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Is Mean Curvature Flow a Gradient Flow?

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Zhonggan Huang

Differential Geometry · Submitted 4 Dec 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this paper we study the mean curvature flow on two nondegenerate metric spaces of simple closed plane curves: the uniformness-preserving metric structure proposed by Shi and Vorotnikov [11] and the curvature-weighted structure proposed by Michor and Mumford [8], and prove that the mean curvature flow is not a gradient flow in either of the spaces.

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arXiv ID2212.03701 (version 1)
Pith IDY2VZAY4C, Pith's permanent record of this paper and its review
CitationZhonggan Huang. "Is Mean Curvature Flow a Gradient Flow?." arXiv:2212.03701 (2022). https://pith.science/paper/Y2VZAY4C
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,741 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.66 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.81 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; .

It is well-known that the mean curvature flow is a formal gradient flow of the perimeter functional. However, by the work of Michor and Mumford [7,8], the formal Riemannian structure that is compatible with the gradient flow structure induces a degenerate metric on the space of hypersurfaces. It is then natural to ask whether there is a nondegenerate metric space of hypersurfaces, on which the mean curvature flow admits a gradient flow structure. In this paper we study the mean curvature flow on two nondegenerate metric spaces of simple closed plane curves: the uniformness-preserving metric structure proposed by Shi and Vorotnikov [11] and the curvature-weighted structure proposed by Michor and Mumford [8], and prove that the mean curvature flow is not a gradient flow in either of the spaces.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 It is well-known that the mean curvature flow is a formal gradient flow of the perimeter functional. background 1.00 0.03 0.02 asserted 1.00
s1 However, by the work of Michor and Mumford [7,8], the formal Riemannian structure that is compatible with the gradient flow structure induces a degenerate metric on the space of hypersurfaces. background 0.91 0.03 0.03 asserted 1.00
s2 It is then natural to ask whether there is a nondegenerate metric space of hypersurfaces, on which the mean curvature flow admits a gradient flow structure. gap 0.91 0.05 0.04 hedged possibility 0.39
s3 In this paper we study the mean curvature flow on two nondegenerate metric spaces of simple closed plane curves: the uniformness-preserving metric structure proposed by Shi and Vorotnikov [11] and the curvature-weighted structure proposed by Michor and Mumford [8], and prove that the mean curvature flow is not a gradient flow in either of the spaces. contribution 0.64 0.96 0.97 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

It is well-known that the mean curvature flow is a formal gradient flow of the perimeter functional. However, by the work of Michor and Mumford [7,8], the formal Riemannian structure that is compatible with the gradient flow structure induces a degenerate metric on the space of hypersurfaces. It is then natural to ask whether there is a nondegenerate metric space of hypersurfaces, on which the mean curvature flow admits a gradient flow structure. In this paper we study the mean curvature flow on two nondegenerate metric spaces of simple closed plane curves: the uniformness-preserving metric structure proposed by Shi and Vorotnikov [11] and the curvature-weighted structure proposed by Michor and Mumford [8], and prove that the mean curvature flow is not a gradient flow in either of the spaces.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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