
==== Front
J Geom Anal
J Geom Anal
Journal of Geometric Analysis
1050-6926
1559-002X
Springer US New York

1683
10.1007/s12220-024-01683-w
Article
Existence of Optimal Flat Ribbons
Blatt Simon 1
http://orcid.org/0000-0002-4908-196X
Raffaelli Matteo matteo.raffaelli@tuwien.ac.at

2
1 https://ror.org/05gs8cd61 grid.7039.d 0000 0001 1015 6330 Department of Mathematics, University of Salzburg, Hellbrunnerstraße 34, 5020 Salzburg, Austria
2 https://ror.org/04d836q62 grid.5329.d 0000 0004 1937 0669 Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstraße 8-10/104, 1040 Vienna, Austria
30 5 2024
30 5 2024
2024
34 8 25023 2 2024
24 4 2024
© The Author(s) 2024
https://creativecommons.org/licenses/by/4.0/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
We apply the direct method of the calculus of variations to show that any nonplanar Frenet curve in R3 can be extended to an infinitely narrow flat ribbon having minimal bending energy. We also show that, in general, minimizers are not free of planar points, yet such points must be isolated under the mild condition that the torsion does not vanish.

Keywords

Bending energy
Developable ribbon
Direct method in the calculus of variation
Gamma-convergence
Mathematics Subject Classification

Primary 49Q10, 53A05
Secondary 49J45, 74B20, 74K20
http://dx.doi.org/10.13039/501100002428 Austrian Science Fund F 77 Raffaelli Matteo TU Wien (TUW)Open access funding provided by TU Wien (TUW).

issue-copyright-statement© Mathematica Josephina, Inc. 2024
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pmcIntroduction and Main Result

In 1930, motivated by the problem of finding the equilibrium shape of a free-standing Möbius band, Sadowsky [13, 18] announced without proof that the bending energy ∫SH2dA of the envelope of the rectifying planes of a C3 unit-speed curve γ:[0,l]→R3 is given by1 w2∫0lκ2+τ22κ2dt;

here w is the width of the ribbon, which is measured in the normal plane of γ and assumed to be infinitely small, while κ>0 and τ are the curvature and torsion of γ, respectively. A proof of Sadowsky’s claim was given by Wunderlich in 1962 [21, 22].

Energy (1) defines a functional on the space of C3 curves γ in R3, a functional that has attracted a renewed wave of interest in the last twenty years; see, e.g., [1–4, 8, 10–12, 15, 19]. On the other hand, given a fixed curve γ, it is well known that if the curvature is nowhere zero, then there exist plenty of (infinitely narrow) flat ribbons along γ. It is therefore natural to interpret Sadowsky’s energy formula, or rather a suitable generalization thereof, as a functional on the set of all such ribbons.

An important first step in this direction was taken in [16], where the author extended Sadowsky’s formula to any flat ribbon along γ. Indeed, he showed that the bending energy, in the limit of infinitely small width, is given by2 w2∫Jκn2+τg22κn2dt,

where κn and τg are the normal curvature and the geodesic torsion of γ, respectively, and where J={t∈[0,l]∣κn(t)≠0}; see also [6, 7, 9].

In this context, a natural question arises: if γ is nonplanar, i.e., when the energy is bounded away from zero, does there exist an optimal flat ribbon along γ, that is, one having minimal bending energy? The purpose of this short note is to answer such question affirmatively when γ is a Frenet curve.

A preliminary step in our analysis consists in transforming (2) into a proper functional. To do so, it is enough to observe that (the normals of) any two flat ribbons along the same curve are related by a rotation θ:[0,l]→R about the common tangent. In particular, when the principal normal P=γ′′/κ is well-defined, the normal curvature and the geodesic torsion of γ with respect to the rotated normal P(θ) can be expressed byκn=κcos(θ),τg=θ′+τ.

Substituting these relations into (2), we thus obtain the functional3 E(θ)=∫0lκ2cos2θ+(θ′+τ)22κ2cos2θdt,

where the integrand is understood to be 0 (resp, +∞) at any point where both the numerator and denominator vanish (resp., the denominator vanishes but the numerator does not).

