Entropy and round-point convergence for the σₖ-curvature flow

Tianci Luo; Yong Wei; Ruijia Zhang; Rong Zhou

Paper · Nov 2, 2026

Entropy and round-point convergence for the $σ_{k}$ -curvature flow

Tianci Luo, Yong Wei, Ruijia Zhang, Rong Zhou

math.DGWider flowsarXiv:2611.03249

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Abstract

We prove that every smooth closed strictly convex hypersurface in $R^{n + 1}$ contracts to a round point under the $σ_{k}$ -curvature flow for $2 \leq k \leq n - 1$ , without additional pinching or symmetry assumptions. After rescaling about the extinction point to mean width two, the hypersurfaces converge smoothly and exponentially in normalized time to the unit sphere. The proof is based on an entropy which is invariant under translations and dilations and admits a representation by weighted boundary areas of orthogonal projections. A strengthened Codazzi inequality yields its monotonicity. An upper bound for the entropy controls the inradius relative to the mean width, while its dissipation controls the spatial variation of the normalized speed. We also obtain two-sided curvature bounds at the natural extinction scale.

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Submitted: Nov 2, 2026
Subjects: math.DG

Comments: 26 pages, comments are welcome!

Other curvature flows Curvature pinching Long-time existence & convergence

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The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

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) Curvature pinching Curvature conditions that Ricci flow preserves and improves. Hamilton–Ivey pinching makes positive curvature dominate in 3D blow-ups, and invariant… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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)

In the guidebook

6A.9
Reaction–Diffusion: Fronts, Patterns and PinchingBook 6A · The Heat Equation and Its Relatives
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.7
Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles
Scalar and tensor maximum principles, and preserved curvature conditions.
11A.10
Hamilton–Ivey PinchingBook 11A · The Ricci Flow: Existence and Maximum Principles
Why high curvature in dimension 3 is almost positive.

[1] 6A.9
Reaction–Diffusion: Fronts, Patterns and PinchingBook 6A · The Heat Equation and Its Relatives
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.

[2] 11A.7
Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles
Scalar and tensor maximum principles, and preserved curvature conditions.

[3] 11A.10
Hamilton–Ivey PinchingBook 11A · The Ricci Flow: Existence and Maximum Principles
Why high curvature in dimension 3 is almost positive.

Entropy and round-point convergence for the $σ_{k}$ -curvature flow

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Entropy and round-point convergence for the σk​-curvature flow

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Entropy and round-point convergence for the $σ_{k}$ -curvature flow