248 problems

Volume rigidity conjecture. The metric $g$ is isometric to the hyperbolic metric $g_H$.

Let $G$ be a connected semisimple Lie group with finite center, all of whose simple factors have real rank at least $2$. Let $\Gamma\subset G$ be a lattice, $M$ be a compact manifo…

Let $T\subset S^3$ be an embedded minimal torus, and let $T_{\mathrm{Cl}}$ denote the Clifford torus in $S^3$. Lawson's conjecture. There exists an isometry $F$ of $S^3$ such that…

Let $M$ be a smooth closed manifold, and let $g$ and $g_0$ be negatively curved metrics, or more generally metrics whose geodesic flows are Anosov. Let $\mathcal L_g$ denote the ma…

Let $X$ be a closed manifold, and let $h_1$ and $h_2$ be Riemannian metrics on $X$ with negative sectional curvature. For each such metric, let $\mathrm{MLS}_h$ assign to every con…

Let $\alpha\colon\mathbb{R}^k\curvearrowright M$ be a higher rank Anosov action, meaning an Anosov action with $k\geq 2$, and suppose that it has no rank $1$ factors. Katok–Spatzie…

Flenner–Zaidenberg rigidity conjecture. The affine Pham–Brieskorn hypersurface $X_{a_0,\dots,a_n}$ is rigid if and only if $\min\{a_0,\dots,a_n\}\geq 2$ and at most one element $i$…

Let $g_0$ be a locally symmetric metric, and let $D_{g_0}^*$ be the operator appearing in the paper. Denote by $\Pi_{\operatorname{Ker}(D_{g_0}^*)}$ the projection onto…

A circle domain $\Omega\subset\widehat{\mathbb C}$ is a domain whose boundary components are points or circles. It is conformally rigid if every conformal map of…

Let $G$ be a countable discrete group, and let $\operatorname{L}(G)$ be its group von Neumann algebra. Write $\operatorname{Char}(G)$ for the group of characters of $G$ and…

Let $\mathrm{Seg}(\mathbb P^2\times\mathbb P^2)\subset\mathbb P^8$ be the Segre variety. Griffiths–Harris rigidity conjecture. The Segre variety can be characterized by a second-or…

Let $(M^3,g)$ be a Riemannian three-manifold with scalar curvature $R_M\geq 0$, and let $\Sigma\subset M$ be a two-sided two-torus. A surface is locally area-minimizing if it minim…

Let $(M,g)$ be a closed Riemannian manifold with all sectional curvatures negative, let $\Gamma=\pi_1(M)$, and let $C_\Gamma$ be the set of conjugacy classes of non-trivial element…

For $p\ge6$, consider the regular tiling ${p,3}$ and the circles inscribed in its faces. A circle packing is strongly solid if it remains solid after any one of its circles is remo…

Connes's conjecture. If

Let $X$ be a compact homogeneous manifold and let $f_0$ be an affine $K$-system on $X$. Assume that $\mathcal{Z}^\infty(f_0)$ contains an embedded subgroup isomorphic to…

Hopf's conjecture. Any metric without conjugate points on $T^n$ must be flat.

Let $P$ be a convex $n$-dimensional polytope in $\mathbb R^n$, equipped with a Riemannian metric $g$. Assume that each face is mean convex with respect to $g$, the dihedral angles…

Simpson's motivicity conjecture. Rigid integrable connections are motivic.

Structural decomposition conjecture.

Katok's Margulis-function conjecture. Unless $M$ is locally symmetric, $c(x)$ is almost always neither constant nor smooth.

A hyperbolic strictly-convex polyhedron is a strictly-convex polyhedron in hyperbolic space, with dihedral angles measured along its edges. Stoker's conjecture. Every hyperbolic st…

Let $S^n$ be the $n$-sphere, and let a metric be defined on the complement of finitely many points in $S^n$. Llarull's theorem is the scalar-curvature rigidity result for metrics o…

Let $\mathfrak{g}$ and $\mathfrak{n}$ be Lie algebras, with $\mathfrak{g}$ reductive and $\mathfrak{n}$ semisimple. Suppose that the pair $(\mathfrak{g},\mathfrak{n})$ admits a pos…

Let $M^n$ be a complete Riemannian manifold with sectional curvature $K\leq 1$, and let $N^n$ be a compact Riemannian manifold with scalar curvature $R\geq -n(n-1)$. For a continuo…