Our main result pertains the functional E and is contained in the following theorem.

Theorem 1.1

There is a minimizer θmin of E on W1,4([0,l]), i.e., we have E(θmin)≤E(θ)for allθ∈W1,4([0,l]).

For any a,b∈R, there is a minimizer of E on the subset {θ∈W1,4([0,l])∣θ(0)=aandθ(l)=b}

of W1,4([0,l]).

Remark 1.2

W1,4((0,l))⊂C0((0,l)) by Sobolev embedding theorem, and we interpret θ∈W1,4([0,l]) as the continuous extension of a function in W1,4((0,l)).

Remark 1.3

The theorem remains valid if one replaces κ∈C1([0,l],R>0) in (3) with any f∈C0([0,l]).

The proof of Theorem 1.1, which will be finalized in Sect. 4, is based on the direct method in the calculus of variation; see Sect. 5 for an alternative proof relying on Γ-convergence. It involves showing the coercivity and the weak sequential lower semicontinuity of E on W1,4([0,l]) or a suitable closed subset therein. As we explain below, each of these tasks presents some challenge.

First of all, our functional is not coercive on W1,4([0,l]), as it is 2π-periodic in θ. Consider, for example, the constant function θn=2πn: it defines an unbounded sequence in W1,4([0,l]), and yet E(θn)=∫0l(κ2+τ2)2κ2dt is (constant and) finite. On the other hand, we can use this periodicity to our advantage: since W1,4([0,l]) embeds into the Hölder space C0,34([0,l])—and hence also in C0([0,l])—the fact that E is 2π-periodic allows us to only consider functions satisfying θ(0)∈[0,2π]. In the next section we show that E is indeed coercive on the closed subsetV:={θ∈W1,4([0,l])∣θ(0)∈[0,2π]}

of W1,4([0,l]).

As for the sequential lower semicontinuity, the main issue is that our integrand is not continuous. To deal with this problem we use an approximation argument. It turns out, as shown in Sect. 3, that the sequential lower semicontinuity of E follows straightforwardly from that of the regular functionalEε(θ)=∫0lκ2cos2θ+(θ′+τ)22κ2cos2θ+ε2dt,

which for ε→0 approximates E monotonically from below.

We emphasize that, precisely because of this discontinuity, the classical indirect method of the calculus of variations does not seem readily applicable in our case. Indeed, to use the Euler–Lagrange equation (in the standard way) one would need to assume that θmin is free of singular points, i.e., that θmin(t)∉π/2+πZ for all t∈[0,l]. Our next result confirms that such assumption is, in general, invalid.

Theorem 1.4

Suppose that the torsion τ is a constant function satisfying|τ|>nπl+max|κ|for somen∈N0.

Then the minimizer θmin of E in W1,4([0,l]) has at least n singular points.

Thus, according to Theorem 1.4, one can enforce the presence of singular points. On the other hand, it turns out that the set of singular points is necessarily discrete when the torsion does not vanish.

Theorem 1.5

Let t0 be a singular point of θmin with τ(t0)≠0. Then t0 is an isolated singular point, i.e., there is an ε>0 such that the ε-neighborhood Iε:=(t0-ε,t0+ε)∩[0,l] around t0 does not contain any other singular point.

It is somewhat surprising that both Theorems 1.4 and 1.5 can be obtained, as we do in Sect. 6, on the basis of such elementary results as the fundamental theorem of calculus, the reverse triangle inequality, and Hölder’s inequality.

Coercivity

Once and for all, letΛ=max{‖τ‖L∞,‖κ‖L∞}.

The purpose of this section is to prove the following lemma.

Lemma 2.1

The functional E is coercive on the closed subsetV:={θ∈W1,4([0,l])∣θ(0)∈[0,2π]}

of W1,4([0,l]), i.e., we have for θ∈V that E(θ)→∞ if ‖θ‖W1,4→∞.

Proof

The strategy is to show that E→∞ if ‖θ′‖L4→∞, and that ‖θ′‖L4→∞ when ‖θ‖W1,4→∞.

First, note that4 Λ2E(θ)≥∫0l(θ′+τ)4cos2θdt.

