15 problems

Let $(M^n,g)$ be an $n$-dimensional Riemannian manifold. A geodesic sphere is sufficiently small if its radius is sufficiently close to zero. Gray–Vanhecke volume conjecture. Suppo…

Let $(M^n,g)$ be a complete Riemannian manifold, and let $(S^n,g_0)$ be a round sphere. Choose the radius of the round sphere so that the filling radius of $(M^n,g)$ equals that of…

Volume-coefficient conjecture. If all coefficients $c_{2k}$ vanish, then $M$ is flat.

Let $(X,\mathsf d,\mathfrak m, \ll, \leq, \tau)$ be a measured Lorentzian space, let $V\subset X$ be the achronal set occurring in the hypotheses of Theorem, and let…

Quantized volume comparison conjecture. For every $m\geq 1$,

Dual higher-order Bol inequality conjecture. One should have

Higher-order Bol inequality conjecture. One should have

Let $(M^n,g)$ be a closed Riemannian manifold. Write $R_g$ for its scalar curvature and $Ric_g$ for its Ricci curvature tensor, and let $(S^n,g_{S^n})$ be the standard $n$-sphere.…

Let $(X,\mathsf{d})$ be the metric limit under consideration, let $k=\dim_{\mathcal{H}}(X,\mathsf{d})$, and let $\mathcal{H}^k$ denote its $k$-dimensional Hausdorff measure. For…

Higher-dimensional Football conjecture. For every such manifold,

Let $M$ be a compact $n$-manifold with Ricci curvature bounded below by $(n-1)H$, where $H\in\{-1,0,1\}$, and let $\widetilde M$ be its Riemannian universal covering space. For…

Ledoux's conjecture. A sharp Euclidean Sobolev inequality, without any curvature assumption, implies a sharp Euclidean volume lower bound.

Higher-dimensional volume comparison conjecture. There exists a positive $\epsilon_0(n)<1$ such that, whenever

Let $\fr{A}$ denote the $2^m$-dimensional real Clifford algebra, and consider spaces of the form $\fr{A}^n$. Clifford-space generalization conjecture. The method of the paper can b…

Let $p,q\ge 3$ with $1/p+1/q\le 1/2$ be given. Let ${\cal G}=({\cal V},{\cal E},{\cal F})$ be a locally tessellating planar graph without cut locus, meaning that it has no cut locu…