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arxiv: 2409.03098 · v2 · submitted 2024-09-04 · 🧮 math.DG · math.AP

Monotonicity of the modulus under curve shortening flow

Arjun Sobnack , Peter M. Topping This is my paper

Pith reviewed 2026-05-23 20:42 UTC · model grok-4.3

classification 🧮 math.DG math.AP
keywords curve shortening flowmodulus of annulusmonotonicitynested curvesconformal modulusgeometric flowsdifferential geometry
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The pith

The modulus of the annulus between two nested curves increases monotonically under curve shortening flow.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that when two disjoint nested embedded closed curves in the plane both evolve under curve shortening flow, the conformal modulus of the annulus they bound grows with time. The same monotonic increase holds for curves on any ambient surface whose curvature is bounded from below. The modulus supplies a scale-invariant measure of how the curves separate, and its monotonicity gives a controlled quantity along a flow that otherwise shrinks lengths. A reader cares because this yields a new monotonic functional for tracking the evolution of nested curves without relying on area or length alone.

Core claim

Given two disjoint nested embedded closed curves in the plane, both evolving under curve shortening flow, the modulus of the enclosed annulus is monotonically increasing in time. An analogous result holds within any ambient surface satisfying a lower curvature bound.

What carries the argument

The modulus of the annulus, the conformal invariant given by the extremal length of the family of curves connecting the two boundary components.

If this is right

  • The conformal separation between the curves cannot decrease while they remain nested and disjoint.
  • The modulus supplies a non-decreasing quantity that controls the geometry of the annulus throughout the evolution.
  • The result extends immediately to surfaces with curvature bounded from below, including the sphere.
  • Any quantity derived from the modulus inherits the same monotonicity along the flow.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The monotonicity might be combined with length or area decrease to obtain upper bounds on the time until the curves collide or shrink to points.
  • Similar monotonicity statements could be tested for other curvature flows such as mean curvature flow for surfaces.
  • The result suggests examining whether the modulus remains monotone when the curves are allowed to touch or when the ambient curvature bound is removed.

Load-bearing premise

The two curves remain disjoint, embedded, closed, and nested for the entire duration of the flow.

What would settle it

An explicit calculation for two concentric circles shrinking by curve shortening flow in which the modulus log(R/r) decreases at some positive time would disprove the claim.

Figures

Figures reproduced from arXiv: 2409.03098 by Arjun Sobnack, Peter M. Topping.

Figure 1
Figure 1. Figure 1: Grim reaper unit speed in a given direction yields a solution to curve shortening flow, mod￾ulo reparametrisation. The standard approach to finding this curve is to write it as a graph of a function x = ϕ(y) and derive an ODE for ϕ that expresses that it translates in the x direction under curve shortening flow. Standard ODE methods then yield that ϕ(y) = − log cos y for y ∈ (− π 2 , π 2 ). In this paper w… view at source ↗
read the original abstract

Given two disjoint nested embedded closed curves in the plane, both evolving under curve shortening flow, we show that the modulus of the enclosed annulus is monotonically increasing in time. An analogous result holds within any ambient surface satisfying a lower curvature bound.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proves that if two disjoint nested embedded closed curves in the plane evolve under curve shortening flow, then the conformal modulus of the annulus they bound is monotonically non-decreasing in time. An analogous monotonicity statement is established when the ambient surface has a lower bound on its Gaussian curvature.

Significance. If correct, the result supplies a new monotone quantity for the curve shortening flow on annular regions. Such quantities are useful for controlling the geometry of the flow and for analyzing long-time behavior or extinction. The extension to surfaces with curvature bounds broadens the applicability beyond the Euclidean plane.

major comments (2)
  1. [§1, Theorem 1.1] §1 (Introduction) and Theorem 1.1: the main theorem is stated under the standing assumption that the two curves remain disjoint, embedded, closed, and nested for the entire existence interval of the flow. The avoidance principle for CSF is standard, but the preservation of nestedness (inner curve stays inside the bounded component of the outer curve) requires a separate argument via the parabolic maximum principle or comparison of enclosed regions; no such argument or reference is supplied in the text, making the geometric hypothesis load-bearing for the claim that the modulus remains defined and the derivative is controlled.
  2. [§3] §3 (Evolution of the modulus): the derivation of dM/dt ≥ 0 relies on an integral identity involving the curvature and the conformal factor. The sign of the resulting expression is asserted to be non-negative, but the step that converts the boundary integrals into a non-negative quantity (presumably via integration by parts or the curvature bound) is only sketched; an explicit verification that no boundary terms can change sign when the ambient curvature is bounded below would strengthen the argument.
minor comments (2)
  1. Notation for the modulus M (or m) is introduced without an explicit formula or reference to the standard definition via extremal length; adding the precise definition (e.g., M = (1/(2π)) log(R/r) in the flat case) would improve readability.
  2. [Theorem 1.2] The statement of the result on surfaces with lower curvature bound should specify whether the bound is on Gaussian curvature or sectional curvature and whether it is strict or non-strict.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper accordingly to improve clarity.