Since |θ′+τ|≥|θ′|2 if |θ′|≥2‖τ‖L∞≤2Λ, equation (4) impliesΛ2E(θ)≥116∫|θ′|≥2Λ(θ′)4dt≥116∫0l(θ′)4dt-Λ4l=116‖θ′‖44-Λ4l,

and so if the homogeneous Sobolev norm ‖θ′‖L4 goes to infinity, then so does E.

Next, let x≠y∈[0,l]. Applying Hölder’s inequality, we obtain the following Morrey estimate:|θ(x)-θ(y)|=|∫[x,y]θ′dt|≤|x-y|34‖θ′‖L4([0,l]).

Together with θ(0)∈[0,l], this implies‖θ‖L4≤‖θ(·)-θ(0)‖L4+‖θ(0)‖L4≤l‖θ′‖L4+2πl14,

from which we conclude that‖θ‖W1,4≤(1+l)‖θ′‖L4+2πl14,

as desired. □

Weak Sequential Lower Semicontinuity

In this section we prove the sequential lower semicontinuity of E on W1,4([0,l]).

Lemma 3.1

The functional E is weakly sequentially lower semicontinuous, i.e., for any sequence of functions θn∈W1,4([0,l]) with weak limit θ∈W1,4([0,l]) we haveE(θ)≤lim infn→∞E(θn).

As already mentioned in the introduction, the plan is to consider for ε>0 the regular functionalEε(θ)=∫0lκ2cos2θ+(θ′+τ)22κ2cos2θ+ε2dt.

Note that, for fixed θ, the integrand is monotonically decreasing in ε. Using Beppo Levi’s monotone convergence theorem, we thus get thatE(θ)=limε↓0Eε(θ)=supε>0Eε(θ).

Now, if we knew that Eε was weakly sequentially lower semicontinuous, then the following well-known lemma would imply the same for E. It simply states the the supremum of any collection of lower semicontinuous functions is again lower semicontinuous.

Lemma 3.2

Let X be a topological space, and let Ai:X→[0,∞], i∈I be a family of lower semicontinuous functions. Then A:X→[0,∞] defined by A(x)=supi∈IAi(x) is lower semicontinuous.

Proof

We need to show that the set A-1((a,∞]) is open for all a∈[0,∞). First, using the fact that A=supi∈IAi, we getA-1((a,∞])=⋃i∈IAi-1((a,∞]).

Since Ai is lower semicontinuous for all i∈I, we observe that A-1((a,∞]) is the union of open sets, and so open itself. □

It remains to show the lower semicontinuity of Eε.

Lemma 3.3

For all ε>0, the functional Eε is weakly sequentially lower semicontinuous on W1,4([0,l]).

As the integrand is convex in θ′, to prove Lemma 3.3 it would be enough to invoke [20, Theorem 1.6]. We nevertheless include an independent proof—which combines Sobolev embeddings with Mazur’s lemma—for the benefit of the reader.

Proof of Lemma 3.3

Let θn converge weakly to θ in W1,4([0,l]). Since W1,4([0,l]) embeds compactly into C0([0,l]) and the image of any weakly convergent sequence under a compact embedding converges strongly, we deduce that θn and (κ2cos2θn+ε2)-1 converge uniformly to θ and (κ2cos2θ+ε2)-1, respectively. Moreover, exchanging θn by a suitable subsequence, we may assume that Eε(θn) converges to the limit inferior of the original sequence.

We now aim to get rid of θn in the expression of Eε(θn) and only keep θn′. To this end, we first rewrite Eε(θn) asEε(θn)=I1(θn)+I2(θn)+I3(θn)+I4(θn),

whereI1(θn)=∫0lκ2cos2θn+(θn′+τ)221κ2cos2θn+ε2-1κ2cos2θ+ε2dt,I2(θn)=∫0lκ4cos4θn-κ4cos4θκ2cos2θ+ε2dt,I3(θn)=∫0l2κ2cos2θn-κ2cos2θ(θn′+τ)2κ2cos2θ+ε2dt,I4(θn)=∫0lκ2cos2θ+(θn′+τ)22κ2cos2θ+ε2dt.

