On n-dimensional complete self-similar solutions to the mean curvature flow in ℝⁿ⁺¹ with nonnegative constant scalar curvature

Yong Luo; Linlin Sun; Jiabin Yin

Paper · May 20, 2021

On $n$ -dimensional complete self-similar solutions to the mean curvature flow in $R^{n + 1}$ with nonnegative constant scalar curvature

Yong Luo, Linlin Sun, Jiabin Yin

math.DGWider flowsarXiv:2105.09610v2

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Abstract

As is well known, self-similar solutions to the mean curvature flow, including self-shrinkers, translating solitons and self-expanders, arise naturally in the singularity analysis of the mean curvature flow. Recently, Guo proved that $n$ -dimensional compact self-shrinkers in $R^{n + 1}$ with scalar curvature bounded from above or below by some constant are isometric to the round sphere $S^{n} (n)$ , which implies that $n$ -dimensional compact self-shrinkers in $R^{n + 1}$ with constant scalar curvature are isometric to the round sphere $S^{n} (n)$ (see also). Complete classifications of $n$ -dimensional translating solitons in $R^{n + 1}$ with nonnegative constant scalar curvature and of $n$ -dimensional self-expanders in $R^{n + 1}$ with nonnegative constant scalar curvature were given by Martín, Savas-Halilaj and Smoczyk and Ancari and Cheng, respectively. In this paper we give complete classifications of $n$ -dimensional complete self-shrinkers in $R^{n + 1}$ with nonnegative constant scalar curvature. We will also give alternative proofs of the classification theorems due to Martín, Savas-Halilaj and Smoczyk and Ancari and Cheng.

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Submitted: May 20, 2021
Last revised: May 24, 2021 (v2)
Subjects: math.DG

Comments: 15 pages. We noted that Theorem 1.8 in our paper has been proved earlier by Ancari and Cheng in a recent paper [AC]

Mean curvature flow Self-shrinkers & translators Singularities Scalar curvature

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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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

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.
9B.6
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.

On $n$ -dimensional complete self-similar solutions to the mean curvature flow in $R^{n + 1}$ with nonnegative constant scalar curvature

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On n-dimensional complete self-similar solutions to the mean curvature flow in Rn+1 with nonnegative constant scalar curvature

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On $n$ -dimensional complete self-similar solutions to the mean curvature flow in $R^{n + 1}$ with nonnegative constant scalar curvature