Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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Embedding Riemann surfaces with isolated punctures into C^2

Frank Kutzschebauch

created by daniele on 07 Feb 2019

12 feb 2019 -- 16:00

Aula D'Antoni, Roma Tor Vergata

Seminario di Analisi Complessa, nell’ambito del progetto MATH@TOV

Abstract.

We enlarge the class of open Riemann surfaces known to be holomorphically embeddable into the plane by allowing them to have additional isolated punctures compared to the known embedding results.

THEOREM The following open Riemann surfaces admit a proper holomorphic embedding into C2:

- the Riemann sphere with a (nonempty) countable closed subset with at most 2 accumulation points removed,

- any compact Riemann surface of genus 1 (torus) with a (nonempty) closed discrete set with at most one accumulation point removed,

- any hyperelliptic Riemann surface with a discrete closed set C removed with the properties that C contains a fibre F=R{-1} (p) (consisting either of two points or a single Weierstrass point) of the Riemann map R and all accumulation points of C are contained in that fibre F.

The same holds if X is as above with additionally a finite number of smoothly bounded regions removed.

The second and the third case with no accumulation points in the closed discrete set correspond to the Theorem of Sathaye.

Joint work with Pierre-Marie Poloni.

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