Since cosθn and (κ2cos2θn+ε2)-1 converge uniformly to cosθ and (κ2cos2θ+ε2)-1, respectively, and∫0lκ2cos2θn+(θn′+τ)22dt≤(Λ-2+ε2)-1supnEε(θn)<∞,

Hölder’s inequality implies that I1(θn), I2(θn), and I3(θn) converge to 0 as n tends to ∞; hence that I4(θn) converges to limn→∞Eε(θn).

To deal with the term I4 we use that the integrand is convex in θn′. Let m∈N. Mazur’s lemma [17, Theorem 3.13] tells us that there are convex combinations ϕn=∑i=mm+nλn,iθi of θm,⋯,θm+n that converge strongly to θ in W1,4([0,l]). Using the convexity of the integrand, we obtainsupi≥mI4(θi)≥∑i=mm+nλn,iI4(θi)≥I4(∑i=mm+nλn,iθi)=I4(ϕn).

As I4 is continuous on W1,4([0,l]) and the convex combinations ϕn converge to θ strongly, this yieldssupi≥mI4(θi)≥I4(θ).

Finally, taking the limit as m→∞, we deduce thatlimm→∞Eε(θm)=limm→∞I4(θm)≥I4(θ).

Hence E4 is weakly sequentially lower semicontinuous on W1,4([0,l]). □

Existence of Minimizers

We are now ready to prove our main result, Theorem 1.1 in the introduction.

Proof of Theorem 1.1

To begin with, letV′={θ∈W1,4([0,l])∣θ(0)=a-2π⌊a⌋andθ(l)=b-2π⌊a⌋},

where ⌊·⌋ denotes the floor function. We are going to show that E has a minimizer on both V and its subset V′, which is also closed and convex in W1,4([0,l]). This way, the statement will follow directly from the periodicity of E.

Let thus θn be a minimizing sequence for E on either V or V′. By Lemma 2.1, the sequence θn is bounded in W1,4([0,l]), and so, according to [14, Lemma 1.13.3], we can assume after passing to a subsequence that it converges weakly to a function θ0∈W1,4([0,l]); in particular, since any closed and convex subset of a Banach space is weakly closed, we deduce that the limit θ0 is contained in either V or V′.

Now, as E is weakly sequentially lower semicontinuos by Lemma 3.3, we haveinfθE(θ)=limn→∞E(θn)≥E(θ0)≥infθE(θ),

where the infimum is taken over V or V′. Hence these inequalities must be equalities, and so θ0 is a minimizer of E on V or V′. □

Γ-Convergence

Here we give an alternative proof of Theorem 1.1 based on the fundamental theorem of Γ-convergence [5, Theorem 7.8].

To begin with, note that Eε is coercive on V, as Eε>E; having already shown that it is weakly sequentially lower semicontinuous, we have the existence of minimizers for any ε>0.

Proposition 5.1

There is a minimizer of Eε on W1,4([0,l]).

For any a,b∈R, there is a minimizer of Eε on the subset Wab1,4([0,l])={θ∈W1,4([0,l])∣θ(0)=aandθ(l)=b}

of W1,4([0,l]).

To apply the fundamental theorem, we first show that Eε→ΓE weakly.

Proposition 5.2

The functionals Eε Γ-converge to E on W1,4([0,l]) with the weak topology as ε goes to 0.

Proof

To show the Γ-convergence of Eε, we have to prove a liminf and a limsup inequality.

For the liminf inequality, suppose that θε converges weakly to θ in W1,4([0,l]), and choose a null sequence εn such thatlimn→∞Eεn(θεn)=lim infε↓0Eε(θε).

Clearly, as the functionals Eεn are weakly sequentially lower semicontinuous,limn→∞Eεm(θεn)≥Eεm(θ)for allm∈N.

Letting m go to infinity and observing that the right-hand side converges to E(θ), we thus obtainlimn→∞Eεn(θεn)≥E(θ),

as desired.

As for the limsup inequality, for θ∈W1,4([0,l]) we simply take θε=θ for all ε>0 as recovery sequence. This we can do, because the monotonicity of the integrand implieslimε↓0Eε(θ)=E(θ)

via Beppo Levi’s monotone convergence theorem. □

Having shown that the approximating functionals Eε have a minimizer and Γ-converge to E, the missing ingredient needed to deduce the existence of a minimizer of E is equicoercivity, which we discuss below.

