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REVIEW 2 major objections 5 minor 3 references

Non-uniqueness of geodesic limits and a question of Grayson and Gage

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read An immortal curve shortening flow on a sphere can have an entire one-parameter family of geodesics as its complete set of subsequential limits, answering Grayson and Gage's uniqueness question in the negative.

desk verdict A credible construction paper that plausibly answers Grayson-Gage negatively; the load-bearing Nash-Moser black box needs a referee check before full certification. read the letter →

arxiv 2608.04855 v1 pith:EEKKADQ5 submitted 2026-08-05 math.DG math.AP

Shrey Aryan , Tang-Kai Lee This is my paper · ORCID
classification math.DGmath.AP MSC 53C4453C22
keywords curveshorteningflowgeodesiclimitsnon-uniquenessGrayson–GagequestionNash–Moserinversefunctiontheoremalmostgeodesicsomega-limitsetFouriermodeanalysis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Curve shortening flow moves a curve on a surface in the direction of its curvature vector, and Grayson and Gage proved that a simple closed curve that flows for all time must accumulate, along some subsequence of times, on a closed geodesic; they asked whether that geodesic is forced to be unique. This paper answers the question in the negative. It constructs a smooth metric on the two-sphere, chosen from an arbitrarily small $C^\infty$ neighborhood of the round metric, and an immortal simple curve shortening flow whose subsequential limits are exactly a prescribed one-parameter family of distinct simple closed geodesics, all of the same length. The result shows that a smooth ambient metric does not by itself force a single asymptotic limit for this gradient-type flow, and that the Grayson–Gage convergence theorem cannot be upgraded to full convergence.

What carries the argument

The construction is carried by three mechanisms working in sequence. First, a Nash–Moser inverse function theorem from [AMN25] is applied to the length functional on a space of graphs over great circles, producing, for any prescribed mean-zero function $W$ on $\mathbb{RP}^2$ and any small $\epsilon>0$, a family of almost-geodesics $\Gamma_{\epsilon,v}$ whose lengths in the conformal metric $g_\epsilon=e^{2\rho_\epsilon}g_{\mathrm{rd}}$ are exactly $L_\epsilon+\epsilon W(v)$; the component of curvature tangent to the family equals $-\nabla_{q_\epsilon}(\epsilon W)$, and the transverse component is smaller by an additional factor of $\epsilon$ (Lemma 2.33). Second, a potential $W$ is engineered (Lemma 3.1) so that negative gradient trajectories of $\epsilon W$ with respect to metrics close to the round one spiral indefinitely around a prescribed critical circle $Z\subset\mathbb{RP}^2$, with speed decaying to zero while its logarithmic time-derivative stays arbitrarily small (Lemma 3.17). Third, every nearby curve is written as a graph $f=\psi_\epsilon(p)+u$ over a reference curve: the first two odd Fourier modes $p$ form a slow center variable that follows the spiral, while the remaining odd modes $u$ — the lowest of which is the third Fourier mode, with eigenvalue $-8$ for the round linearized operator $\partial_x^2+1$ — decay much faster (Lemma 4.28), so the full flow tracks the spiral and accumulates on exactly the geodesics indexed by $Z$. Grayson's uniform derivative decay ([Gra89, Theorem 7.2]) then promotes the finite-regularity limits to $C^\infty$ limits.

What would settle it

A direct numerical check: integrate the two-dimensional center-mode system (3.21) for the potential of Lemma 3.1 with a perturbation $e(t)$ of relative size $\eta$ as in (3.18), and test whether the tube $C_S$ is forward-invariant with the constants of Lemma 3.17; a single trajectory starting in $C_{\mathrm{in}}$ that exits through a lateral boundary $z=\pm h(s)/5$ would break the trapping argument. For the full geometric claim, run the curve shortening flow on the constructed metric starting from $\gamma_0$ and look for any subsequential limit curve at positive distance from the family $\{\Gamma(z)\}$; finding one would contradict the asserted equality $\omega_{C^\infty}(\gamma)=\{\Gamma(z)\}$.

