Embeddings through discrete sets of balls
This paper in several complex variables makes a contribution to the circle of theorems that revolve around perturbation of proper holomorphic embeddings of Stein manifolds into complex Euclidean space armed with the Andersén-Lempert theory of biholomorphisms thereof as advanced by Forstnerič, Rosay, and others. There are two main theorems, and an open question ends the paper. Theorem~I. If a complex space \(X\) of complex dimension \(k\) admits a proper holomorphic embedding into some \({\mathbb C}^n\) and \(0<k<n/2\), then for any discrete and closed sequence \(D_j\), \(j\geq1\), of \(k\)-dimensional balls in \({\mathbb C}^n\) and any constants \(\varepsilon_j>0\), \(j\geq1\), there is a proper holomorphic embedding of \(X\) into \({\mathbb C}^n\) whose image contains an \(\varepsilon_j\)-perturbation in the normal direction of the ball \(D_j\) for every \(j\geq1\). Theorem~II. For \(n/2\leq k<n\) there are a discrete and closed sequence \(D_j\), \(j\geq1\), of \(k\)-dimensional balls in \({\mathbb C}^n\) and constants \(\varepsilon_j>0\), \(j\geq1\), such that any Stein manifold is \((n-k)\)-Eisenman hyperbolic if it admits a proper holomorphic embedding into \({\mathbb C}^n\) that contains an \(\varepsilon_j\)-perturbation in the normal direction of the ball \(D_j\) for every \(j\geq1\). The proof of Theorem~I closely follows [J. Geom. Anal. 9, No.~1, 93--117 (1999; Zbl 0963.32006)] by \textit{F. Forstnerič}, and that of Theorem~II is based upon F. Forstnerič [loc. cit.] again and [Trans. Am. Math. Soc. 310, No.~1, 47--86 (1988; Zbl 0708.58003)] by \textit{J. P. Rosay} and \textit{W. Rudin}. The paper ends with the question whether, given any discrete and closed sequence of discs \(D_j\) in \({\mathbb C}^2\) and constants \(\varepsilon_j>0\) for \(j\geq1\), there is a proper holomorphic embedding of the unit disc into \({\mathbb C}^2\) whose image contains an \(\varepsilon_j\)-perturbation in the normal direction of the disc \(D_j\) for every \(j\geq1\). The material is well presented and pleasant to read.
embedding Stein manifolds in Euclidean space
scientific article; zbMATH DE number 5357904
scientific article