34 problems

Thurston's bending conjecture. The data of these two pleating loci determine the geometry of the entire manifold $M$.

Gromov's conjecture. Geometric manifolds with large automorphism groups should be classifiable.

Let $M^{3}$ be a closed, prime, atoroidal $3$-manifold. A locally homogeneous Riemannian metric is a Riemannian metric such that for any two points $p$ and $q$ there is an isometry…

The paper considers compact $3$-manifolds and canonical decompositions into geometric pieces. Geometric decomposition conjecture. All compact $3$-manifolds admit canonical decompos…

Let $M$ be a complete finite-volume hyperbolic $3$-manifold, and consider the space of its Dehn filling deformations. Hodgson–Kerckhoff conjecture. This space is connected and smoo…

Let $N$ be a compact and complete affine manifold, and let $\widetilde{N}\to N$ be a finite Galois cover. Affine-map conjecture. The cover $\widetilde{N}$ is the domain of a nontri…

Sullivan–Thurston conjecture. Every $3$-manifold admits some analytic geometric structure.

Quotient conjecture for compact integral-integral affine manifolds. Every compact integral-integral affine manifold arises as a quotient of $\mathbb{R}^n$ by a free and proper acti…

A closed manifold is locally modelled on a reductive homogeneous space if it has a geometric structure locally isomorphic to a reductive homogeneous space $G/H$. Markus' conjecture…

Let $S$ be a closed surface, let $\mathrm{Hit}_{2n}(S)$ and $\mathrm{QHit}_{2n}(S)$ denote the Hitchin and quasi-Hitchin representation spaces, respectively, and consider the geome…

Let $S$ be the closed surface underlying the quasi-Hitchin representations under consideration, let $M_{\mathbb C}$ be the manifold associated with a quasi-Hitchin representation i…

Differential 1-form representation conjecture. In the continuous limit, a smectic is faithfully represented by a differential 1-form.

Projection conjecture. Compatible structures for Lie systems can be obtained very generally via projections onto quotient spaces of automorphic Lie systems admitting VG Lie algebra…

Dloussky's conjecture. Every surface of class VII admits a $\bigl(\operatorname{Bir}(\mathbb{P}^2),\mathbb{P}^2\bigr)$-structure.

Let $S$ be a surface, and let $T^1S$ carry a convex foliated $\mathbb{RP}^3$-structure. Its holonomy is allowed to lie in an arbitrary subgroup of…

Local homogeneity conjecture. Any such structure $\phi$ is locally homogeneous. This implies that if $\phi$ is rigid, then the fundamental group of $X$ is infinite.

Local homogeneity conjecture. Any holomorphic Riemannian metric on $X$ is locally homogeneous.

Guichard-Wienhard conjecture. There exists a generalized flag variety $X$ and a compact fiber bundle $M\to S$ such that $\overline{\rho}$ is the holonomy of a locally homogeneous…

Hitchin's conjecture. Hitchin components are geometrically significant and parametrize geometric structures.

Let $\underline{G}$ be one of the Lie groups considered in the source, let $M$ be a compact manifold, and consider $(\underline{G}\times\underline{G},\underline{G})$-structures on…

Let an MGHC anti-de Sitter spacetime be a maximal globally hyperbolic spatially compact spacetime modeled on anti-de Sitter space, and let $\Lambda$ be its associated achronal subs…

A closed anti-de Sitter 3-manifold is a closed $3$-manifold equipped with a Lorentz metric of constant negative sectional curvature; it is standard if its anti-de Sitter structure…

Soit $M$ une variété compacte munie d'une structure affine, c'est-à-dire d'une structure modelée sur $(\operatorname{Aff}(\mathbb{R}^n),\mathbb{R}^n)$. Une forme volume parallèle e…

Let $n>2$, let $\mathbf{X}=G=\mathrm{PO}(n,1)$, and let $\mathbf{G}=G\times G$ act on $\mathbf{X}$ by left and right multiplication. For a compact manifold $N$, a…

A geometric structure on a manifold is a $G$-structure when it is locally homogeneous at the first order. Suppose that the automorphism group of the manifold acts with a dense orbi…