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

Abstract

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 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 $W_k$ and $W_\ell$ for smooth bounded convex domains whose boundaries have nonnegative sectional curvature, whenever $0\leq\ell<k\leq n$ and $k+\ell\leq n$. 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 $\mathbb H^6$. We also construct counterexamples showing that $k+\ell\leq n$ is the optimal range for preservation of positive sectional curvature by these flows.