Immersion of open Riemann surfaces

Noncritical holomorphic functions on Stein manifolds

“…Theorem 2.5 also holds for q = dim X = 1 and is due in this case to Gunning and Narasimhan who proved that for every nonvanishing holomorphic one-form θ on an open Riemann surface there exists a holomorphic function w such that e w θ = df is exact holomorphic [GN,p. 107].…”

Holomorphic maps from 𝐶ⁿ to 𝐶ⁿ

“…The desired map $ will carry each T^fc in Qy\Qo into P, thus assuring conclusion (iii) of the lemma, and will have the form (4) $ = Ta o * where * = 04 o 03 o cr2 ° ay will be the composition of four shears ai, chosen so that (5) \z-9(z)\<e/2 HzEQo, which implies that *(Qo) C Qy, and so that (6) *(rifc)C (^mjk\ey+P0 for each T^fc c Qy\Qo; here m,jk is some integer.…”

“…For some s, FS(K) C rB. Hence (6) shows that The region that is attracted to the origin by this F is all of C2. To get away from this, put G(z, w) = (az + (pw + z2)2,Pw + z2).…”

“…Ifn>l then C" contains continuum many discrete sets no two of which are Aut(Cn)-equivalent. 6. Tame sets that are not very tame.…”

Closed holomorphic 1-forms without zeros on Stein manifolds

“…The existence of a holomorphic function f : X → C without critical points, proved by Forstnerič in [4], implies that in the zero class there is a closed holomorphic 1-form without zeros, namely df . (For the case when X is an open Riemann surface see also [10].) Our goal in this paper is to show that a closed holomorphic 1-form without zeros can be chosen in every cohomology class.…”