A positive compact operator

I give an example of a positive compact operator on L²[0,1] which is not representable by any Lebesgue measurable function on [0,1]². This example can be adapted to answer a question of H.H.Schaefer (§4 below).     

Involutions of compact Klein surfaces

Partially supported by British-Spanish Joint Research Program     

Minimal genus problem for pseudo-real Riemann surfaces

A Riemann surface is said to be pseudo-real if it admits an antiholomorphic automorphism but not an antiholomorphic involution (also known as a symmetry). The importance of such surfaces comes from the fact that in the moduli space of compact Riemann surfaces of given genus, they represent the points with real moduli. Clearly, real surfaces have real moduli. However, as observed by Earle, the converse is not true. Moreover, it was shown by Seppälä that such surfaces are coverings of real surfaces. Here we prove that the latter may always be assumed to be purely imaginary. We also give a characterization of finite groups being groups of automorphisms of pseudo-real Riemann surfaces. Finally, we solve the minimal genus problem for the cyclic case.     

The minimal genus problem for elliptic surfaces

We solve a certain case of the minimal genus problem for embedded surfaces in elliptic 4-manifolds. The proofs involve a restricted transitivity property of the action of the orientation preserving diffeomorphism group on the second homology. In the case we consider we get the minimal possible genus allowed by the adjunction inequality.     

Sets of periods for automorphisms of compact Riemann surfaces

Let G=〈f〉 be a finite cyclic group of order N that acts by conformal automorphisms on a compact Riemann surface S of genus g≥2. Associated to this is a set A of periods defined to be the subset of proper divisors d of N such that, for some x∈S, x is fixed by f^d but not by any smaller power of f. For an arbitrary subset A of proper divisors of N, there is always an associated action and, if g_A denotes the minimal genus for such an action, an algorithm is obtained here to determine g_A. Furthermore, a set A_max is determined for which g_A is maximal.     

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.     

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.     

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.     

Logarithmic foliations on compact algebraic surfaces

LetF be a holomorphic singular foliation with an invariant compact curveS on a compact algebraic surfaceX. In this work we prove thatF is logarithmic under some generic conditions in the singularities ofF inS.Supported by CNPq (Brazil).     

A variational proof for the existence of a conformal metric with preassigned negative Gaussian curvature for compact Riemann surfaces of genus > 1

Given a smooth functionK < 0 we prove a result by Berger, Kazhdan and others that in every conformal class there exists a metric which attains this function as its Gaussian curvature for a compact Riemann surface of genusg > 1. We do so by minimizing an appropriate functional using elementary analysis. In particular forK a negative constant, this provides an elementary proof of the uniformization theorem for compact Riemann surfaces of genusg > 1. An erratum to this article is available at .