Geodesic
Proper Holomorphic Mappings onto Symmetric Products of a Riemann Surface
Documenta mathematica, Tome 23 (2018), pp. 1291-1311.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

We show that the structure of proper holomorphic maps between the $n$-fold symmetric products, $n\geq 2$, of a pair of non-compact Riemann surfaces $X$ and $Y$, provided these are reasonably nice, is very rigid. Specifically, any such map is determined by a proper holomorphic map of $X$ onto $Y$. This extends existing results concerning bounded planar domains, and is a non-compact analogue of a phenomenon observed in symmetric products of compact Riemann surfaces. Along the way, we also provide a condition for the complete hyperbolicity of all $n$-fold symmetric products of a non-compact Riemann surface.
Classification : 32H35, 30F99
Keywords: proper holomorphic maps, Riemann surfaces, symmetric product 
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     title = {Proper {Holomorphic} {Mappings} onto {Symmetric} {Products} of a {Riemann} {Surface}},
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Bharali, Gautam; Biswas, Indranil; Divakaran, Divakaran; Janardhanan, Jaikrishnan. Proper Holomorphic Mappings onto Symmetric Products of a Riemann Surface. Documenta mathematica, Tome 23 (2018), pp. 1291-1311. http://geodesic.mathdoc.fr/item/DOCMA_2018__23__a24/