read point-by-point responses

  1. Referee: [§1, Theorem 1.1] §1 (Introduction) and Theorem 1.1: the main theorem is stated under the standing assumption that the two curves remain disjoint, embedded, closed, and nested for the entire existence interval of the flow. The avoidance principle for CSF is standard, but the preservation of nestedness (inner curve stays inside the bounded component of the outer curve) requires a separate argument via the parabolic maximum principle or comparison of enclosed regions; no such argument or reference is supplied in the text, making the geometric hypothesis load-bearing for the claim that the modulus remains defined and the derivative is controlled.

    Authors: We agree that an explicit justification for the preservation of nestedness is needed to make the standing assumption fully rigorous. While disjointness follows from the standard avoidance principle for CSF, nestedness can be preserved by applying the parabolic maximum principle to a suitable distance function or by comparing the evolution of the enclosed regions. We will add a brief argument with a reference (e.g., to Grayson's work or standard comparison principles for CSF) in the revised introduction and Theorem 1.1 statement. revision: yes

  2. Referee: [§3] §3 (Evolution of the modulus): the derivation of dM/dt ≥ 0 relies on an integral identity involving the curvature and the conformal factor. The sign of the resulting expression is asserted to be non-negative, but the step that converts the boundary integrals into a non-negative quantity (presumably via integration by parts or the curvature bound) is only sketched; an explicit verification that no boundary terms can change sign when the ambient curvature is bounded below would strengthen the argument.

    Authors: We thank the referee for pointing this out. In the derivation, integration by parts produces boundary terms whose non-negativity follows from the lower bound on Gaussian curvature combined with the conformal factor. We will expand Section 3 with a fully explicit computation of these terms, verifying that the curvature bound ensures the expression remains non-negative in all cases, including when the bound is saturated. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper states a direct theorem: given two disjoint nested embedded closed curves evolving under curve shortening flow, the modulus of the enclosed annulus is monotonically increasing. The abstract and description present this as a proved result using standard geometric analysis tools (likely evolution equations or maximum principles on the modulus). No self-definitional reduction, fitted parameters renamed as predictions, or load-bearing self-citations appear. The preservation of disjointness/nestedness is a standard hypothesis maintained by avoidance principles in CSF, not a circular input. The result does not reduce to its own assumptions by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities can be extracted.

pith-pipeline@v0.9.0 · 5548 in / 965 out tokens · 23401 ms · 2026-05-23T20:42:50.680682+00:00 · methodology

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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    Gage and R

    M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves. J. Differential Geom. 23 (1986) 69–96. https://doi.org/10.4310/jdg/1214439902

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    M. A. Grayson, The heat equation shrinks embedded plane curves to round points. J. Differential Geom. 26 (1987) 285–314. https://doi.org/10.4310/jdg/1214441371

    work page doi:10.4310/jdg/1214441371 1987

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    M. A. Grayson, Shortening embedded curves. Annals of Math. 129 (1989), 71–111. https://doi.org/10.2307/1971486

    work page doi:10.2307/1971486 1989

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    Sobnack, ‘Geometric regularity properties of the cur ve shortening flow.’ PhD thesis

    A. Sobnack, ‘Geometric regularity properties of the cur ve shortening flow.’ PhD thesis. University of Warwick. September 2024

    work page 2024

  5. [5]

    Sobnack and P

    A. Sobnack and P. M. Topping, Delayed parabolic regularity for curve short- ening flow. Preprint (2024). arXiv:2408.04049

    work page arXiv 2024

  6. [6]

    P. M. Topping and H. Yin, Sharp Decay Estimates for the Logarithmic Fast Diffusion Equation and the Ricci Flow on Surfaces. Annals of PDE 3 (2017). https://doi.org/10.1007/s40818-017-0024-x 7 AS: arjun.sobnack@warwick.ac.uk PT: https://homepages.warwick.ac.uk/~maseq/ Mathematics Institute, University of W arwick, Coventry, C V4 7AL, UK. 8

    work page doi:10.1007/s40818-017-0024-x 2017