Nonflat Graphical Special Lagrangian Cones in Higher Dimensions

Definitions and mathematical statement

Identify ${\mathbb{C}}^n$ with ${\mathbb{R}}_x^n\times{\mathbb{R}}_y^n$, write $z_j=x_j+iy_j$, and put \[ \omega=\sum_{j=1}^n{\,\mathrm{d}} x_j\wedge{\,\mathrm{d}} y_j, \qquad \Omega={\,\mathrm{d}} z_1\wedge\cdots\wedge{\,\mathrm{d}} z_n. \] A smooth oriented real $n$-dimensional submanifold $M\subset{\mathbb{C}}^n$ is special Lagrangian of phase $\theta\in{\mathbb{R}}$ if \[ \omega|_M=0,\qquad \operatorname{Im}(e^{-i\theta}\Omega)|_M=0,\qquad \operatorname{Re}(e^{-i\theta}\Omega)|_M=\operatorname{vol}_M. \] A set $L\subset{\mathbb{C}}^n$ is conical if $tL=L$ for every real $t>0$. For a special Lagrangian cone, the special Lagrangian conditions hold on its smooth locus with one constant phase. Here nonflat means that $L$ is not contained in a real $n$-dimensional linear subspace of ${\mathbb{C}}^n$.

A graphical description relative to the displayed splitting has the form \[ \{x+iF(x):x\in D\}, \] where $D\subset{\mathbb{R}}^n$ is a domain and $F:D\to{\mathbb{R}}^n$. When this graph is conical, $D$ is invariant under positive dilations and $F(tx)=tF(x)$ whenever $x\in D$ and $t>0$. The vertex may need to be adjoined separately.

Question supplied by the record. For every integer $n\geq5$, does there exist a nonflat graphical special Lagrangian cone in ${\mathbb{C}}^n$? The phase is not prescribed in the supplied question.

Unresolved qualifications. The supplied record does not specify the graph domain, whether graphicality is relative to the standard real plane or another plane, the required regularity, or the allowed singularities. In particular, it does not say whether the graph must be defined over all of ${\mathbb{R}}^n\setminus\{0\}$, extend over the origin, or be defined over a smaller dilation-invariant domain. These choices are material to the existence question. The graphical description above explains the terminology without selecting one of these domains. No smoothness at the vertex or absence of other singularities is imposed by this draft. Consequently, the question retains an unresolved admissible-class qualification and requires the missing source context before it becomes a fully specified existence assertion.

Short English statement

The supplied question asks whether every complex dimension $n\geq5$ admits a nonflat special Lagrangian cone that is a graph. The graph domain and permitted regularity and singularities are unspecified and require review. The record labels the problem open; that label is supplied curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30006405$, source problem code catalogue entry, title: "Nonflat Graphical Special Lagrangian Cones in Higher Dimensions." Attribution is to the Oberwolfach Reports Open Problems collection; no individual proposer is supplied. The recorded proposed year is 2025.
• Supplied report citation: Partial Differential Equations (2026), Oberwolfach Reports, DOI 10.4171/owr/2025/33, pp. 1753--1822. https://doi.org/10.4171/owr/2025/33. The citation's title year is reproduced as supplied; its relation to the 2025 report designation requires verification.
• Supplied literature source: "Oberwolfach Report 33/2025," https://ems.press/content/serial-article-files/52247.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem open in dimensions $n\geq5$. This is supplied dated curation, not independently verified here. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry marks the statement corrected_verified and reports that the official report was checked and its notation repaired. These are supplied verification claims, not verification performed for this draft. The supplied original-statement field says that "Problem 1" remains open for $n\geq5$, but does not reproduce that problem's defining context.