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On bounds of entropy and total curvature for ancient curve shortening flows

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Wei-Bo Su, Kai-Wei Zhao

Differential Geometry · Submitted 2 Oct 2024

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

The authors' main claim We show that these two hypotheses are equivalent for the class of ancient complete embedded smooth planar curve shortening flows, which are one-dimensional mean curvature flows.

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arXiv ID2410.01924 (version 1)
Pith IDZVMB5EJV, Pith's permanent record of this paper and its review
CitationKai-Wei Zhao, Wei-Bo Su. "On bounds of entropy and total curvature for ancient curve shortening flows." arXiv:2410.01924 (2024). https://pith.science/paper/ZVMB5EJV
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,933 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.22 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.16 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

Bounds of total curvature and entropy are two common conditions placed on mean curvature flows. We show that these two hypotheses are equivalent for the class of ancient complete embedded smooth planar curve shortening flows, which are one-dimensional mean curvature flows. As an application, we give a short proof of the uniqueness of tangent flow at infinity of an ancient smooth complete non-compact curve shortening flow with finite entropy embedded in $\mathbb{R}^2$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Bounds of total curvature and entropy are two common conditions placed on mean curvature flows. background 1.00 0.04 0.03 asserted 1.00
s1 We show that these two hypotheses are equivalent for the class of ancient complete embedded smooth planar curve shortening flows, which are one-dimensional mean curvature flows. result 0.91 0.96 0.92 asserted 1.00
s2 As an application, we give a short proof of the uniqueness of tangent flow at infinity of an ancient smooth complete non-compact curve shortening flow with finite entropy embedded in $\mathbb{R}^2$. result 0.63 0.95 0.15 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

Bounds of total curvature and entropy are two common conditions placed on mean curvature flows. We show that these two hypotheses are equivalent for the class of ancient complete embedded smooth planar curve shortening flows, which are one-dimensional mean curvature flows. As an application, we give a short proof of the uniqueness of tangent flow at infinity of an ancient smooth complete non-compact curve shortening flow with finite entropy embedded in $\mathbb{R}^2$.

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