14 problems

Let $(M, \langle\;, \: \rangle)$ be a complete Riemannian manifold with quasi-positive curvature, meaning that it has non-negative sectional curvature and there is a point at which…

Let $M$ be an $n$-dimensional closed Riemannian manifold with almost nonnegative Ricci curvature, satisfying … The first Betti number is denoted by $b_1(M)$, and the $b_1(M)$-torus…

Let $(M,g)$ be a complete Riemannian manifold with non-negative sectional curvature, and suppose that there is a point $p\in M$ whose two-planes have positive sectional curvature.…

Let $K(x,y)$ denote the sectional curvature in two dimensions for the Fisher–Rao geometry of Dirichlet distributions, with the curvature function defined by the metric under consid…

Let $M_{p,q}=\mathbf S^p\times\mathbf S^q\times[-1,1]$ be a Clifford band, and let $g$ be a smooth metric on $M_{p,q}$. Clifford-band width conjecture. If … then … This is a propos…

Let $M=T^2\times[-1,1]$ and let $g$ be a smooth metric on $M$. Gromov's conjecture. If … then … This is a proposed sharp width estimate for bands with positive sectional curvature,…

Sha–Shen finite topological type conjecture. Every complete manifold $M^n$ with nonnegative Ricci curvature and quadratically asymptotically nonnegative curvature should be of fini…

Yau's rigidity conjecture. If $(X,\omega)$ has negative Riemannian sectional curvature, then $X$ is rigid.

Let $M$ be a simply connected Einstein four-manifold with positive scalar curvature and non-negative sectional curvature. Folklore conjecture. The manifold must be one of the follo…

Let $(M,g)$ be a smooth four-dimensional Einstein manifold satisfying … for a constant $\lambda>0$, and suppose that its sectional curvature is non-negative. The metrics $g_0$,…

Positivity conjecture. The $H^{1/2}$-metric has positive sectional curvature.

Let $N$ be an arbitrary Riemannian manifold, and let the $L^2$-metric be the metric on the relevant diffeomorphism group induced by the Riemannian metric on $N$. The sectional curv…

Let $M$ be a compact Kähler manifold and let $\mathcal{H}$ be the space of Kähler potentials endowed with the Dirichlet metric. When $M$ is a Riemann surface, this metric is flat.…

Let $X$ be an Alexandrov space with a lower sectional-curvature bound, and let $\gamma$ be a minimizing geodesic in $X$. Burago's tangent-cone conjecture. Tangent cones at any two…