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

@misc{luo2026curvature,
  title = {{Curvature quotient flows and Alexandrov--Fenchel inequalities in hyperbolic space}},
  author = {Tianci Luo and Yong Wei and Ruijia Zhang and Rong Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35707},
  eprint = {2610.35707},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35707},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Curvature quotient flows and Alexandrov--Fenchel inequalities in hyperbolic space
AU  - Tianci Luo
AU  - Yong Wei
AU  - Ruijia Zhang
AU  - Rong Zhou
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35707
UR  - https://arxiv.org/abs/2610.35707
AB  - We study quermassintegral-preserving curvature flows in ℍ^n+1, n≥3, driven by the curvature quotient (E_k/E_ℓ)^1/(k−ℓ). For 1≤ℓ<k with k+ℓ≤ 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≤ℓ<k≤ n and k+ℓ≤ 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≤ n is the optimal range for preservation of positive sectional curvature by these flows.
ER  -