Contraction of surfaces in hyperbolic space and in sphere

Yingxiang Hu; Haizhong Li; Yong Wei; Tailong Zhou

Paper · Apr 1, 2019

Contraction of surfaces in hyperbolic space and in sphere

Yingxiang Hu, Haizhong Li, Yong Wei, Tailong Zhou

math.DGWider flowsarXiv:1904.00572v3

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Abstract

In this paper, we consider the contracting curvature flow of smooth closed surfaces in $3$ -dimensional hyperbolic space and in $3$ -dimensional sphere. In the hyperbolic case, we show that if the initial surface $M_{0}$ has positive scalar curvature, then along the flow by a positive power $α$ of the mean curvature $H$ , the evolving surface $M_{t}$ has positive scalar curvature for $t > 0$ . By assuming $α \in [1, 4]$ , we can further prove that $M_{t}$ contracts a point in finite time and become spherical as the final time is approached. We also show the same conclusion for the flows by powers of scalar curvature and by powers of Gauss curvature provided that the power $α \in [1/2, 1]$ . In the sphere case, we show that the flow by a positive power $α$ of mean curvature contracts strictly convex surface in $S^{3}$ to a round point in finite time if $α \in [1, 5]$ . The same conclusion also holds for the flow by powers of Gauss curvature provided that the power $α \in [1/2, 1]$ .

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Submitted: Apr 1, 2019
Last revised: Sep 27, 2020 (v3)
Subjects: math.DG, math.AP
Journal: Calculus of Variations and Partial Differential Equations, Vol.59, Article no. 172 (2020)
DOI: 10.1007/s00526-020-01826-1

Comments: 29 pages, v3, accepted version for Calculus of Variations and Partial Differential Equations; v2, add more details in section 7

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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.

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In the guidebook

6C.4
The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.
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Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.
6A.13
Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

[1] 6C.4
The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

[2] 9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.

[3] 6A.13
Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

Contraction of surfaces in hyperbolic space and in sphere

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