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Alexandre Eremenko

Publications and source records attributed to Alexandre Eremenko.

At least 37 records · Page 2Linked to original sources
arXiv2021

Moduli spaces for Lamé functions and Abelian integrals of the second kind

The space of Lamé functions of order m is isomorphic to the space of pairs (elliptic curve, Abelian differential) where the differential has a single zero of order 2m at the origin and m double poles with vanishing residues. We describe the topology of this space: it is a Riemann surface of finite type; we find the number of components and the genus and Euler characteristic of each component. As an application we find the degrees of Cohn's polynomials confirming a conjecture by Robert Maier. As another application we partially describe the degeneration locus of the space of spherical metrics on tori with one conic singularity where the conic angle is an odd multiple of 2$π$.

math.CV↗
arXiv2021

Stability in the Marcinkiewicz theorem

Ostrovskii's generalization of the Marcinkiewicz theorem implies that if an entire characteristic functions of a probability distribution satisfies $\log^+\log|f(z)|=o(|z|),\; z\to\infty,$ and is zero-free then the distribution is normal. We show that under the same growth condition, absence of zeros in a wide vertical strip implies that the distribution is close to a normal one. This generalizes and simplifies a recent result of Michelen and Sahasrabudhe.

math.PR↗
arXiv2021

Metrics of constant positive curvature with conic singularities. A survey

We consider conformal metrics of constant curvature 1 on a Riemann surface, with finitely many prescribed conic singularities and prescribed angles at these singularities. Especially interesting case which was studied by C. L. Chai, C. S Lin and C. L. Wang is described in some detail, with simplified proofs.

math.DG↗
arXiv2021

Meromorphic functions with three radially distributed values

We consider transcendental meromorphic functions for which the zeros, 1-points and poles are distributed on three distinct rays. We show that such functions exist if and only if the rays are equally spaced. We also obtain a normal family analogue of this result.

math.CV↗
arXiv2020

Moduli of spherical tori with one conical point

In this paper we determine the topology of the moduli space $\mathcal{MS}_{1,1}(\vartheta)$ of surfaces of genus one with a Riemannian metric of constant curvature $1$ and one conical point of angle $2π\vartheta$. In particular, for $\vartheta\in (2m-1,2m+1)$ non-odd, $\mathcal{MS}_{1,1}(\vartheta)$ is connected, has orbifold Euler characteristic $-m^2/12$, and its topology depends on the integer $m>0$ only. For $\vartheta=2m+1$ odd, $\mathcal{MS}_{1,1}(2m+1)$ has $\lceil{m(m+1)/6}\rceil$ connected components. For $\vartheta=2m$ even, $\mathcal{MS}_{1,1}(2m)$ has a natural complex structure and it is biholomorphic to $\mathbb{H}^2/G_m$ for a certain subgroup $G_m$ of $\mathrm{SL}(2,\mathbb{Z})$ of index $m^2$, which is non-normal for $m>1$.

math.DG↗
arXiv2020

Interactions between function theory and holomorphic dynamics

This is a survey talk on the ICMS conference "Resonances of complex dynamics", 9-13 July 2018. Three examples which show how dynamical considerations lead to new results in complex function theory are discussed. These examples are: a) a refinement of of the Wiman-Valiron method, b) the proof of Hayman's conjecture on the value distribution of derivatives of meromorphic functions, and c) a new simplified proof of the result of Lin and Wang on the number of critical points of the Green function on a torus.

math.CV↗
arXiv2020

The space of Schwarz-Klein triangles

We describe the space of spherical triangles (in the sense of Schwarz and Klein). It is a smooth connected orientable 3-manifold, homotopy equivalent to the 1-skeleton of the cubic partition of the closed first octant in $\mathbf{R}^3$. The angles and sides are real analytic functions on this manifold which embed it to $\mathbf{R}^6$.

math.GT↗
arXiv2020

Entire functions with separated zeros and $1$-points

We consider transcendental entire functions of finite order for which the zeros and $1$-points are in disjoint sectors. Under suitable hypotheses on the sizes of these sectors we show that such functions must have a specific form, or that such functions do not exist at all.

math.CV↗
arXiv2019

Radially distributed values and normal families, II

We consider the family of all functions holomorphic in the unit disk for which the zeros lie on one ray while the 1-points lie on two different rays. We prove that for certain configurations of the rays this family is normal outside the origin.

math.CV↗
arXiv2018

Fuchsian Equations with Three Non-Apparent Singularities

We show that for every second order Fuchsian linear differential equation $E$ with $n$ singularities of which $n-3$ are apparent there exists a hypergeometric equation $H$ and a linear differential operator with polynomial coefficients which maps the space of solutions of $H$ into the space of solutions of $E$. This map is surjective for generic parameters. This justifies one statement of Klein (1905). We also count the number of such equations $E$ with prescribed singularities and exponents. We apply these results to the description of conformal metrics of curvature $1$ on the punctured sphere with conic singularities, all but three of them having integer angles.

math.CA↗
arXiv2018

Entire functions with two radially distributed values

We study entire functions whose zeros and one-points lie on distinct finite systems of rays. General restrictions on these rays are obtained. Non-trivial examples of entire functions with zeros and one-points on different rays are constructed, using the Stokes phenomenon for second order linear differential equations.

math.CV↗
arXiv2018

Radially distributed values and normal families

Let $L_0$ and $L_1$ be two distinct rays emanating from the origin and let ${\mathcal F}$ be the family of all functions holomorphic in the unit disk ${\mathbb D}$ for which all zeros lie on $L_0$ while all $1$-points lie on $L_1$. It is shown that ${\mathcal F}$ is normal in ${\mathbb D}\backslash\{0\}$. The case where $L_0$ is the positive real axis and $L_1$ is the negative real axis is studied in more detail.

math.CV↗
arXiv2017

PT-symmetric eigenvalues for homogeneous potentials

We consider one-dimensional Schrödinger equations with homogeneous potential, under appropriate PT-symmetric boundary conditions. We prove the phenomenon which was discovered by Bender and Boettcher by numerical computation: as the degree of the potential changes, the real spectrum suddenly becomes non-real in the sense that all but finitely many eigenvalues become non-real. We find the limit arguments of these non-real eigenvalues as they tend to infinity.

math-ph↗
arXiv2017

Co-axial monodromy

For Riemannian metrics of constant positive curvature on a punctured sphere with conic singularities at the punctures and co-axial monodromy of the developing map, possible angles at the singularities are completely described. This completes the recent result of Mondello and Panov. The related problem of describing possible multiplicities of critical points of logarithmic potentials of finitely many charges is also solved.

math.MG↗