arXiv subjects

Alvaro Alvarez-Parrilla

Publications and source records attributed to Alvaro Alvarez-Parrilla.

7 recordsLinked to original sources

arXiv2026

Tessellations and Speiser graphs arising from meromorphic functions on simply connected Riemann surfaces

Motivated by W. P. Thurston, we ask: What is the shape of a meromorphic function on a simply connected Riemann surface $Ω_z$? We consider Speiser functions, i.e. meromorphic functions on a simply connected Riemann surface, that have a finite number $q$ at least 2 of singular (critical or asymptotic) values. As a first result, we make precise the correspondence between: Speiser functions $w(z)$, Speiser Riemann surfaces $R_w(z)$, Speiser $q$-tessellation, and analytic Speiser graphs of index $q$. As the second main result, we characterize tessellations with alternating colors (equivalently abstract pre-Speiser graphs) that are realized by Speiser functions on $Ω_z$. The characterization is in terms of the $q$-regular extension problem of bipartite planar graphs. As third main results, the Speiser Riemann surface $R_w(z)$ can be constructed by isometric glueing of a finite number of types of sheets, where each sheet is a maximal domain of single-valuedness of the inverse of $w(z)$. Furthermore, a unique decomposition of $R_w(z)$ into maximal logarithmic towers and a soul is provided. Using vector fields we recognize that logarithmic towers come in two flavors: exponential or $h$-tangent blocks, directly related to the exponential or the hyperbolic tangent functions on the upper half plane. The surface $R_w(z)$ of a finite Speiser function is characterized by surgery of a rational block and a finite number of exponential or $h$-tangent blocks.

math.CV↗

arXiv2025

Tessellations of rational complex functions and the Riemann's existence theorem

A complex rational function R, of degree n>1, on a compact Riemann surface M provided with a cyclic order of its q critical values, determines an homogeneous tessellation of the Riemann surface M, whose 2n tiles are topological q-gons with alternating colors.The tessellation provides a simple and straighforward visual description of the rational function R. Conversely, assume a possibly non homogeneous tessellation T of a compact differentiable surface M' with tiles of alternating colors and a suitable labelling in the vertices of its tiles. Non homogeneous means that the tiles of T are r-gons, for different values of r. Then there exists a Riemann surface structure M on M', a complex rational function R and a cyclic order of its critical values, such that the tessellation of R on M topologically coincides with the original T.

math.CV↗

arXiv2024

Geometry of transcendental singularities of complex analytic functions and vector fields

On Riemann surfaces $M$, there exists a canonical correspondence between a possibly multivalued function $Ψ_X$ whose differential is single valued ($i.e.$ an additively automorphic singular complex analytic function) and a vector field $X$. From the point of view of vector fields, the singularities that we consider are zeros, poles, isolated essential singularities and accumulation points of the above. The theory of singularities of the inverse function $Ψ_X^{-1}$ is extended from meromorphic functions to additively automorphic singular complex analytic functions. The main contribution is a complete characterization of when a singularity of $Ψ_X^{-1}$ is either algebraic, logarithmic or arises from a zero with nonzero residue of $X$. Relationships between analytical properties of $Ψ_X$, singularities of $Ψ_X^{-1}$ and singularities of $X$ are presented. Families and sporadic examples showing the geometrical richness of vector fields on the neighbourhoods of the singularities of $Ψ_X^{-1}$ are studied. As applications we have; a description of the maximal univalence regions for complex trajectory solutions of a vector field $X$, a geometric characterization of the incomplete real trajectories of a vector field $X$, and a description of the singularities of the vector field associated to the Riemann $ξ$ function.

math.CV↗

arXiv2021

Dynamics of singular complex analytic vector fields with essential singularities II

The singular complex analytic vector fields $X$ on the Riemann sphere $\widehat{\mathbb C}_z$ belonging to the family ${\mathscr E}(r,d)=\left\{ X(z)=\frac{1}{P(z)} e^{E(z)}\frac{\partial }{\partial z}\ \Big\vert \ P, E\in\mathbb{C}[z]\right\}$, where $P$ is monic, $deg(P)=r$, $deg(E)=d$, $r+d\geq 1$, have a finite number of poles on the complex plane and an isolated essential singularity at infinity (for $d\geq 1$). Our aim is to describe geometrically $X$, particularly the singularity at infinity. We use the natural one to one correspondence between $X$, a global singular analytic distinguished parameter $Ψ_X(z)=\int^z P(ζ) e^{-E(ζ)}dζ$, and the Riemann surface ${\mathcal R}_X$ of this distinguished parameter. We introduce $(r,d)$-configuration trees which are weighted directed rooted trees. An $(r,d)$-configuration tree completely encodes the Riemann surface ${\mathcal R}_X$ and the singular flat metric associated on ${\mathcal R}_X$. The $(r,d)$-configuration trees provide "parameters" for the complex manifold ${\mathscr E}(r,d)$, which give explicit geometrical and dynamical information; a valuable tool for the analytic description of $X\in{\mathscr E}(r,d)$. Furthermore, given $X$, the phase portrait of the associated real vector field $Re(X)$ on the Riemann sphere is decomposed into $Re(X)$-invariant components: half planes and finite height strips. The germ of $X$ at infinity is described as a combinatorial word (consisting of hyperbolic, elliptic, parabolic and entire angular sectors having the point at infinity of $\widehat{\mathbb C}_z$ as center). The structural stability, under perturbation in ${\mathscr E}(r,d)$, of the phase portrait of $Re(X)$ is characterized by using the $(r,d)$-configuration trees. We provide explicit conditions, in terms of $r$ and $d$, as to when the number of topologically equivalent phase portraits of $Re(X)$ is unbounded.

