6 problems

Let $(M^n,g)$ be a complete Riemannian manifold with infinite filling radius. Let $\omega_n$ be the volume of the unit $n$-ball, and let $V(R)$ denote the volume of a radius-$R$ ba…

Let $(M^n,g)$ be a complete Riemannian manifold with scalar curvature … Here $\operatorname{FillRad}(M)$ denotes the filling radius, and $c_n$ is a universal constant depending onl…

Let $(M^n,g)$ be a complete Riemannian manifold whose scalar curvature is at least that of the unit $n$-sphere. Gromov's filling-radius conjecture. The filling radius of $(M,g)$ sh…

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…

Optimal intrinsic filling radius constant conjecture. The optimal value for the intrinsic filling radius inequality is

Generalized filling radius conjecture. The filling radius of $(M,g)$ is no greater than $C(n)$.