Proper Holomorphic Maps in Euclidean Spaces Avoiding Unbounded Convex Sets

The Journal of Geometric Analysis (2023) 33:170 https://doi.org/10.1007/s12220-023-01222-z Proper Holomorphic Maps in Euclidean Spaces Avoiding Unbounded Convex Sets Barbara Drinovec Drnovšek1,2 · Franc Forstnerič1,2 Received: 4 January 2023 / Accepted: 8 February 2023 © The Author(s) 2023 Abstract We show that if E is a closed convex set in Cn (n > 1) contained in a closed halfspace H such that E ∩ bH is nonempty and bounded, then the concave domain  = Cn \E contains images of proper holomorphic maps f : X → Cn from any Stein manifold X of dimension < n, with approximation of a given map on closed compact subsets of X . If in addition 2 dim X +1 ≤ n then f can be chosen an embedding, and if 2 dim X = n, then it can be chosen an immersion. Under a stronger condition on E, we also obtain the interpolation property for such maps on closed complex subvarieties. Keywords Stein manifold · Holomorphic embedding · Oka manifold · Minimal surface · Convexity Mathematics Subject Classification 32H02 · 32Q56; 52A20 · 53A10 1 Introduction Let X be a Stein manifold. Denote by O(X , Cn ) the Frechet space of holomorphic maps X → Cn endowed with the compact-open topology and write O(X , C) = O(X ). A theorem of Remmert [36], Narasimhan [35], and Bishop [7] states that almost proper maps are residual in O(X , Cn ) if dim X = n, proper maps are dense if dim X < n, In memoriam Nessim Sibony B Franc Forstnerič Barbara Drinovec Drnovšek 1 Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia 2 Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia 0123456789().: V,-vol 123 170 Page 2 of 22 B. D. Drnovšek, F. Forstneriˇ c proper immersions are dense if 2 dim X ≤ n, and proper embeddings are dense if 2 dim X < n. A proof is also given in the monograph [29] by Gunning and Rossi. It is natural to ask how much space proper maps need. We pose the following question. Problem 1.1 For which domains  ⊂ Cn are proper holomorphic maps (immersions, embeddings) X → Cn as above, with images contained in , dense in O(X , )? It is evident that  cannot be contained in a halfspace of Cn since every holomorphic map from C to a halfspace lies in a complex hyperplane. In this paper, we give an affirmative answer for concave domains of which complement E = Cn \  satisfies the following condition. Definition 1.2 A closed convex set E in a finite dimensional affine space V has bounded convex exhaustion hulls (BCEH) if for every compact convex set K in V the set h(E, K ) = Conv(E ∪ K ) \ E is bounded. (1.1) Here, Conv denotes the convex hull. The following is our first main result. Theorem 1.3 Let E be an unbounded closed convex set in Cn (n > 1) with bounded convex exhaustion hulls. Given a Stein manifold X with dim X < n, a compact O(X )convex set K in X , and a holomorphic map f 0 : K → Cn with f 0 (bK ) ⊂  = Cn \E, we can approximate f 0 uniformly on K by proper holomorphic maps f : X → Cn satisfying f (X \ K̊ ) ⊂ . The map f can be chosen an embedding if 2 dim X < n and an immersion if 2 dim X ≤ n. In this paper, a map f : K → Cn from a compact set K is said to be holomorphic if it is the restriction to K of a holomorphic map on an open neighbourhood of K . In particular, if f 0 (K ) ⊂ , then the theorem gives uniform approximation of f 0 by proper holomorphic maps f : X → Cn with f (X ) ⊂ . If bE is of class C 1 and strictly convex near infinity, we obtain an analogue of Theorem 1.3 with additional interpolation on a closed complex subvariety X  of X such that f 0 : X  → Cn is proper holomorphic; see Theorem 4.2. Without the condition on the range, interpolation of proper holomorphic embeddings X → Cn on a closed complex subvariety was obtained by Acquistapace et al. [1] in 1975. The analogue of the BCEH condition for unbounded closed sets in Stein manifolds, with the convex hull replaced by the holomorphically convex hull, is used in holomorphic approximation theory of Arakelyan and Carleman type; see the survey in [18]. It is evident that a closed convex set E ⊂ Rn has BCEH if and only if there is an increasing sequence K 1 ⊂ K 2 ⊂ · · · of compact convex sets exhausting Rn such that the set h(E, K j ) (see (1.1)) is bounded for every j = 1, 2, . . .. In particular, BCEH is a condition at infinity which is invariant under perturbations supported on a compact subset. For compact convex sets E ⊂ Cn , Theorem 1.3 was proved in [24]; in this case BCEH trivially holds. We show in Sect. 3 that a closed convex set E in Rn has BCEH if and only if E is continuous in the sense of Gale and Klee [26]; see Proposition 3.3. If E has BCEH, 123 Proper Holomorphic Maps in Euclidean Spaces Avoiding... Page 3 of 22 170 then Conv(E ∪ K ) is closed for any compact convex set K ⊂ Rn (see [26, Theorem 1.5]). If such E is unbounded, which is the main case of interest, there are affine coordinates (x, y) ∈ Rn−1 × R such that E = E φ = {(x, y) ∈ Rn : y ≥ φ(x)} is the epigraph of a convex function φ : Rn−1 → R+ = [0, +∞) growing at least linearly near infinity (see Proposition 3.4). In particular, an unbounded closed convex set E ⊂ Cn with BCEH is of the form E = E φ = {z = (z  , z n ) ∈ Cn : z n ≥ φ(z  , z n )}, (1.2) in some affine complex coordinates z = (z  , z n ) on Cn , with φ as above. (Here, and denote the real and the imaginary part.) For a convex function φ of class C 1 , we give a differential characterization of the BCEH condition on its epigraph E φ ; see Proposition 3.8. The BCEH property holds if the radial derivative of φ tends to infinity; see Corollary 3.9. On the other hand, there are convex functions of linear growth of which epigraphs have BCEH; see Example 3.10. By Proposition 3.11, a convex function φ with at least linear growth at infinity can be approximated uniformly on compacts by functions ψ ≤ φ of the same kind of which epigraphs E ψ have BCEH. This allows us to extend Theorem 1.3 as follows; see Sect. 4 for the proof. Corollary 1.4 The conclusion of Theorem 1.3 holds for any convex epigraph E φ of the form (1.2) such that φ ≥ 0 and the set {φ = 0} is nonempty and compact. A closed convex set E ⊂ Cn with BCEH does not contain any affine real line (see Proposition 3.4), and for n > 1, its complement  = Cn \ E is an Oka domain according to Wold and the second named author; see [25, Theorem 1.8]. This fact plays an important role in our proof of Theorem 1.3, given in Sect. 4. (The precise result from Oka theory which we shall use is stated as Theorem 4.1.) Among closed convex epigraphs (1.2), the class of sets with Oka complement is strictly bigger than the class of sets with BCEH. In particular, the former class contains many sets containing boundary lines, which is impossible for a set with BCEH. Problem 1.5 Is there a (not necessarily convex) set E φ ⊂ Cn of the form (1.2) with φ ≥ 0 of sublinear growth for which Theorem 1.3 holds? Is there a set of this kind in C2 such that C2 (...truncated)