On knotted self expanders of the mean curvature flow in R³

Alexander Mramor; Shengwen Wang

Paper · Nov 2, 2026

On knotted self expanders of the mean curvature flow in $R^{3}$

Alexander Mramor, Shengwen Wang

math.DGarXiv:2611.02829

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Abstract

In this article we produce many examples of knotted self expanders. For example, there is an expander isotopic to a trefoil knot with three half cylinders attached.

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Submitted: Nov 2, 2026
Subjects: math.DG

Comments: Comments welcome!

Mean curvature flow Ricci solitons Self-shrinkers & translators

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

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Topics Self-shrinkers & translators · Shared terms “self expanders”
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Topics Self-shrinkers & translators · Shared terms “self expanders”
Poincaré inequality and topological rigidity of translators and self-expanders for the mean curvature flow
Debora Impera and Michele Rimoldi2023Wider flows

Topics Self-shrinkers & translators · Shared terms “self expanders”
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Topics Self-shrinkers & translators, Ricci solitons · Shared terms “self expanders”
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Topics Self-shrinkers & translators · Shared terms “self expanders”

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Background reading

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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)

In the guidebook

11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
12B.2
The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.
6A.10
Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

[1] 11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

[2] 12B.2
The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

[3] 6A.10
Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

On knotted self expanders of the mean curvature flow in $R^{3}$

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On knotted self expanders of the mean curvature flow in R3

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On knotted self expanders of the mean curvature flow in $R^{3}$