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

We study quermassintegral-preserving curvature flows in Hⁿ⁺¹, n≥3, driven by the curvature quotient (Eₖ/E₎1/(k-). For 1≤<k with k+≤ n, we prove that positive sectional curvature is preserved. Starting from a smooth closed strictly convex hypersurface with positive sectional curvature, the flow exists for all time and converges smoothly and exponentially to a geodesic sphere. As an application, we establish sharp Alexandrov--Fenchel inequalities between Wₖ and W_ for smooth bounded convex domains whose boundaries have nonnegative sectional curvature, whenever 0≤<k≤ n and k+≤ n. Equality holds if and only if the domain is a geodesic ball. Together with previously known cases, these inequalities give the full family of Alexandrov--Fenchel inequalities for this class of domains in H⁶. We also construct counterexamples showing that k+≤ n is the optimal range for preservation of positive sectional curvature by these flows.