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Extended reading notes

Core claim

The paper's central discovery is that the $\omega$-limit set of an immortal curve shortening flow on a closed surface can be a whole circle of geodesics rather than a single one. Theorem 5.1 states that for every $C^\infty$ neighborhood of the round metric on $\mathbb{S}^2$ there exist an antipodally invariant metric $g$ in that neighborhood, an embedded initial curve $\gamma_0$, and an embedded circle $Z\subset\mathbb{RP}^2$ with a smooth injective map $\Gamma$ from $Z$ into the space of smooth unparametrized embedded curves, such that every $\Gamma(z)$ is a simple closed $g$-geodesic, all the $\Gamma(z)$ have equal length, the curve shortening flow $\gamma(t)$ starting from $\gamma_0$ exists and remains embedded for all $t\geq 0$, and the $C^\infty$ $\omega$-limit set of $\gamma$ is exactly $\{\Gamma(z):z\in Z\}$. In particular, for every $z$ there is a sequence of times along which the flow converges smoothly to $\Gamma(z)$, and no curve outside the family is a subsequential limit. Because the metric can be taken arbitrarily close to the round metric, the non-uniqueness is a smooth phenomenon rather than a consequence of extreme geometry.

Load-bearing premise

Everything depends on the Nash–Moser inverse function theorem from [AMN25] still producing the prescribed nearly-geodesic family after the problem is restricted to antipodally symmetric data; if that restriction breaks the theorem's key technical identity, the family of almost-geodesics on which the whole construction rests does not exist.

Editorial extensions

If this is right

  • Grayson and Gage's convergence theorem is sharp: subsequential convergence to a closed geodesic is the best general statement available, and convergence to a unique geodesic cannot be asserted for arbitrary smooth metrics.
  • Non-uniqueness occurs in every $C^\infty$ neighborhood of the round metric, so positive curvature and near-round geometry do not restore uniqueness.
  • The limiting geodesics all have equal length and are parametrized by an embedded circle, so the $\omega$-limit set is uncountably large yet rigidly structured.
  • The monotonicity of Gage's set $G(g,t)$ of geodesics meeting the flow in at least four points cannot detect the phenomenon: in this example the evolving curve eventually meets each limiting geodesic in exactly two points, so $G(g,t)$ eventually contains none of them (Remark 5.6).
  • The construction runs inside antipodally invariant metrics, so even a large symmetry group of the ambient metric does not force a unique limit, in contrast with the round metric, where every immortal flow converges to a unique great circle.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The center-mode/stable-mode template is general: any flow admitting a finite-dimensional family of almost-stationary solutions with a spectral gap after removing finitely many modes, plus a gradient spiral in those modes, could in principle exhibit a circle-valued $\omega$-limit set — a pattern that may transfer to higher-order curve flows or to mean curvature flow near families of minimal hypersu
  • If the expected uniqueness for real-analytic metrics is correct, then the metrics constructed here cannot be chosen real-analytic; the smoothing machinery would be genuinely necessary rather than a convenience.
  • A quantitative signature of the construction is that length should approach the common geodesic length at a subexponential rate while the curve's position keeps rotating through the family — a rate pattern one could look for numerically as a diagnostic of non-uniqueness.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper constructs a smooth antipodally symmetric Riemannian metric g on S^2, a one-parameter family of distinct simple closed geodesics all of the same length, and an immortal embedded curve-shortening flow whose C^infinity omega-limit set is exactly that family. The proof has three parts: an almost-geodesic family with a prescribed length profile is built using the Ambrozio-Marques-Neves Nash-Moser inverse function theorem (Proposition 2.8); a potential W is chosen so that its negative gradient flow spirals into a prescribed curve Z (Section 3); and a spectral-gap/bootstrap analysis of the graph equation (4.13) shows that the stable Fourier modes decay while the center modes follow the spiraling gradient flow (Proposition 4.34). The final section upgrades finite-regularity convergence to smooth convergence via Grayson's derivative decay and proves the main theorem.

