11 problems

Let $M$ be a compact simply connected differentiable manifold, and fix $\delta\in(0,1]$. Consider all $\delta$-pinched Riemannian structures on $M$, and let $i(M,g)$ denote the inj…

Let $M^n$ be a manifold and let $\delta$-pinched metrics on $M^n$ mean metrics whose sectional curvatures satisfy the pinching condition determined by $\delta$. Consider the inject…

Little Loop Lemma conjecture. The Little Loop Lemma should be true for all solutions of the Ricci flow on compact $3$-manifolds.

Injectivity-radius conjecture. The displayed equalities hold for the respective ranges of $\beta$. The result gives only an interval estimate when $\beta\in(\frac13,\frac23)$ and o…

Absil et al.'s injectivity-radius conjecture. For the canonical metric on the Stiefel manifold, the injectivity radius is

Let $\mathrm{St}(n,p)$ be the Stiefel manifold with $2\leq p\leq n-2$, equipped with a metric from the one-parameter family indexed by $\beta>0$. Let $t^{\mathrm{r}}_\beta$ be the…

Gromov's noncompact injectivity radius conjecture. Then

Let $(M^n,g)$ be a complete Riemannian manifold. Write $\operatorname{Inj}(M)$ for its injectivity radius and $\operatorname{Rad}_{\mathbb S^n}(M)$ for its spherical radius. The in…

Ammann–Lauter–Nistor's conjecture. The injectivity radius of a connected manifold with Lie structure at infinity is positive.

Let $M$ be a closed Riemannian $n$-manifold, let $\inj(M)$ denote its injectivity radius, let $\alpha_n$ be the volume of the canonical $n$-sphere, and let $B(r)$ be a ball of radi…

Let $M$ be a closed arithmetic hyperbolic $3$-manifold, and let $\operatorname{inj}(M)$ denote its injectivity radius. Geometric Lehmer conjecture. There is a universal constant…