Existence of self-shrinkers to the degree-one curvature flow with a rotationally symmetric conical end

Siao-Hao Guo

Paper · Nov 12, 2016

Existence of self-shrinkers to the degree-one curvature flow with a rotationally symmetric conical end

Siao-Hao Guo

math.DGWider flowsarXiv:1611.03933v2

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Abstract

Given a smooth, symmetric, homogeneous of degree one function $f (λ_{1}, \dots, λ_{n})$ satisfying $\partial_{i} f > 0$ for all $i = 1, \dots, n$ , and a rotationally symmetric cone $C$ in $R^{n + 1}$ , we show that there is a $f$ self-shrinker (i.e. a hypersurface $Σ$ in $R^{n + 1}$ which satisfies $f (κ_{1}, \dots, κ_{n}) + \frac{1}{2} X \cdot N = 0$ , where $X$ is the position vector, $N$ is the unit normal vector, and $κ_{1}, \dots, κ_{n}$ are principal curvatures of $Σ$ ) that is asymptotic to $C$ at infinity.

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Submitted: Nov 12, 2016
Last revised: Aug 24, 2017 (v2)
Subjects: math.DG

Comments: arXiv:1604.08577v1 was split into arXiv:1604.08577v2 and this paper; published in Journal of Differential Equations

Other curvature flows Self-shrinkers & translators Weak & singular flows Homogeneous spaces 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) 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

In the guidebook

11A.13
Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles
A computable laboratory: the flow on Thurston’s geometries.
12C.9
After PerelmanBook 12C · Extinction and the Proof
Singular flows, the Smale conjecture, and Bamler’s theory.

[1] 11A.13
Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles
A computable laboratory: the flow on Thurston’s geometries.

[2] 12C.9
After PerelmanBook 12C · Extinction and the Proof
Singular flows, the Smale conjecture, and Bamler’s theory.

Existence of self-shrinkers to the degree-one curvature flow with a rotationally symmetric conical end

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