39 problems

Let $(M,g,J,\omega)$ be a compact Einstein almost-Kähler manifold. Goldberg's conjecture. The almost-Kähler structure is Kähler; equivalently, the almost complex structure $J$ is i…

A positive quaternion-Kähler manifold is a complete quaternion-Kähler manifold $(M^{4n},g)$ with positive scalar curvature; the known examples are the Wolf spaces, which are compac…

A noncompact Einstein homogeneous space is a noncompact homogeneous Riemannian manifold whose metric is Einstein, and an isometry group acts simply transitively when through every…

Standardness conjecture. Solvmanifolds exhaust the class of noncompact homogeneous Einstein manifolds.

Let $g(t)$ be a Ricci flow solution on a compact manifold in the expanding setting, and consider the rescaled metrics $g(t)/t$ as $t\to\infty$. Asymptotic convergence conjecture. T…

Let $M$ be a compact Einstein contact metric manifold. Contact Goldberg conjecture. $M$ is Sasakian–Einstein. This is proposed as the contact-metric analogue of the Goldberg conjec…

Let $(M_i,g_i)$ be Kähler-Einstein manifolds with positive scalar curvature, let $M_\infty$ be their limit, and let $S$ be the singular set so that $M_\infty\backslash S$ is the re…

Let $M_\infty$ be a metric limit of Einstein manifolds with special holonomy, with singular set $S$ stratified into strata $S_i$. At almost all points of $S_{n-4}$, suppose the tan…

Let $M_\infty$ be the limit of a sequence of Einstein manifolds $(M_i,g_i)$ with the same special holonomy and uniformly bounded diameter. Let $S\subset M_\infty$ be the rectifiabl…

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 $(X,d)$ be a noncollapsed Gromov–Hausdorff limit of Einstein $5$-manifolds, and let $\mathcal{S}_{\mathrm{orb}}$ denote its orbifold singular set. Einstein metrics are real-ana…

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…

Let $(M,g)$ be a compact Riemannian $n$-manifold with $obreakoperatorname{Ric}ggeq -(n-1)$ and bounded diameter, and let $widetilde{B1(x)}$ denote the local universal cover of the…

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…

Uniqueness conjecture. Tangent cones are unique for noncollapsed Einstein manifolds.

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

Let $(M^{n},g)$ be a closed connected Einstein space of dimension $n\geq 3$, and let $\kappa_{0}$ be the lower bound of its sectional curvature. Let $\lambda$ be an eigenvalue of t…

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 $T^*\mathbb{S}^2$ be the cotangent bundle of the two-sphere, and let $\mathbb{H}^4\slash\mathbb{Z}_2$ be the hyperbolic orbifold obtained by antipodal identification in a globa…

A Ricci-flat ALE space is a complete Ricci-flat manifold that is asymptotically locally Euclidean; the paper refers to gravitational instantons and their Kähler quotients as the tw…

Classification conjecture. The manifold $(M,g)$ must be either $\mathbb{S}^{4}$, $\mathbb{C}\operatorname{P}^{2}$, $\mathbb{S}^{2}(1)\times\mathbb{S}^{2}(1)$, or a quotient of one…

A Einstein metric is a Riemannian metric satisfying … for some $\lambda\in\mathbb{R}$. In dimension four, a Riemannian manifold is said to have positive sectional curvature when ev…

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…