159 problems

Let $M$ be an open manifold with nonnegative Ricci curvature, and let $\widetilde M$ be its universal cover. The manifold $\widetilde M$ has Euclidean volume growth if there is a c…

Ricci-flat tangent cone conjecture. If one tangent cone at infinity is Ricci flat, then $M$ is Ricci flat. Consequently, the tangent cone at infinity should be unique; analogous qu…

The curvature-dimension condition $\operatorname{CD}(K,N)$ concerns metric measure spaces with a lower Ricci curvature bound $K\in\mathbb{R}$ and an upper dimension bound…

Ricci lower-bound conjecture. Then

Yau's finite-dimensionality conjecture. These spaces are finite dimensional.

Yau's conjecture. The scalar curvature should satisfy

Let $M$ be a complete noncompact Kähler manifold with positive bisectional curvature and maximal Euclidean volume growth. Say that scalar curvature decays quadratically in the aver…

Gromov's conjecture. There exists a dimensional constant $C(n)$ such that, for every $p\in M$,

Let $(M^3,g)$ be a complete, non-compact Riemannian manifold. Say that $(M^3,g)$ is Ricci pinched when it satisfies the Ricci-pinching condition used in Hamilton's conjecture. Hami…

Let $n\ge 4$ and let $(M^n,g)$ be a complete Riemannian manifold with $\operatorname{Ric}>0$. A free abelian subgroup of $\pi_1(M^n)$ is an abelian subgroup isomorphic to…

Let $(M^n,g)$ be a compact space with nonnegative Ricci curvature, and let $\pi_1(M)$ denote its fundamental group. A group is uniformly almost abelian if it has an abelian subgrou…

A uniformly Ricci pinched Riemannian manifold is a complete, connected 3-dimensional Riemannian manifold satisfying the uniform Ricci pinching condition. Hamilton–Lott conjecture.…

Splitting conjecture. Every tangent cone at infinity $M_\infty$ splits isometrically as

Let $(M,\xi)$ be a contact 3-manifold equipped with a compatible metric $g$. Write $Ricci(g)>0$ for positivity of its Ricci curvature, and let $(\mathbb{S}^3,\xi_{std})$ denote the…

A Riemannian manifold has lower bounded Ricci curvature if its Ricci curvature is bounded below by a fixed constant. The Margulis lemma asserts that sufficiently small loops genera…

Let $(M_i,p_i)$ be $n$-dimensional complete pointed Riemannian manifolds with a uniform lower Ricci-curvature bound, and suppose that a subsequence converges in the pointed measure…

Let $(X,d,\mu)$ be a measured Gromov--Hausdorff limit of $n$-dimensional compact Riemannian manifolds $M_i$ with $\operatorname{Ric}_{M_i}\ge (n-1)k$, equipped with normalized Riem…

Let $(M_i,p_i)$ be $n$-dimensional complete pointed Riemannian manifolds with $\operatorname{Ric}_{M_i}\ge -(n-1)$, and suppose that a subsequence converges in the pointed Gromov--…

Let $(M_i,p_i)$ be $n$-dimensional complete pointed Riemannian manifolds with a uniform lower Ricci-curvature bound, and suppose that a subsequence converges in the pointed Gromov-…

Fukaya–Yamaguchi's fibering conjecture. A finite cover of $M$ fibers over a nilmanifold with a fiber that has nonnegative Ricci curvature and finite fundamental group.

Gromov's conjecture. There is a positive number $\epsilon_n$ depending only on $n$ such that $\pi_1(M)$ is almost nilpotent, meaning that it contains a nilpotent subgroup of finite…

Let $M$ be complete, let $O$ be a fixed origin, and suppose that for some $r_0>0$ the Ricci curvature is bounded below by $(n-1)/(4r^2)$ whenever $r\geq r_0$. Higher-dimensional Ri…

Let $(M,g)$ be a complete, noncompact manifold with only one end: outside a compact set it is diffeomorphic to a spherical shell $S^{n-1}\times\mathbb{R}$. Let $g_0$ denote the Euc…

A plurisubharmonic function with sub-logarithmic growth on a complete noncompact Kähler manifold is understood as a plurisubharmonic function satisfying the stated sub-logarithmic…

Ricci sphere-threshold conjecture. The result remains true for