Complete meromorphic curves with Jordan boundaries

DOI10.1016/J.JMAA.2023.127850arXiv2303.01814MaRDI QIDQ6428275

Publication date: 3 March 2023

Abstract: We prove that given a finite set

E

in a bordered Riemann surface

m a t h c a l R

, there is a continuous map

h c o l o n o v e r l i n e m a t h c a l R s e t m i n u s E o m a t h b b C^{n}

(

n g e q 2

) such that

h |_{m a t h c a l R s e t m i n u s E} c o l o n m a t h c a l R s e t m i n u s E o m a t h b b C^{n}

is a complete holomorphic immersion (embedding if

n g e q 3

) which is meromorphic on

m a t h c a l R

and has effective poles at all points in

E

, and

h |_{b o v e r l i n e m a t h c a l R} c o l o n b o v e r l i n e m a t h c a l R o m a t h b b C^{n}

is a topological embedding. In particular,

h (b o v e r l i n e m a t h c a l R)

consists of the union of finitely many pairwise disjoint Jordan curves which we ensure to be of Hausdorff dimension one. We establish a more general result including uniform approximation and interpolation.

Mathematics Subject Classification ID

Manifolds and measure-geometric topics (49Qxx) Holomorphic mappings and correspondences (32Hxx) Holomorphic convexity (32Exx)