2001/01/31 by Miran Cerne, Franc Forstneric
Mathematics · #math.CV #msc:30F60 #msc:32G15 #msc:32H02
published as Math. Res. Lett., 9 (2002), 683--696
arxiv created 2003/05/16 · arxiv updated 2009/11/30
We study the existence of proper holomorphic embeddings of bordered Riemann surfaces into the complex plane C2. Denote by M(R) the moduli space consisting of all equivalence classes of complex structures J on a given smooth oriented bordered surface R. We introduce a class F(R) in M(R)with the following properties: (1) F(R) is nonempty and open (in a natural topology on M(R)); (2) The interior of any Riemann surface (R,J) in the class F(R) admits a proper holomorphic embedding in C2; (3) If R is a finitely connected planar domain then F(R)=M(R); (4) Each hyperelliptic bordered Riemann surface (R,J) belongs to the class F(R) and hence admits a proper holomorphic embedding in C2. Part (3) above is equivalent to the theorem of Globevnik and Stensones (Holomorphic embeddings of planar domains into C2, Math. Ann. 303, 579-597, 1995). Our approach builds upon the earlier work of Cerne and Globevnik (On holomorphic embedding of planar domains into C2, J. d'Analyse Math. 8, 269-282, 2000).