Embedding Certain Infinitely Connected Subsets of Bordered Riemann Surfaces Properly into ℂ2

Abstract

We prove that certain infinitely connected domains D in a bordered Riemann surface, which admits a holomorphic embedding into C 2 , admit a proper holomorphic embedding into C 2 . We also prove that certain infinitely connected open subsets D ⊂ C admit a proper holomorphic embedding into C 2 .

Complex Geometry and Dynamics

“…By introducing the technique of exposing boundary points alluded to above, combined with the Andersén-Lempert theory of holomorphic automorphisms of C n for n > 1, Forstnerič and Wold proved in 2009 that, if a compact bordered Riemann surface M admits a (nonproper) holomorphic embedding in C 2 then its interior M admits a proper holomorphic embedding in C 2 [48]. Further applications of their technique can be found in [69] and [49]. For example, every circular domain in the Riemann sphere admits a proper holomorphic embedding in C 2 [49].…”

“…This list does not include papers on embedding Riemann surfaces in C 2 where different techniques are used; see e.g. [48,49,55,69]. Some work has also been done on almost complex Stein manifolds of real dimension 4 [28].…”

Untitled

“…This result was extended to some infinitely connected Riemann surfaces by I. Majcen [2009] under a nontrivial additional assumption on the accumulation set of the boundary curves. (These results can also be found in [Forstnerič 2011, Chapter 8].)…”

“…The Muskat problem and related problems [Saffman and Taylor 1958] have been studied recently [Constantin and Pugh 1993;Siegel et al 2004;Córdoba and Gancedo 2007;2009;Córdoba et al 2011]. The first natural question is about the evolution of the system (existence of solutions), at least for a short time t > 0, and the persistence of smoothness of the interface S(t) if we begin with a smooth enough surface at time t = 0.…”

Proper Holomorphic Embeddings of complements of large Cantor sets in $\Bbb C^2$

“…Other remarkable results include proper holomorphic embeddings of certain Riemann surfaces into C 2 with interpolation (see [13]), deformation of continuous mappings f : S → X between Stein manifolds into proper holomorphic embeddings under certain hypothesis on the dimension of the spaces (see [2]), embeddings of infinitely connected planar domains into C 2 (see [11]), homotopy of continuous mappings f : D → C×C * into proper holomorphic embedding whenever D is a finitely connected planar domain without punctures (see [20]), existence of proper holomorphic embeddings of the unit ball B into connected pseudoconvex Runge domains Ω ⊂ C n (when n ≥ 2) whose image contains arbitrarily fixed discrete subsets of Ω (see [9]), approximation of proper embeddings on smooth curves contained in a finitely connected planar domain D into C n (with n ≥ 2) by proper holomorphic embeddings f : D ֒→ C 2 (see [15]), proper holomorphic embeddings into C 2 of certain infinitely connected domains Ω lying inside a bordered Riemann Surface D knowing to admit a proper holomorphic embedding into C 2 [16].…”