Stable Embedded Minimal Hypersurfaces Below Dimension Eight

Definitions and mathematical statement

An affine hyperplane in ${\mathbb{R}}^n$ is a set $\{x\in{\mathbb{R}}^n:a\cdot x=b\}$, where $a\in{\mathbb{R}}^n\setminus\{0\}$ and $b\in{\mathbb{R}}$. In the smooth setting, an embedded hypersurface $\Sigma\subset{\mathbb{R}}^n$ has dimension $n-1$ and carries the induced Euclidean metric. It is minimal if its mean curvature vector vanishes identically.

For a smooth minimal hypersurface, stability under compactly supported normal variations means that \[ Q_\Sigma(V):=\int_\Sigma\left(\lvert\nabla^\perp V\rvert^2-\lvert A_\Sigma\rvert^2\lvert V\rvert^2\right)\,\mathrm d\mu_\Sigma\geq0 \] for every smooth compactly supported normal vector field $V$. Here $A_\Sigma$ is the second fundamental form, $\nabla^\perp$ is the normal connection, and $\mu_\Sigma$ is induced volume. If a boundary is allowed, this definition uses fields supported in the interior. In the two-sided case, a global unit normal $\nu$ exists, and writing $V=\varphi\nu$ gives the equivalent inequality \[ \int_\Sigma\lvert\nabla_\Sigma\varphi\rvert^2\,\mathrm d\mu_\Sigma\geq\int_\Sigma\lvert A_\Sigma\rvert^2\varphi^2\,\mathrm d\mu_\Sigma \quad\text{for every }\varphi\in C_c^\infty(\Sigma). \] These definitions explain the usual smooth interpretation; the supplied record does not specify the admissible global class or the precise stability convention.

The supplied question is: for ambient dimension $n\leq7$, are all stable embedded minimal hypersurfaces in ${\mathbb{R}}^n$ affine hyperplanes? More explicitly, if $\mathcal C_n$ denotes the intended class of admissible embedded hypersurfaces, the proposed classification is \[ \forall n\leq7,\quad\forall\Sigma\in\mathcal C_n,\qquad \left(\Sigma\text{ is minimal and stable}\right) \Longrightarrow \left(\exists a\in{\mathbb{R}}^n\setminus\{0\}\ \exists b\in{\mathbb{R}}:\ \Sigma=\{x:a\cdot x=b\}\right), \] where $n$ ranges over the intended integer ambient dimensions. The source leaves $\mathcal C_n$ unresolved: completeness, absence of boundary, connectedness, two-sidedness, regularity, properness, and any growth assumptions are not stated. In particular, completeness and absence of boundary must not be silently inserted. The displayed implication records the intended classification target but is not a fully specified conjecture until these hypotheses are settled. The background mentions a fractional analogue but supplies no precise fractional problem.

Short English statement

For ambient dimensions $n\leq7$, the source asks whether every stable embedded minimal hypersurface in ${\mathbb{R}}^n$ is an affine hyperplane. Essential global hypotheses, including completeness, boundary conditions, and connectedness, are unspecified. The supplied open label is dated curation, not an independently verified status.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30003176$, title Stable Embedded Minimal Hypersurfaces Below Dimension Eight, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2016 and names no proposer.
• Supplied report citation: Calculus of Variations (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/34, pp. 1943--2008. https://doi.org/10.4171/owr/2016/34. The supplied background describes this collection as drawn from workshop reports and their open-problem discussions.
• Davila--del Pino--Wei, Stable nonlocal minimal cones, supplied literature reference: https://arxiv.org/abs/1402.4173.
• Cabre--Cinti--Serra, Stable nonlocal cones, supplied literature reference: https://arxiv.org/abs/1710.08722.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, states that the classical stable-hypersurface classification in ambient dimension at most seven remains open, and describes the fractional analogue as having only partial cone classifications with counterexamples or restrictions in special regimes. This is supplied dated curation derived from the OWR record, not an independently verified assessment; time-sensitive status and cited claims require independent checking.
• The supplied registry labels the statement corrected_verified and says its mathematical content was preserved while notation, typography, or omitted definitions were repaired. This records a source-side qualification and does not establish that the missing hypotheses have been resolved.