Definition 5.3

A family of functionals Fα:X→R, α∈I on a normed vector space X is said to be equicoercive if the set⋃α∈I{Fα≤t}

is bounded for all t∈R.

Lemma 5.4

The family of functionals Eε, 0<ε≤1 is equicoercive on V.

Proof

As Eε is pointwise nonincreasing in ε>0, we haveEε≥E1,

and so⋃ε∈(0,1]{Eε≤t}⊂{E1≤t}.

But the set on the right-hand side is bounded, as we know that E1 is coercive on W1,4([0,l]). □

Applying [5, Theorem 7.8], we finally get the existence of a minimizer of E.

Theorem 5.5

The infimum of E is attained and we haveminθE(θ)=limε↓0minθEε(θ),

where the minima are taken over W1,4([0,l]) or Wab1,4([0,l]).

To keep the exposition as self-contained as possible, we close this section by giving an independent proof of Theorem 5.5.

Proof of Theorem 5.5

From Eε≤E we immediately get the inequality5 infθE(θ)≥limε↓0minθEε(θ).

Moreover, as E(0)=∫0lκ2+τ2dt≤Λ2l+‖τ‖22 and Eε is monotonically decreasing in ε>0, we have the uniform boundminθEε(θ)≤Λ2l+‖τ‖22.

To show the existence of a minimizer of E, let εn>0 be a null sequence, and choose minimizers θεn∈V (resp., θεn∈V′) of Eεn. As Eεn(θεn)≤Λ2l+‖τ‖L22, Lemma 5.4 ensures that the sequence θεn is bounded in W1,4([0,l]). Hence, exactly as in the proof of Theorem 1.1, we can assume that it converges weakly to some θ0∈V (resp., θ0∈V′) in W1,4([0,l]).

Now, applying the liminf inequality, we obtaininfθE(θ)≤E(θ0)≤lim infn→∞Eεn(θεn)=limε↓0minθEε(θ),

where the infimum and minimum are taken over V (resp., V′). Together with (5), this implieslimε↓0minθEε(θ)≤infθE(θ)≤E(θ0)≤lim infn→∞Eεn(θεn)=limε↓0minθEε(θ),

and so this series of inequalities must hold with equality. Especially, we haveE(θ0)=minθE(θ)=limε↓0minθEε(θ),

and the theorem follows by periodicity. □

Number of Singular Points

Let t∈[0,l]. We say that t is a singular point of θ∈W1,4([0,l]) if θ(t)∈π2+πZ, i.e., if the denominator of our integrand vanishes. Clearly, when E(θ)<∞, singular points of θ correspond to planar points of the associated flat ribbon. The purpose of this section is to show that under certain assumptions, a minimizer has “many" singular points; besides, we will see that if τ(t)≠0, then t is at most an isolated singular point.

We begin with a lemma. To state it, let us introduce for any compact interval I⊂[0,l] the energyEI(θ)=∫Iκ2cos2θ+(θ′+τ)22κ2cos2θdt.

Lemma 6.1

Let a, b such that 0≤a<b≤l, and suppose that the torsion τ never vanishes in [a, b]. Then6 |θ(a)-θ(b)|≥A(b-a)-B12(b-a)34EI(θ)14,

whereA=mint∈[a,b]|τ(t)|

andB=maxt∈[a,b]|κ(t)cosθ(t)|.

Proof

First, an application of the fundamental theorem of calculus and the reverse triangle inequality yields|θ(a)-θ(b)|=|∫abθ′dt|=|∫ab-τ+(θ′+τ)dt|≥|∫abτdt|-∫ab|θ′+τ|dt.

Note that, as τ never vanishes in [0, l], it must have a sign there. Hence|∫abτdt|≥(b-a)mint∈[a,b]|τ(t)|=(b-a)A.