Significance. If valid, the result settles the Grayson-Gage uniqueness question in the negative, in a strong form: non-uniqueness occurs for metrics arbitrarily C^infinity-close to the round metric, and the entire omega-limit set is a circle of equal-length geodesics. The construction is detailed and the spectral-gap estimate (4.29)-(4.30) together with the bootstrap in Proposition 4.34 are explicit and checkable. The main risk is the applicability of the AMN Nash-Moser theorem to the antipodally projected operators, on which the existence of the almost-geodesic family rests. The argument does not assume its conclusion; the length profile W is chosen independently and the metric is built around it.

major comments (2)
  1. [Section 2, Proposition 2.8, equations (2.23)-(2.25)] The entire construction of the almost-geodesic family {Gamma_{epsilon,v}} with exact length profile L_epsilon + epsilon W(v) is obtained by applying [AMN25, Theorem 5.1 and Corollary 5.2] to the reduced map Lambda^+(z,tau)=Lambda(z)-tau(W,0) on the projected tame spaces X^+ x R and Y^+. The manuscript verifies the equivariance identity (2.20), the quadratic right-inverse identity (2.23), and the cancellation (2.24), and it asserts that the projected maps V^+ and Q^+ are smooth and tame because they are compositions of smooth tame maps with the tame projections (2.19). What is not shown is that the projected spaces X^+, Y^+ X^+ x R are tame spaces of the exact type required by [AMN25, Section 5.1], and that the tame estimates, derivative-gain orders, and smoothing operators for V and Q survive composition with Pi_X^+ and Pi_Y^+. Since Sections 3 through 5 and the main theorem rest entirely on Proposition 2.8, a failure of any of these hypotheses would invalidate the result. Please add a complete verification of every hypothesis of the AMN theorem as applied to the reduced map, or state and prove a variant that accepts the quadratic right-inverse identity in the projected categories.
  2. [Section 5, proof of Theorem 5.1, equations (5.3)-(5.5)] The upgrade from omega_{C^{m-1}}(gamma) to omega_{C^infinity}(gamma) rests on the invocation of [Gra89, Theorem 7.2] and on the claim that the decay of |nabla^r_s kappa| gives uniform bounds for every derivative of constant-speed parametrizations of gamma(t). The paper does not state the exact hypotheses of [Gra89, Theorem 7.2] or explain in detail how the decay of curvature derivatives along the geometric flow yields smooth convergence of unparametrized curves in the topology of E(S^2). This step is load-bearing because it converts the finite-regularity identification (5.3) into the C^infinity statement (5.5). Please provide the precise argument, including the role of the uniform length bounds and the diagonal Arzela-Ascoli application.
minor comments (5)
  1. [Lemma 3.1, equation (3.2)] The domain of Theta is written as 's > s_0 - 2' in (3.2) but as 's >= s_0 - 2' elsewhere in the proof; please unify the notation.
  2. [Lemma 3.1, estimate (3.8)] The verification of (3.8) for the function B(s) at k=0 is only summarized; the one-line computation using the decay of alpha(s) = exp(-e^{s/2}) would make the proof easier to follow.
  3. [Section 2, equation (2.17)] The identity C(-Psi circ bA) = C(Psi) is stated without derivation; since this is the first nontrivial use of the center-mode extraction, a short explanation would improve readability.
  4. [Title page] The word 'Grayson' is split across lines in the title; please fix the line break in the journal version.
  5. [Lemma 4.6, proof] The phrase 'by the C^1-stability of embeddings on compact sets' is acceptable but could be replaced by an explicit uniform inverse-function-theorem statement as epsilon varies; the current wording is slightly informal.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the construction freely prescribes W and builds the metric and geodesic family around it; the only load-bearing external input is the Nash-Moser theorem of [AMN25], cited as an external theorem and not assumed to contain the conclusion.

