Minimal Ambient Dimension for Complete Bounded Complex Submanifolds

Definitions and mathematical statement

Let $k\in{\mathbb{Z}}$ with $k>1$, and let $N\geq k$ be an integer. A complete bounded embedded complex $k$-dimensional submanifold of ${\mathbb{C}}^N$ means a nonempty complex submanifold $M\subset{\mathbb{C}}^N$ of complex dimension $k$, whose inclusion is a holomorphic embedding, such that \[ \exists R>0\ \forall p\in M,\qquad \|p\|<R, \] and such that the Riemannian metric induced on $M$ by the standard Euclidean metric of ${\mathbb{C}}^N$ is complete on every connected component. Here completeness refers to the intrinsic Riemannian distance, rather than the restriction of the ambient distance. Submanifolds are understood to have no boundary, as in the usual convention; the supplied statement does not explicitly specify this convention or the metric.

Define \[ \mathcal D_k=\{N\in{\mathbb{Z}}:N\geq k\text{ and such a submanifold }M\subset{\mathbb{C}}^N\text{ exists}\}. \] The corrected supplied statement records the existence assertion $2k\in\mathcal D_k$ for every integer $k>1$. Taking that assertion as supplied background, the question is whether \[ \forall k\in{\mathbb{Z}},\quad k>1\ \Longrightarrow\ \min\mathcal D_k=2k. \] Equivalently, for each integer $k>1$, does there fail to exist a complete bounded embedded complex $k$-dimensional submanifold of ${\mathbb{C}}^N$ for every integer $N$ satisfying $k\leq N<2k$?

The original supplied wording makes existence in ${\mathbb{C}}^{2k}$ conditional and does not explicitly state $k>1$; the formulation above follows the corrected statement and retains the original qualification here. Existence and the reported open status are supplied source assertions, not independently verified conclusions of this draft.

Short English statement

For each integer $k>1$, the supplied corrected record states that complete bounded embedded complex $k$-dimensional submanifolds exist in ${\mathbb{C}}^{2k}$. The question asks whether any such submanifold can exist in a smaller ambient complex dimension, with completeness understood for the induced Euclidean Riemannian metric.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002771$, source problem code catalogue entry, titled Minimal Ambient Dimension for Complete Bounded Complex Submanifolds. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2015; no individual proposer is supplied.
• Geometric Methods of Complex Analysis (2015), Oberwolfach Reports, pp. 235--283, DOI 10.4171/owr/2015/4. Supplied report reference: https://doi.org/10.4171/owr/2015/4. The supplied background describes these reports as documenting workshop discussions and open problems. No precise page or passage identifying this problem within the report is supplied.
• Supplied literature reference labeled "Globevnik, Annals of Mathematics": https://doi.org/10.4007/annals.2015.182.3.4.
• Supplied literature reference labeled "Alarcon, Yang problem survey": https://arxiv.org/abs/2212.08521.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, reports the existence of complete bounded embedded complex $k$-dimensional examples in ambient complex dimension $2k$, and describes minimality of $2k$ as open in the survey literature. This is supplied dated curation derived from the OWR record, not independently verified here; its status and cited claims require independent checking.
• The registry assigns statement status corrected_verified and states that the official report passage was checked directly and its question, definitions, and notation restored without strengthening the claim. This is a supplied verification assertion, not a source visit or verification performed for this draft.