Acyclic embeddings of open Riemann surfaces into new examples of elliptic manifolds

Some remarks on the correspondence between elliptic curves and four points in the Riemann sphere

In this paper we relate some classical normal forms for complex elliptic curves in terms of 4-point sets in the Riemann sphere. Our main result is an alternative proof that every elliptic curve is isomorphic as a Riemann surface to one in the Hesse normal form. In this setting, we give an alternative proof of the equivalence between the Edwards and the Jacobi normal forms. Also, we give a geometric construction of the cross ratios for 4-point sets in general position.

奇异椭圆曲线的邻域与无穷远处的黎曼-希尔伯特对应

Neighborhoods of singular elliptic curves and the Riemann-Hilbert correspondence at infinity

本文研究奇异椭圆曲线邻域的解析分类，通过双重覆盖与光滑情形不变量关联，并推测其与Painlevé VI的无穷远黎曼-希尔伯特对应有关。

Operators associated to the Cauchy-Riemann operator in elliptic complex numbers

In this article we provide a generalized version of the result of L.H. Son and W. Tutschke \cite{tut} on the solvability of first order systems on the plane whose initial functions are arbitrary holomorphic functions. This is achieved by considering the more general concept of holomorphicity with respect to the structure polynomial $X^2 + βX + α$. It is shown that the Son-Tutschke lemma on the construction of complex linear operators associated to the Cauchy-Riemann operator remains valid when interpreted for a large class of real parameters $α$ and $β$ including the elliptic case but also cas…

关于椭圆和拟正则椭圆流形

On elliptic and quasiregularly elliptic manifolds

本文研究黎曼流形的椭圆性和拟正则椭圆性，证明了闭流形的基本群必须是几乎阿贝尔的，并给出了开流形中两者不等价的例子。

A strong Oka principle for embeddings of some planar domains into CxC*

Gromov, in his seminal 1989 paper on the Oka principle, introduced the notion of an elliptic manifold and proved that every continuous map from a Stein manifold to an elliptic manifold is homotopic to a holomorphic map. We show that a much stronger Oka principle holds in the special case of maps from certain open Riemann surfaces called circular domains into CxC*, namely that every continuous map is homotopic to a proper holomorphic embedding. An important ingredient is a generalisation to CxC* of recent results of Wold and Forstneric on the long-standing problem of properly embedding open Rie…

Nonlinear steepest descent approach to orthogonality on elliptic curves

We consider the recently introduced notion of denominators of Padé--like approximation problems on a Riemann surface. These denominators are related as in the classical case to the notion of orthogonality over a contour. We investigate a specific setup where the Riemann surface is a real elliptic curve and the measure of orthogonality is supported on one of the two real ovals. Using a characterization in terms of a Riemann--Hilbert problem, we determine the strong asymptotic behaviour of the corresponding orthogonal functions for large degree. The theory of vector bundles and the non-abelian C…