Complete embedded complex curves in the ball of ℂ2 can have any topology

Abstract

In this paper we prove that the unit ball B of C 2 admits complete properly embedded complex curves of any given topological type. Moreover, we provide examples containing any given closed discrete subset of B.

Complex curves in pseudoconvex Runge domains containing discrete subsets

“…We therefore obtain the following corollary. Theorem 1.1 is already known in the particular cases when the domain Ω is C 2 (see Ritter [25]) and when it is the open unit ball in C 2 (see Globevnik and the author [1]). We point out that the assumptions on Ω (i.e., pseudoconvexity and having the Runge property) cannot be entirely removed from the statement of the theorem.…”

Holomorphic Embeddings and Immersions of Stein Manifolds: A Survey

“…In [6] it was shown that any bordered Riemann surface admits a proper complete holomorphic immersion into B 2 and embedding into B 3 (no change of the complex structure on the surface is necessary). In [9] the authors showed that properly embedded complete complex curves in the ball B 2 can have any topology, but their method (using holomorphic automorphisms) does not allow one to control the complex structure of the examples. Drinovec Drnovšek [38] proved that every strongly pseudoconvex domain embeds as a complete complex submanifold of a high dimensional ball.…”

Complete densely embedded complex lines in ℂ²

“…This is no longer true for complex curves in C 2 (a special case of minimal surfaces in R 4 ). Indeed, there exist complete embedded complex curves in C 2 with arbitrary topology which are bounded and hence non-proper (see [3]; the case of finite topology was previously shown in [4]). Furthermore, every relatively compact domain in C admits a complete nonproper holomorphic embedding into C 2 (see [2,Corollary 4.7]).…”

“…It is known that for any n > 1 the unit ball B n of C n contains complete properly embedded complex hypersurfaces (see [6,16,4] and the references therein); this settles in an optimal way a problem posed by Yang in 1977 about the existence of complete bounded complex submanifolds of C n (see [20,21]). Moreover, given a discrete subset Λ ⊂ B 2 there are complete properly embedded complex curves in B 2 containing Λ (see [17] for discs and [3] for examples with arbitrary topology). It remained and open problem whether B n also admits complete densely embedded complex submanifolds.…”