Paper · Oct 26, 2026

Curvature quotient flows and Alexandrov–Fenchel inequalities in hyperbolic space

Tianci Luo, Yong Wei, Ruijia Zhang, Rong Zhou

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Abstract

We study quermassintegral-preserving curvature flows in $H^{n + 1}$ , $n \geq 3$ , driven by the curvature quotient $(E_{k} / E_{ℓ})^{1/ (k - ℓ)}$ . For $1 \leq ℓ < k$ with $k + ℓ \leq n$ , 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 $W_{ℓ}$ for smooth bounded convex domains whose boundaries have nonnegative sectional curvature, whenever $0 \leq ℓ < k \leq n$ and $k + ℓ \leq n$ . 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 $H^{6}$ . We also construct counterexamples showing that $k + ℓ \leq n$ is the optimal range for preservation of positive sectional curvature by these flows.

Comments: 36 pages. Comments are welcome!

Other curvature flows Sphere theorems Long-time existence & convergence Geometric inequalities

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Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Geometric inequalities Sharp inequalities between area, volume and integrals of curvature, with equality only for round spheres. Start with: Mean curvature flow and geometric inequalities (Topping, 1998)

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11A.11
Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles
Positive Ricci curvature in dimension 3 flows to a round metric.

Curvature quotient flows and Alexandrov–Fenchel inequalities in hyperbolic space

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