The octonionic Calabi–Yau conjecture for compact sixteen-dimensional manifolds

Let $M^{16}$ be a compact $GL_2(\mathbb{O})$-manifold. Let $G_0$ be a $C^\infty$-smooth positive section of the bundle $\underline{{\cal H}_2}(M)$ satisfying the condition that, locally, $G_0=\operatorname{Hess}_{\mathbb{O}}u$ for some smooth function $u$. Let $f\colon M\longrightarrow\mathbb{R}$ be a $C^\infty$-smooth function.

Octonionic Calabi–Yau conjecture. There exist a $C^\infty$-smooth function $u\colon M\longrightarrow\mathbb{R}$ and a constant $A>0$ such that

$$\det\bigl(G_0+\operatorname{Hess}_{\mathbb{O}}(\phi)\bigr)=A e^f\det(G_0)\quad\text{on }M.$$

This conjecture is motivated by the Calabi–Yau theorem for Kähler manifolds. The paper proves a special case of the conjecture; the general solvability statement remains open in the source.