The argument principle and holomorphic extendibility

LetD be a bounded domain in the complex plane whose boundary consists of finitely many pairwise disjoint simple closed curves. GivebD the standard orientation, and letA(D) be the algebra of all continuous functions on which are holomorphic onD. We prove that a continuous functionf onbD extends to a function inA(D) if and only if for eachg∈A(D) such thatf+g≠0 onbD, the change of argument off+g alongbD is nonnegative.     

The equivalence classes of holomorphic mappings of genus 3 Riemann surfaces onto genus 2 Riemann surfaces

Denote the set of all holomorphic mappings of a genus 3 Riemann surface S ₃ onto a genus 2 Riemann surface S ₂ by Hol(S ₃, S ₂). Call two mappings f and g in Hol(S ₃, S ₂) equivalent whenever there exist conformal automorphisms α and β of S ₃ and S ₂ respectively with f ? α = β ? g. It is known that Hol(S ₃, S ₂) always consists of at most two equivalence classes.We obtain the following results: If Hol(S ₃, S ₂) consists of two equivalence classes then both S ₃ and S ₂ can be defined by real algebraic equations; furthermore, for every pair of inequivalent mappings f and g in Hol(S ₃, S ₂) there exist anticonformal automorphisms α? and β? with f ? α? = β? ? g. Up to conformal equivalence, there exist exactly three pairs of Riemann surfaces (S ₃, S ₂) such that Hol(S ₃, S ₂) consists of two equivalence classes.     

Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians

Two compactifications of the space of holomorphic maps of fixed degree from a compact Riemann surface to a Grassmannian are studied. It is shown that the Uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the Grothendieck Quot Scheme. The latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by Bradlow. This gives an effective way of computing certain intersection numbers (known as ``Gromov invariants') on the space of holomorphic maps into Grassmannians. We carry out these computations in the case where the Riemann surface has genus one.

     

The argument principle and integral curvature of minimal surfaces

In this paper we present a generalization of the argument principle on meromorphic curves and with its help we establish some relationships among the geometrical and topological characteristics of generalized meromorphic minimal surfaces in Rⁿ.Translated from Matematicheskie Zametki, Vol. 15, No. 5, pp. 691–700, May, 1974.In conclusion the author expresses his thanks to B. V. Shabat for valuable remarks made as my paper was being prepared for publication.     

An extremal problem for Dirichlet-finite holomorphic functions on Riemann surfaces

Iff is a nonconstant holomorphic function with finite Dirichlet integralD(f) on a Riemann surfaceR, then |f|² has the least harmonic majorantf ₂ onR. We show Σf ₂(a ≦π ⁻¹ D(f)), wherea runs over all the roots off = 0 onR. The equality holds if and only iff is of type ℬℓ₁ fromR onto a disk of center 0. A consideration is proposed for the non-Euclidean case.     

Bounds on the number of holomorphic maps of compact Riemann surfaces

We give bounds on the number of nonconstant holomorphic maps of compact Riemann surfaces of genera 1$">.

     

A conformally invariant metric on Riemann surfaces associated with integrable holomorphic quadratic differentials

In this paper, we define a conformally invariant (pseudo-)metric on all Riemann surfaces in terms of integrable holomorphic quadratic differentials and analyze it. This metric is closely related to an extremal problem on the surface. As a result, we have a kind of reproducing formula for integrable quadratic differentials. Furthermore, we establish a new characterization of uniform thickness of hyperbolic Riemann surfaces in terms of invariant metrics.     

Zeta-determinant and torsion functions on Riemann surfaces of finite volume

We define ζ-determinant andL ²-analytic torsion functions for a Riemann surface of finite volume. We use the Selberg trace formula to express these determinant and torsion functions in terms of four Zeta functions which are related to the structure of discrete groups. A new invariant is also obtained.     

The holomorphic flow of the Riemann zeta function

The flow of the Riemann zeta function, , is considered, and phase portraits are presented. Attention is given to the characterization of the flow lines in the neighborhood of the first 500 zeros on the critical line. All of these zeros are foci. The majority are sources, but in a small proportion of exceptional cases the zero is a sink. To produce these portraits, the zeta function was evaluated numerically to 12 decimal places, in the region of interest, using the Chebyshev method and using Mathematica.

The phase diagrams suggest new analytic properties of zeta, of which some are proved and others are given in the form of conjectures.

     

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.