Curvature quotient flows and Alexandrov--Fenchel inequalities in hyperbolic space

We study quermassintegral-preserving curvature flows in $\mathbb H^{n+1}$, $n\geq3$, driven by the curvature quotient $(E_k/E_\ell)^{1/(k-\ell)}$. For $1\leq\ell<k$ with $k+\ell\leq n$, we prove that positive sectional curvature is preserved. Starting from a smooth closed strictly convex hypersurfac