Hartogs type extension theorems
This paper has been withdrawn by the authors due to an error.
math.CV↗
arXiv subjects
Miran Cerne
Publications and source records attributed to Miran Cerne.
Embedding some bordered Riemann surfaces in the affine plane
We study the existence of proper holomorphic embeddings of bordered Riemann surfaces into the complex plane C^2. 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 C^2; (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 C^2. Part (3) above is equivalent to the theorem of Globevnik and Stensones (Holomorphic embeddings of planar domains into C^2, Math. Ann. 303, 579-597, 1995). Our approach builds upon the earlier work of Cerne and Globevnik (On holomorphic embedding of planar domains into C^2, J. d'Analyse Math. 8, 269-282, 2000).
math.CV↗
Nonlinear Riemann-Hilbert problem for bordered Riemann surfaces
Let S be a bordered Riemann surface with genus g and m boundary components. For a smooth family of smooth Jordan curves in the complex plane parametrized by the boundary of S and such that all curves contain 0 in their interior we show that there exists a holomorphic solution of the corresponding Riemann-Hilbert problem with at most 2g+m-1 zeros on S.
math.CV↗