10 problems

Codimension-5 conjecture. The set $\mathcal{S}_{\mathrm{orb}}$ is countable. In particular, away from a countable set of points, $(X,d)$ has the structure of a smooth Einstein $5$-…

Let $(M^n,g_E)$ be a closed Einstein $n$-manifold with $n\geq 3$ and negative scalar curvature $R^*<0$. Let $g$ be another metric on $M$ satisfying … Here $g\leq g_E$ means that…

Let $M$ be a noncompact Einstein almost Kähler manifold, and suppose that every conformal vector field on $M$ is Killing. Rigidity Conjecture. Then $M$ must be isometric to complex…

Anderson's conjecture. For each fixed $\Lambda$, the boundary map $\Pi$ is regular at generic metrics, meaning that its linearization is surjective with split kernel at those metri…

Splitting conjecture. $M^n$ splits isometrically as a product of a compact Einstein manifold and a non-compact homogeneous Einstein space.

The conjecture. The only compact locally homogeneous simple pseudo-Riemannian Einstein manifolds are of the form $\Gamma\backslash G/H$, determined by either some standard triple…

Let $(M_o,g_o)$ be a singular spherical or hyperbolic orbifold, and let $(M_i,g_i)$ be smooth Einstein manifolds. Spherical and hyperbolic orbifold limit conjecture. Singular spher…

Let $M$ be a simply connected Einstein four-manifold with positive scalar curvature and non-negative sectional curvature. Folklore conjecture. The manifold must be one of the follo…

Let $(M,g)$ be a smooth four-dimensional Einstein manifold satisfying … for a constant $\lambda>0$, and suppose that its sectional curvature is non-negative. The metrics $g_0$,…

A homogeneous manifold is a manifold equipped with a transitive group action by isometries, and an Einstein manifold has Ricci tensor proportional to its metric. Its scalar curvatu…