Infinite-Topology Types of Embedded Minimal Surfaces

Definitions and mathematical statement

A topological type means a homeomorphism class. Under the conventional interpretation that a surface is a connected, Hausdorff, second-countable two-dimensional manifold without boundary, a noncompact surface has finite topological type if it is homeomorphic to a compact surface without boundary with finitely many points removed. It has infinite topology if it is not of finite topological type. The supplied source does not explicitly specify connectedness, boundary conditions, or its definition of infinite topology; the formulation below is conditional on these conventions.

Equip a surface $S$ with a compatible smooth structure. A smooth embedding $f:S\to{\mathbb{R}}^3$ is proper if $f^{-1}(K)$ is compact in $S$ for every compact subset $K\subseteq{\mathbb{R}}^3$. Its image is minimal if its mean curvature vector vanishes everywhere, with respect to the metric induced from Euclidean ${\mathbb{R}}^3$.

Classification problem. Determine exactly which homeomorphism classes of noncompact orientable surfaces $S$ of infinite topology admit a compatible smooth structure and a proper smooth embedding $f:S\to{\mathbb{R}}^3$ whose image is minimal. Equivalently, characterize the class \[ \left\{[S]_{\cong}: S\text{ is noncompact, orientable, and of infinite topology, and admits a proper minimal embedding into }{\mathbb{R}}^3\right\}, \] where $[S]_{\cong}$ denotes the homeomorphism class of $S$, subject to the surface conventions stated above. If surfaces with boundary or disconnected surfaces were intended, their admissibility and the corresponding boundary conditions remain unspecified by the supplied record.

Short English statement

Determine precisely which noncompact orientable surfaces of infinite topology can be realized, up to homeomorphism, as properly embedded minimal surfaces in Euclidean ${\mathbb{R}}^3$. The formulation uses explicitly stated surface conventions that require confirmation from the source.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001357$, title Infinite-Topology Types of Embedded Minimal Surfaces, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified. The record lists proposed year $2009$.
• Supplied source citation: Arbeitsgemeinschaft: Minimal Surfaces (2010), Oberwolfach Reports, pp. 2545--2584, DOI 10.4171/owr/2009/45. https://doi.org/10.4171/owr/2009/45. The supplied background describes these reports as documenting workshops and their open-problem discussions. The citation's parenthetical year is retained despite the different year in the DOI and proposed-year field.
• Supplied literature reference: Collin--Kusner--Meeks--Rosenberg, "properly embedded surfaces." https://arxiv.org/abs/math/0401267.
• Supplied literature reference: Meeks--Pérez, "survey of conjectures." https://www.intlpress.com/site/pub/files/_fulltext/journals/sdg/2016/0021/0001/SDG-2016-0021-0001-a006.pdf.
• Supplied literature reference: "original OWR report." https://doi.org/10.4171/owr/2009/45.
• Literature assessment checked at 2026-08-21: this is supplied dated curation, not independently verified here. It labels the extracted infinite-topology question open and states that classification results restrict limit ends and provide constructions without establishing a full existence characterization. The supplied background requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, explaining only that TeX notation was placed inside math delimiters without changing mathematical content. This supplied label does not establish mathematical correctness or independent source verification.