11 problems

Let $\mathcal{M}$ be a homogeneous quaternionic Kähler manifold with negative Ricci curvature. The manifold is called normal if it admits a transitive solvable group of isometries…

Let $M^{4m}$ be a compact spin quaternionic Kähler manifold of positive scalar curvature $\kappa$, and let $D$ be its Dirac operator. For an eigenspinor of $D$ with eigenvalue…

Non-solvsoliton conjecture. If the one-loop deformation has $c>0$ and dimension $4n>4$, then every hypersurface orbit of its cohomogeneity-one action, with the induced metric, is n…

Lichnerowicz–Salamon conjecture. The only complete quaternionic Kähler manifolds of positive scalar curvature are symmetric spaces of compact type.

Conjecture on negative quaternionic–Kähler manifolds. There are no closed simply-connected negative quaternionic–Kähler manifolds other than the known symmetric examples.

Wolf-space exhaustion conjecture. Wolf spaces exhaust all complete quaternionic Kähler manifolds with positive scalar curvature.

Regular $3$-Sasakian manifolds arise as certain principal ${\rm SO}(3)$ or ${\rm Sp}(1)$ bundles over a quaternionic Kähler manifold with positive scalar curvature. An…

Canonical Hamiltonian conjecture. The canonical choice of Hamiltonian yields the Ferrara–Sabharwal metric, up to a factor.

A Fano contact manifold is a Fano complex contact manifold, and a twistor space of a quaternionic-Kähler manifold is the twistor space associated with a quaternionic-Kähler manifol…

Let $M$ be a quaternionic Kähler manifold with positive scalar curvature. The conjecture concerns whether $M$ is a symmetric space $G/H$ with equal ranks of the groups $G$ and $H$.…

Determinant-form metric conjecture. The M-theory metric can be written as