full rationale

The derivation is self-contained relative to the external Nash-Moser theory of Ambrozio-Marques-Neves. Proposition 2.8 starts from an arbitrary mean-zero W and solves the reduced equation bLambda+(z,tau)=Lambda(z)-tau(W,0)=0, using the quadratic right-inverse identity (2.25) derived from (2.15), (2.20), and (2.24). The output is the family {Gamma_epsilon,v} with exact length profile L_epsilon+epsilon W(v) (2.10) and Lambda_2=0 (2.11); the target geodesics are produced by the construction, not presupposed. Lemma 2.33 then derives the curvature decomposition Y_epsilon=-grad_{q_epsilon}(epsilon W) and the epsilon-factor decay of R_epsilon from the variation of this exact length profile, so no fitted parameter is later relabeled as a prediction. Section 3 freely chooses W so that a robust negative-gradient trajectory spirals around a prescribed curve Z, and Section 4 proves that the center mode p(t) follows Y_epsilon while the stable modes decay by the spectral gap. Theorem 5.1 therefore follows from previously established estimates. The only author self-citation, [LZ26], appears in the introduction's survey list of finite-dimensional gradient systems and is not used anywhere in the proof; it is not load-bearing. The principal caveat is whether the projected operators V^+ and Q^+ satisfy all tameness and derivative-gain hypotheses assumed in [AMN25, Theorem 5.1]; the paper gives the algebraic identities but the full verification is plausibly implicit. Even if that external applicability failed, the defect would be an unverified external hypothesis, not circular reasoning in which a conclusion is assumed as an input.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central construction is a proof, so the ledger contains no fitted parameters and no invented physical entities. The load-bearing external inputs are the AMN Nash-Moser theorem and Grayson's curvature decay theorem, both cited as axioms rather than re-derived. The constants in Section 3 are existential choices made to satisfy inequalities and do not function as free parameters.

assumptions (3)
  • domain assumption The Nash-Moser inverse function theorem of Ambrozio-Marques-Neves ([AMN25], Theorems 5.1, 7.6, Corollaries 5.2, 8.9 and Proposition 2.4) applies to the functional Lambda and the symmetric tame spaces X^+, Y^+ defined in Section 2.
    Used to construct the almost-geodesic family with prescribed length profile and the estimates (2.12) and (2.35) in Proposition 2.8 and Lemma 2.33. This is external machinery, not proved in the paper.
  • domain assumption Grayson's theorem ([Gra89], Theorem 7.2 and Lemma 1.4) gives uniform decay to zero of all derivatives of geodesic curvature for an immortal simple closed curve shortening flow on a closed surface, and continuation of the flow as long as the curvature is bounded.
    Used in Theorem 5.1 to upgrade C^{m-1} convergence to C-infinity and in Proposition 4.34 for long-time existence.
  • standard math Standard short-time existence, uniqueness, and parabolic regularity for curve shortening flow, including Angenent's zero-counting principle ([Ang88], [Ang91]).
    Used throughout Section 4 and in Remark 5.6 to control intersections and to justify the continued existence of the flow.

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Pith. "Pith review of Non-uniqueness of geodesic limits and a question of Grayson and Gage." pith.science (2026). https://pith.science/paper/EEKKADQ5

@misc{pith2026260804855,
  author       = {Pith},
  title        = {Pith review of: Non-uniqueness of geodesic limits and a question of Grayson and Gage},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EEKKADQ5}},
  note         = {Machine review of arXiv:2608.04855}
}
abstract

Grayson and Gage proved that an immortal curve shortening flow of simple closed curves on a closed surface converges subsequentially to a closed geodesic, and they asked whether this limiting geodesic is unique. We answer this question negatively by constructing a smooth Riemannian metric on $\mathbb{S}^2$ and an immortal simple closed curve shortening flow that converges along different sequences of times to every geodesic in a one-parameter family of distinct simple closed geodesics.

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Works this paper leans on

3 extracted references · 1 canonical work pages

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