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PreprintCC BY 4.0arXiv

Alexandrov--Fenchel inequalities for convex domains in the sphere

  • Tianci Luo
  • Yong Wei
  • Rong Zhou
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Abstract

We prove the Alexandrov--Fenchel inequalities between any two spherical quermassintegrals for smooth weakly convex domains contained in an open hemisphere, with equality if and only if the domain is a geodesic ball. For strictly convex hypersurfaces, we introduce a globally constrained curvature flow which preserves one quermassintegral and decreases the next one. We establish uniform curvature estimates by combining a pinching estimate, a support function argument, and spherical polarity. As a consequence, the flow exists for all time and converges smoothly and exponentially to a geodesic sphere. The monotonicity of the quermassintegrals gives the full family of Alexandrov--Fenchel inequalities. A short-time mean curvature flow approximation and a localized rigidity argument extend the result, including the equality characterization, to weakly convex domains.

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Cite

BibTeXRIS
BibTeX
@misc{luo2026alexandrovfenchel,
  title = {{Alexandrov--Fenchel inequalities for convex domains in the sphere}},
  author = {Tianci Luo and Yong Wei and Rong Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13829},
  eprint = {2609.13829},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13829},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Alexandrov--Fenchel inequalities for convex domains in the sphere
AU  - Tianci Luo
AU  - Yong Wei
AU  - Rong Zhou
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.13829
UR  - https://arxiv.org/abs/2609.13829
AB  - We prove the Alexandrov--Fenchel inequalities between any two spherical quermassintegrals for smooth weakly convex domains contained in an open hemisphere, with equality if and only if the domain is a geodesic ball. For strictly convex hypersurfaces, we introduce a globally constrained curvature flow which preserves one quermassintegral and decreases the next one. We establish uniform curvature estimates by combining a pinching estimate, a support function argument, and spherical polarity. As a consequence, the flow exists for all time and converges smoothly and exponentially to a geodesic sphere. The monotonicity of the quermassintegrals gives the full family of Alexandrov--Fenchel inequalities. A short-time mean curvature flow approximation and a localized rigidity argument extend the result, including the equality characterization, to weakly convex domains.
ER  -