8 problems

Thurston's Elliptization Conjecture. Every closed $3$-manifold with finite fundamental group can be modeled on the spherical geometry $(S^3,SO(4))$.

Let $M$ be a closed, orientable, atoroidal 3-manifold, meaning that it contains no essential torus. Thurston's hyperbolization conjecture. If $M$ has infinite fundamental group, th…

Thurston's geometrization conjecture. Each component resulting from the geometric decomposition of $\mathcal{M}$ is geometric.

Let $M$ be a compact, orientable, prime $3$-manifold. The eight Thurston geometries are the homogeneous geometries that admit compact quotient manifolds. Thurston's geometric decom…

Let $(M^4,J)$ be a compact Class $\mathrm{VII}_+$ surface, let $\omega_0$ be a pluriclosed metric on $M$, and let $p\in M$ be a generic point as described in the surrounding discus…

Let $(M^4,J)$ be an Inoue surface, and let $\omega_0$ be a pluriclosed metric on $M$. Inoue surface geometrization conjecture. The solution $\omega_t$ to pluriclosed flow with this…

Let $(M^4,J)$ be a properly elliptic surface with $\operatorname{kod}(M)=1$ and odd first Betti number, and let $\omega$ be a pluriclosed metric. Properly elliptic geometrization c…

Let $M$ be an irreducible orientable compact 3-manifold with possibly empty boundary consisting of tori. A geometric 3-manifold is one whose interior has a finite-volume complete g…