math.DS↗

arXiv2020

On the geometry, flows and visualization of singular complex analytic vector fields on Riemann surfaces

Motivated by the wild behavior of isolated essential singularities in complex analysis, we study singular complex analytic vector fields $X$ on arbitrary Riemann surfaces $M$. By vector field singularities we understand zeros, poles, isolated essential singularities and accumulation points of the above kind. In this framework, a singular analytic vector field $X$ has canonically associated; a 1-form, a quadratic differential, a flat metric (with a geodesic foliation), a global distinguished parameter or $\mathbb{C}$-flow box $Ψ_X$, a Newton map $Φ_X$, and a Riemann surface $\mathcal{R}_X$ arising from the maximal $\mathbb{C}$-flow of $X$. We show that every singular complex analytic vector field $X$ on a Riemann surface is in fact both a global pullback of the constant vector field under $Ψ_X$ and of the radial vector field on the sphere under $Φ_X$. As a result of independent interest, we show that the maximal analytic continuation of the a local $\mathbb{C}$-flow of $X$ is univalued on the Riemann surface $\mathcal{R}_{X} \subset M \times \mathbb{C}_t$, where $\mathcal{R}_X$ is the graph of $Ψ_{X}$. Furthermore we explore the geometry of singular complex analytic vector fields and present a geometrical method that enables us to obtain the solution, without numerical integration, to the differential equation that provides the $\mathbb{C}$-flow of the vector field. We discuss the theory behind the method, its implementation, comparison with some integration-based techniques, as well as examples of the visualization of complex vector fields on the plane, sphere and torus. Applications to visualization of complex valued functions is discussed including some advantages between other methods.

math.DS↗

arXiv2019

Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere

We consider the family $\mathcal{E}(s,r,d)=\Big\{ X(z)=\frac{Q(z)}{P(z)}\ e^{E(z)} \frac{\partial}{\partial z} \Big\},$ with $Q, P, E$ polynomials, deg${Q}=s$, deg$(P)=r$ and deg$(E)=d$, of singular complex analytic vector fields $X$ on the Riemann sphere $\widehat{\mathbb{C}}$. For $d\geq1$, $X\in\mathcal{E}(s,r,d)$ has $s$ zeros and $r$ poles on the complex plane and an essential singularity at infinity. Using the pullback action of the affine group $Aut(\mathbb{C})$ and the divisors for $X$, we calculate the isotropy groups $Aut(\mathbb{C})_{X}$ and the discrete symmetries for $X\in\mathcal{E}(s,r,d)$. Each subfamily $\mathcal{E}(s,r,d)_{id}$, of those $X$ with trivial isotropy group in $Aut(\mathbb{C})$, is endowed with a holomorphic trivial principal $Aut(\mathbb{C})$-bundle structure. Necessary and sufficient conditions in order to ensure the equality $\mathcal{E}(s,r,d)=\mathcal{E}(s,r,d)_{id}$ and those $X\in\mathcal{E}(s,r,d)$ with non-trivial isotropy are realized. Explicit global normal forms for $X\in\mathcal{E}(s,r,d)$ are presented. A natural dictionary between vector fields, 1-forms, quadratic differentials and functions is extended to include the presence of non-trivial discrete symmetries $Γ<Aut(\mathbb{C})$.

math.DS↗

arXiv2018

Classification of rational differential forms on the Riemann sphere, via their isotropy group

We classify the rational differential 1-forms with simple poles and simple zeros on the Riemann sphere according to their isotropy group; when the 1-form has exactly two poles the isotropy group is isomorphic to $\mathbb{C}^{*}$, namely $\{z\mapsto az\ \vert\ a\in\mathbb{C}, a\neq0\}$, and when the 1-form has $k\geq 3$ poles the isotropy group is finite. In particular we show that all the finite subgroups of $PSL(2,\mathbb{C})$ are realizable as isotropy groups for a rational 1-form on $\widehat{\mathbb{C}}$. We also present local and global geometrical conditions for their classification. The classification result enables us to describe the moduli space of rational 1-forms with finite isotropy that have exactly $k$ simple poles and $k-2$ simple zeros on the Riemann sphere. Moreover, we provide sufficient (geometrical) conditions for when the 1-forms are isochronous. Concerning the recent work of J.C.~Langer, we reflect on the strong relationship between our work and his and provide a partial answer regarding polyhedral geometries that arise from rational quadratic differentials on the Riemann sphere.

math.GT↗