Contracting axially symmetric hypersurfaces by powers of the σₖ-curvature

Haizhong Li; Xianfeng Wang; Jing Wu

Paper · May 14, 2019

Contracting axially symmetric hypersurfaces by powers of the $σ_{k}$ -curvature

Haizhong Li, Xianfeng Wang, Jing Wu

math.DGWider flowsarXiv:1905.05571

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Abstract

In this paper, we investigate the contracting curvature flow of closed, strictly convex axially symmetric hypersurfaces in $R^{n + 1}$ and $S^{n + 1}$ by $σ_{k}^{α}$ , where $σ_{k}$ is the $k$ -th elementary symmetric function of the principal curvatures and $α \geq 1/ k$ . We prove that for any $n \geq 3$ and any fixed $k$ with $1 \leq k \leq n$ , there exists a constant $c (n, k) > 1/ k$ such that that if $α$ lies in the interval $[1/ k, c (n, k)]$ , then we have a nice curvature pinching estimate involving the ratio of the biggest principal curvature to the smallest principal curvature of the flow hypersurface, and we prove that the properly rescaled hypersurfaces converge exponentially to the unit sphere. In the case $1 < k \leq n \leq k^{2}$ , we can choose $c (n, k) = \frac{1}{k - 1}$ . Our results provide an evidence for the general convergence result without initial curvature pinching conditions.

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Submitted: May 14, 2019
Subjects: math.DG

Comments: 40 pages

Other curvature flows Curvature pinching Long-time existence & convergence Rotational symmetry & warped products

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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) 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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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.

Contracting axially symmetric hypersurfaces by powers of the $σ_{k}$ -curvature

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Contracting axially symmetric hypersurfaces by powers of the $σ_{k}$ -curvature