64 problems

Let $(M^n,g,f)$ be a CPE metric, meaning that $(M^n,g)$ is a closed, oriented Riemannian manifold of dimension $n\geq 3$ with constant scalar curvature and $f:M^n\to\mathbb{R}$ is…

Let $M$ be a compact nilmanifold and let $g$ be an Einstein metric on $M$. The Einstein metric conjecture. The metric $g$ is flat, and $M$ is a torus. This concerns the possible Ei…

Let $(M,g)$ be a compact Riemannian manifold admitting a smooth non-constant solution of … with $(\mu,\lambda)=(1,-R/n)$, so that $(M,g)$ is a critical point equation (CPE) metric.…

Sasakian–Einstein conjecture. Every such homotopy sphere admits a Sasakian–Einstein metric.

Finiteness conjecture. This set is finite.

Let $G/H$ be a compact homogeneous space whose isotropy representation consists of pairwise inequivalent irreducible summands; in particular, this includes the case where…

Let $M$ be a closed 3-manifold with metric $g$, scalar curvature $s$, and scalar-curvature tensor $z$. Let $L^{*}$ denote the formal adjoint of the linearization of the scalar-curv…

Let $M$ be a closed smooth $4$-manifold. Uniqueness conjecture. $M$ admits Einstein metrics for at most one sign of the scalar curvature. The conjecture proposes that, for a fixed…

Finiteness conjecture. This set is finite.

An odd-dimensional sphere is the standard sphere $S^{2n+1}$, and a parallelizable manifold is a manifold with trivial tangent bundle. A Sasakian-Einstein metric is a Sasakian metri…

A homotopy sphere is a smooth manifold homotopy equivalent to a sphere, and a Sasakian-Einstein metric is a Sasakian metric whose associated Riemannian metric is Einstein. The prec…

Let $\overline M$ be a smooth compact manifold with boundary, let $M$ be its interior, and let $g$ be a conformally compact Einstein metric on $M$. A smooth conformal infinity is t…

Let $E$ be an ALE degeneration with end modeled on $S^3/{\mathbb Z}_k$, where $k$ is an integer, and suppose the signature and curvature estimates yield…

Let $g\in E_{AH}$ be an asymptotically hyperbolic Einstein metric in dimension $n+1$. The strong control property means that (4.3) holds for some $(k,\beta)$ and $(m,\alpha)$ satis…

Let $B^4$ be the $4$-ball, or more generally let $B^{n+1}$ be the $(n+1)$-ball, equipped with the setting of asymptotically hyperbolic Einstein metrics discussed above. Ball cusp-f…

Nonexistence conjecture. No such strictly globally static asymptotically hyperbolic Einstein solutions exist.

Kählerness conjecture. Einstein metrics on irrational surfaces are always Kähler.

Higher-order vanishing conjecture. The component of the Einstein equation to order $4$ on $\Gamma(\operatorname{Sym}^{2,+}TM)$ should admit an analogue of the order-three identity,…

Let $M$ be a closed hyperbolic manifold of dimension $n\geq 4$, and let $X$ be a nontrivial finite covering of $M$ branched along a closed totally geodesic submanifold of codimensi…

Einstein conjecture. Every solution of the corresponding critical point equation is an Einstein metric.

Stability conjecture. Every Einstein metric of non-positive scalar curvature is stable.

Let $M=\mathsf G/\mathsf H$ be one of the low-dimensional full flag manifolds listed in Theorem, with $\mathsf G$ a compact simple Lie group of type ${\rm A}_n$, ${\rm B}_n$,…

Let $M=\mathsf G/\mathsf H$ be a compact homogeneous space whose isotropy representation consists of pairwise inequivalent irreducible summands. In particular, this includes the ca…

A homogeneous Einstein manifold is a Riemannian manifold with a transitive isometry group and a metric satisfying $\operatorname{Ric}_g=\lambda g$ for a constant $\lambda$. Aleksee…

Let $d_1,d_2$ be positive integers and set $n=d_1+d_2+1$. Let $A(\alpha)$ and $\Omega(\omega)$ be the curves whose intersections correspond to…