Rigidity of Complete Embedded Minimal Surfaces

Definitions and mathematical statement

Equip a smooth connected embedded surface $M\subset{\mathbb{R}}^3$ with the induced Riemannian metric $g=\iota^*g_{\mathrm{Euc}}$, where $\iota:M\hookrightarrow{\mathbb{R}}^3$ is inclusion. Completeness means completeness of the induced Riemannian distance, and minimality means that the mean curvature vector vanishes everywhere. An intrinsic isometry is a diffeomorphism $\phi:M\to M$ satisfying $\phi^*g=g$. A Euclidean isometry has the form $A(x)=Qx+b$, where $Q\in O(3)$ and $b\in{\mathbb{R}}^3$.

The extension clause of the supplied question asks: for every connected complete embedded minimal surface $M\subset{\mathbb{R}}^3$ and every intrinsic isometry $\phi:M\to M$, do there exist $Q\in O(3)$ and $b\in{\mathbb{R}}^3$ such that \[ \iota(\phi(p))=Q\iota(p)+b\qquad\text{for every }p\in M? \] This formulation uses the usual convention that a surface has no boundary. The record does not explicitly specify the boundary convention; whether surfaces with boundary are intended requires clarification. No properness, orientability, finite-genus, or periodicity hypothesis is supplied.

The source additionally asks for "minimal rigidity unless $M$ is a helicoid." Here a helicoid means a surface congruent to \[ H_a=\{(s\cos t,s\sin t,at):s,t\in{\mathbb{R}}\},\qquad a\in{\mathbb{R}}\setminus\{0\}. \] The source does not define minimal rigidity or its comparison class. If it means rigidity among all isometric minimal immersions of the same Riemannian surface, the additional assertion would be: whenever $M$ is not a helicoid, every smooth minimal immersion $F:M\to{\mathbb{R}}^3$ satisfying $F^*g_{\mathrm{Euc}}=g$ satisfies \[ F=A\circ\iota \] for some Euclidean isometry $A$. This is an explicitly conditional interpretation requiring source verification: the supplied record does not determine whether the comparison maps must be immersions, embeddings, or members of a deformation family, or whether reparametrization is allowed in the congruence relation. Consequently, the full rigidity clause cannot yet be assigned a unique precise formulation. The helicoid exception is attached to this additional rigidity clause, not to the stated intrinsic-isometry extension clause.

Short English statement

The question asks whether every intrinsic symmetry of a connected complete embedded minimal surface in ${\mathbb{R}}^3$ comes from a Euclidean motion. It also asks for minimal rigidity away from helicoids, but the intended meaning of rigidity and the boundary convention require clarification. The supplied status is partially solved; this draft does not independently verify that assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001358$, titled "Rigidity of Complete Embedded Minimal Surfaces," in the Oberwolfach Reports Open Problems collection; exact source problem code catalogue entry. Project registry locations are data/unsolvedmath/problems.json and the supplementary catalog/problems.json; no independent registry inspection is claimed.
• Supplied source citation: Arbeitsgemeinschaft: Minimal Surfaces (2010), Oberwolfach Reports, DOI 10.4171/owr/2009/45, pp. 2545--2584. https://doi.org/10.4171/owr/2009/45. The record gives proposed year 2009 and names no proposer. Its background describes Oberwolfach Reports as documenting mathematical workshops and open-problem discussions.
• Supplied literature reference: Meeks--Pérez, minimal-surface survey. https://www.intlpress.com/site/pub/files/_fulltext/journals/sdg/2016/0021/0001/SDG-2016-0021-0001-a006.pdf.
• Supplied literature reference: Collin--Kusner--Meeks--Rosenberg, end restrictions. https://arxiv.org/abs/math/0401267.
• Supplied literature reference: original OWR report. https://doi.org/10.4171/owr/2009/45.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports that the Isometry Conjecture is proved in several substantial classes, including finite-genus and periodic cases, while the general infinite-genus case remains open. This is supplied dated curation derived from the OWR record, not independently verified here; its time-sensitive status and cited claims require independent verification.
• The supplied statement-status label is corrected_verified. Its accompanying verification note says only that TeX notation was enclosed in mathematical delimiters without changing mathematical content; it does not establish mathematical correctness or source verification.