Moreover, by Hölder’s inequality,∫ab|θ′+τ|dt≤(b-a)34(∫ab|θ′+τ|4dt)14≤maxt∈[a,b]|κ(t)cosθ(t)|12(b-a)34(∫ab|θ′+τ|4κ2cos2θdt)14≤maxt∈[a,b]|κ(t)cosθ(t)|12(b-a)34EI(θ)14=B12(b-a)34EI(θ)14.

Summing up, we get|θ(a)-θ(b)|≥A(b-a)-B12(b-a)34EI(θ)14,

which is the desired conclusion. □

Applying Lemma 6.1 for a=0 and b=l, we can now deduce that minimizers of E are generally not free of singular points. In fact, by making sure that the right-hand side of (6) is large enough, one can enforce the presence of any given number of singular points—as explained by Theorem 1.4 in the introduction (reproduced below for the reader’s convenience).

Theorem 1.4

Suppose that the torsion τ is a constant function satisfying|τ|>nπl+max|κ|for somen∈N0.

Then the minimizer θmin of E in W1,4([0,l]) has at least n singular points.

Proof

Let K=maxt∈[0,l]|κ(t)|. Using the linear function θ0(t)=-∫0tτ(s)ds as competitor, we obtainE(θmin)≤E(θ0)=∫0lκ2cos2θdt≤K2l.

Now we apply Lemma 6.1 to the complete interval, i.e., for a=0 and b=l. Since A=mint∈[0,l]|τ(t)|=|τ| and B=maxt∈[0,l]|κ(t)cos(θ(t))|≤K, equation (6) yields|θmin(0)-θmin(l)|≥l|τ|-K12l34K12l14=l(|τ|-K),

which is larger than nπ if |τ|>nπl+K. □

Although, as we just saw, singularities abound, the following generalized version of Theorem 1.5 shows that they typically form a discrete set. The proof is again based on Lemma 6.1.

Theorem 6.2

Suppose that E(θ)<∞, and let t0 be a singular point of θ with τ(t0)≠0. Then t0 is an isolated singular point, i.e., there is an ε>0 such that the ε-neighborhood Iε:=(t0-ε,t0+ε)∩[0,l] around t0 does not contain any other singular point.

The simple heuristic behind the proof is that, in the neighborhood of a singular point, we must haveθ′+τ≈0,

as otherwise the energy cannot be bounded. So when τ does not vanish, the function θ must be strictly monotone.

Proof of Theorem 6.2

Let t0 be a singular point of θ, and choose ε>0 such that |θ(t)-θ(t0)|≤π for all t∈Iε. Noting that, by Hölder’s inequality,|θ(x)-θ(y)|≤|x-y|34‖θ′‖L4([x,y])≤|x-y|34Λ12E[x,y]14(θ),

we first obtain, using that cos is Lipschitz continuous with Lipschitz constant one,|cos(θ(t))|=|cos(θ(t))-cos(θ(t0))|≤|θ(t)-θ(t0)|≤|t-t0|34Λ12E[t0,t]14(θ)≤|t-t0|34Λ12E[t-ε,t+ε]14(θ).

Then, asB=maxx∈[t0,t]|κ(x)cos(θ(x))|≤|t-t0|34Λ12E14(θ),

an application of Lemma 6.1 with [a,b]=[t0,t] gives|θ(t)-θ(t0)|≥|t-t0|minx∈[t0-ε,t0+ε]∩[0,l]|τ(x)|-Λ14|t-t0|34+38E14+18(θ)=|t-t0|(minx∈[t0-ε,t0+ε]∩[0,l]|τ(x)|-Λ14|t-t0|18E38(θ))≥|t-t0|(minx∈[t0-ε,t0+ε]∩[0,l]|τ(x)|-Λ14ε18E38(θ)).

Finally, sinceminx∈[t0-ε,t0+ε]∩[0,l]|τ(x)|-Λ14ε18E38(θ)→|τ(t0)|>0asε→0,

it follows that there is an ε>0 such that|θ(t)-θ(t0)|>0for allt∈Iεwitht≠t0.

This shows that Iε does not contain any other singular point, as desired. □

Acknowledgements

We thank an anonymous referee for several corrections.

Funding

Open access funding provided by TU Wien (TUW).

Data availability

Not applicable.

The second-named author was supported by Austrian Science Fund (FWF) project F 77.

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