Paper · Aug 11, 2026
The quermassintegral preserving curvature flow for horo-convex hypersurfaces in the sphere
As on arXiv
Abstract
We introduce fully nonlinear curvature flows of horo-convex hypersurfaces in the sphere that preserve an arbitrarily prescribed spherical quermassintegral. The non-local normalization is defined relative to a fixed ambient origin, and we prove that the evolution is governed by a single smooth fixed-origin equation for all time. For monotone, homogeneous curvature functions satisfying concavity and inverse concavity, we establish preservation of horo-convexity, uniform curvature pinching, a direct estimate for the non-local coefficient. A Tso-type argument then yields global curvature bounds and long-time existence. We further prove exponential decay of the traceless second fundamental form and exponential convergence to the geodesic sphere centered at the fixed origin whose radius is determined by the preserved quermassintegral.
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 6A.9Reaction–Diffusion: Fronts, Patterns and Pinching
Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.
- 11A.7Maximum Principles under Ricci Flow
Scalar and tensor maximum principles, and preserved curvature conditions.
- 11A.10Hamilton–Ivey Pinching
Why high curvature in dimension 3 is almost positive.