Hyperbolic comparison and rigidity for the hyperspherical radius

We prove Gromov’s hyperbolic hyperspherical radius comparison and rigidity conjectures in all dimensions $n\ge 3$, without the spin assumption. Let $(X^n,g)$ be a smooth, oriented, connected, compact Riemannian manifold with nonempty smooth connected boundary $S$. If ${\mathrm{S}\mathrm{c}}_g\ge -n(n-1)$ and $H_S\ge (n-1)\coth R$ for some $R>0$, then $\mathrm{R}\mathrm{a}\mathrm{d}(S,g_S)\le \sinh R$, with equality if and only if $(X^n,g)$ is isometric to the closed hyperbolic geodesic ball of radius $R$. Under the same curvature bounds, every smooth $1$-Lipschitz map of nonzero degree from $S$ to the boundary of the hyperbolic geodesic ball of radius $R$ extends to an isometry of the fillings. Here, $g_S$ is the induced boundary metric, $H_S$ is the mean curvature of $S$ with respect to the outward unit normal, and $\mathrm{R}\mathrm{a}\mathrm{d}(S,g_S)$ denotes the hyperspherical radius.