Garofalo–Vassilev conjecture on the octonionic Heisenberg group

The Garofalo–Vassilev conjecture asserts that, on groups of Heisenberg type, the functions $K^\varepsilon$ are the only nontrivial entire Yamabe-equation solutions, up to translations and dilations. The octonionic Heisenberg group is the exceptional $15$-dimensional case.

Known results

• Garofalo and Vassilev: partial symmetry implies cylindrical symmetry, and the cylindrically symmetric solutions are classified for Iwasawa groups.
• Frank and Lieb, 2014: sharp Hardy–Littlewood–Sobolev results and extremizers in part of the octonionic range, without resolving the conjecture.
• Yang, January 9, 2023: determines the optimal $L^2$ Folland–Stein constant and partially confirms the conjecture; later versions extend some results to octonionic groups.

September 2026 claimed resolution

A September 15, 2026 report links Daowen Lin’s preprint, which claims to classify all finite-energy positive solutions and the associated Folland–Stein–Sobolev extremals in the $15$-dimensional octonionic case. If correct, this resolves the conjecture in that exceptional dimension; the claim remains unverified.

Current status (as of September 2026): The conjecture is claimed solved for the $15$-dimensional octonionic Heisenberg group, but that resolution is unverified; broader groups or dimensions are not settled by this claim.