Stability properties of complete self-shrinking surfaces in ℝ³

Hilário Alencar; Gregório Silva Neto; Detang Zhou

Paper · Jun 16, 2021

Stability properties of complete self-shrinking surfaces in $R^{3}$

Hilário Alencar, Gregório Silva Neto, Detang Zhou

math.DGWider flowsarXiv:2106.09165v2

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Abstract

This paper studies rigidity for immersed self-shrinkers of the mean curvature flow of surfaces in the three-dimensional Euclidean space $R^{3} .$ We prove that an immersed self-shrinker with finite $L$ -index must be proper and of finite topology. As one of consequences, there is no stable two-dimensional self-shrinker in $R^{3}$ without assuming properness. We conclude the paper by giving an affirmative answer to a question of Mantegazza.

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Submitted: Jun 16, 2021
Last revised: Apr 29, 2022 (v2)
Subjects: math.DG, math.AP, math.SP

Comments: We made a mistake in the proof of Lemma 2.4, we thanks PR. Márcio Batista (UFAL) for pointed us out this

Mean curvature flow Self-shrinkers & translators 3-manifolds Surfaces Stability & uniqueness

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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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
11A.12
Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles
Uniformization by Ricci flow, and Hamilton’s entropy.
10A.1
A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization
The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

[1] 5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

[2] 11A.12
Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles
Uniformization by Ricci flow, and Hamilton’s entropy.

[3] 10A.1
A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization
The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

Stability properties of complete self-shrinking surfaces in $R^{3}$

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Stability properties of complete self-shrinking surfaces in R3

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Stability properties of complete self-shrinking surfaces in $R^{3}$