20 problems

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…

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…

Let $M$ be an odd-dimensional homotopy sphere that bounds a parallelizable manifold. Sasakian–Einstein metric conjecture. Every such $M$ admits a Sasakian–Einstein metric. This wou…

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…

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 Kähler manifold with Kähler metric $\omega$, and let $\alpha$ be a real $(1,1)$-form representing $c{1}(M)$. Calabi's conjecture. There exists a unique Kähl…

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…

Let $(X_j,g_j,p_j)$ be a sequence of Einstein metrics converging with non-collapsing volume to a singular Einstein metric $(X_\infty,g_\infty,p_\infty)$, and let…

Finiteness conjecture. Only finitely many even-dimensional spheres admit non-round Einstein metrics of cohomogeneity one.

Let $G/H$ be a connected homogeneous space of dimension $n$. An Einstein metric has Ricci tensor proportional to the metric, and negative scalar curvature means that this proportio…

Let $(M_o,\mathbf{g}_o)$ be a singular spherical or hyperbolic compact orbifold. A sequence of smooth Einstein metrics is said to converge to $(M_o,\mathbf{g}_o)$ in the Gromov–Hau…

Let $L_\infty$ be the limiting linear map appearing in the desingularization and obstruction setup, where the theorem was originally stated under the assumption…

Let $\mathbf{E}(M^4)$ be the moduli space of Einstein metrics on a fixed smooth four-manifold, let $\partial_o\mathbf{E}(M^4)$ denote the subspace of singular Einstein orbifold met…

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 $P_L$ be the linearization, at a Riemannian Kottler metric with negative cosmological constant, of the $TT$-gauge-fixed Einstein operator. Here $K$ is the constant sectional cu…

Let $M=G/H$ be a homogeneous space, let $H$ be maximal in $G$, and let $\mathcal{M}$ denote the space of Riemannian metrics under consideration. Let $g_1\in\mathcal{M}$, and let…

A CPE metric is a 3-tuple $(M^n,g,f)$, where $(M^n,g)$ is a compact oriented Riemannian manifold of dimension $n\geq 3$ with constant scalar curvature, and $f$ is a smooth function…

The Einstein CLW metric is a distinguished Einstein metric arising in the context above. A conformal perturbation means a variation of the metric obtained by multiplying it infinit…

Hermitian Einstein minimizer conjecture. The conformal class $[g]$ is an absolute minimizer of $\mathscr{W}$. Moreover, for the given $M$, every absolute minimizer arises from a me…

Let $X$ be a compact 4-manifold admitting a positive definite connection, meaning a connection whose curvature components form an oriented frame for $\Lambda^+$. Donaldson's positi…