields. We first give general density results about function fields and show how they apply to ridge function fields. We next discuss the parameterization of sets of ridge function fields and give additional density results for a class of continuous ridge function fields that admits a weak parameterization. Finally, we discuss the construction of the elements in that class.     

Riemann surfaces defined over the reals

The known (explicit) examples of Riemann surfaces not definable over their field of moduli are those with that field being a subfield of the reals but which cannot be defined over the reals. In this paper we provide explicit families of Riemann surfaces which are definable over the reals but cannot be defined over the field of moduli.     

Meromorphic differentials with twisted coefficients on compact Riemann surfaces

We apply the Dirichlet’s principle to a modified energy functional on Riemann surfaces to reprove the existence of harmonic metrics with certain prescribed singularities due to Simpson, Sabbah and Biquard–Boalch, and hence of differentials with twisted coefficients of the second and third kinds. As a by-product, this generalizes the classical theory of Abelian differentials on a compact Riemann surface to the case of twisted coefficients. This also proposes a more natural approach for general existence of harmonic metrics in the higher dimensional case. The author supported partially by NSF of China (No. 10471105, 10771160).     

Branched affine and projective structures on compact Riemann surfaces

It is first established that there exist linear manifolds of branched affine structures having certain nonpolar branch divisors and simple polar divisors on an arbitrary compact Riemann surface M of genus g≤1. When ≥2, it is shown that these linear manifolds form a complex analytic vector bundle over the manifold of simple polar divisors on M. When g=1, elliptic functions are used to construct certain projective structures on M. A partial determination is made as to which of these projective structures are affine and which are not.     

Metacyclic groups as automorphism groups of compact Riemann surfaces

Let X be a compact Riemann surface of genus \(g\ge 2\), and let G be a subgroup of \(\mathrm{Aut}(X)\). We show that if the Sylow 2-subgroups of G are cyclic, then \(|G|\le 30(g-1)\). If all Sylow subgroups of G are cyclic, then, with two exceptions, \(|G|\le 10(g-1)\). More generally, if G is metacyclic, then, with one exception, \(|G|\le 12(g-1)\). Each of these bounds is attained for infinitely many values of g.     

On extendability of group actions on compact Riemann surfaces

The question of whether a given group which acts faithfully on a compact Riemann surface of genus is the full group of automorphisms of (or some other such surface of the same genus) is considered. Conditions are derived for the extendability of the action of the group in terms of a concrete partial presentation for associated with the relevant branching data, using Singerman's list of signatures of Fuchsian groups that are not finitely maximal. By way of illustration, the results are applied to the special case where is a non-cyclic abelian group.

     

Uniqueness theorem for algebraic curves on compact Riemann surfaces

In this paper, we prove a uniqueness theorem for algebraic curves from a compact Riemann surface into complex projective spaces.     

A matrix representation for automorphisms of compact Riemann surfaces

In this paper we prove a result which has as corollaries theorems of Hurwitz, Accola, Grothendieck, and Serre on automorphisms of Riemann surfaces.