Calculus of variations manuscript No. (will be inserted by the editor)

Friedemann Schuricht^{1} · Heiko von der Mosel^{2}

Characterization of ideal knots

Received: date / Revised version: date – \begin{matrix}c \\ ◯ \end{matrix} Springer-Verlag 2003

Abstract. We present a characterization of ideal knots, i.e., of closed knotted curves of prescribed thickness with minimal length, where we use the notion of global curvature for the definition of thickness. We show with variational methods that for an ideal knot γ, the normal vector γ′′(s) at a curve point γ(s) is given by the integral over all vectors γ(τ) − γ(s) against a Radon measure, where |γ(τ) − γ(s)|/2 realizes the given thickness. As geometric consequences we obtain in particular, that points without contact lie on straight segments of γ, and for points γ(s) with exactly one contact point γ(τ) we have that γ′′(s) points exactly into the direction of γ(τ) − γ(s). Moreover, isolated contact points lie on straight segments of γ, and curved arcs of γ consist of contact points only, all realizing the prescribed thickness with constant (maximal) global curvature.

Mathematics Subject Classification (2000): 53A04, 57M25, 74K05, 74M15, 92C40

1. Introduction

Consider a closed knotted curve γ : [0, 1] → ℝ^{3} such that the tubular neighbourhood with given radius θ > 0 does not intersect itself. Then we shrink the length L(γ) of the curve until we arrive at some minimal length L_{0} where the tube surrounding the shortened curve γ_{0}([0, 1]) forms a completely tight knot which cannot be shortened anymore. The middle curve γ_{0}([0, 1]) of such a tight tube is an ideal knot in the sense of [2], [9] and [11].

For a long time an accessible formulation of the problem was not available. The main difficulty was to find a suitable analytical condition preventing self-intersection of the tubular neighbourhood of the curve. In the work of Kusner and Sullivan [10] several notions of thickness based, e.g., on the critical self-distance introduced by Litherland et al. in [11], or on Gromov’s concept of distortion were investigated and related to each other. At present, however, it seems to be most convenient to use a lower bound

Mathematisches Institut, Universität zu Köln, Weyertal 86–90, D-50931 Köln,

Germany, e-mail: fs@math.uni-koeln.de

Mathematisches Institut, Universität Bonn, Beringstraße 1, D-53115 Bonn, Ger-

many, e-mail: heiko@math.uni-bonn.de

Send offprint requests to: Heiko von der Mosel

on the global radius of curvature to prescribe the thickness for a curve, a notion which was introduced by Gonzalez and Maddocks in [6] and further analyzed in [7] and [12]; see also [3]. In fact, a lower bound θ > 0 on the global radius of curvature means that any open ball of radius θ placed tangent at any point of the curve does not intersect the curve [7, Lemma 3 (i)], which corresponds to a given thickness θ of the curve.

Numerical simulations based on various notions of thickness indicated some interesting features of possible ideal configurations; see, e.g., the various contributions in [15]. However, there was no satisfactory interpretation of these numerical results, since

(a) the existence of ideal knots was not verified yet, and

(b) there was no equation (or any other analytical or geometric characterization) of ideal knots, which could be checked to verify that the numerically computed configurations are indeed close to being ideal.

Meanwhile the existence of ideal knots for each knot type was shown independently in [3],[7] and [8]. Furthermore it turned out that the minimizing curves belong to C^{1,1}. The present paper provides necessary conditions for optimality as a characterization of ideal knots and confirms a number of conjectures found in the literature (cf. [10],[3],[16]). In [6] such conditions are studied under the assumption that ideal knots are C^{2}-curves, which we believe cannot be expected in general. In fact, there are examples of ideal links, exhibited by Cantarella et al. in [3], which are only C^{1,1}, and there seems to be strong numerical evidence that one can also find ideal knots which are not C^{2}-smooth; see [16]. In the present work we show that, for a.e. s ∈ [0, 1], the curvature vector γ′′(s) of any ideal knot γ belongs to the convex cone spanned by the vectors γ(τ) − γ(s), where γ(τ) realizes the contact of the surrounding tubular neighbourhood. In fact, we prove an integral representation formula for γ′′(s) in terms of the contact chords γ(τ)−γ(s); see Theorem 1 in Section 3. This result implies several geometric consequences for the shape of an ideal knot. Parts of the ideal configuration where the tubular neighbourhood does not exhibit self-contact have to be straight (which may be considered as a special case of the cone condition for γ′′ with a trivial cone). Consequently, there has to be self-contact on curved parts of the tubular neighbourhood which implies a constant global curvature function there. These results are derived under the assumption that the local curvature of the ideal knot is strictly less than its global curvature, and it seems that this condition is not violated for a large class of ideal knots. Let us finally mention that our methods apply also to ideal links (i.e., several closed knotted curves that are linked with each other) and provide analogous results in this more general situation.

The proof essentially adapts the variational methods developed in [13], where the Euler-Lagrange equations are derived for elastic rods with selfcontact. There, a similar variational problem involving an upper bound on global curvature as a nonsmooth side condition was treated by a Lagrange multiplier rule of Clarke’s nonsmooth calculus of generalized gradients (cf.

[4]). Furthermore the concept of framed curves, i.e., curves associated with an orthonormal frame at each point, played an important role. The key idea to transfer these methods to the present setting of ideal knots is that C^{1,1}-curves can be represented as framed curves. This way the freedom of different parameterizations of a curve is removed in a natural way, thus avoiding unpleasant expressions for derivatives and artificial singularities. Speaking in mechanical terms we identify C^{1,1}-curves with a tubular elastic string which resists extension but without bending and torsional stiffness. From this point of view the above mentioned optimality conditions express the fact that the resultant self-contact force at a cross section is directed as the curvature vector of the middle curve, and that bending can only occur at points of self-contact.

Section 2 introduces the notion of global curvature and summarizes a number of properties needed for our analysis. In particular a restricted continuity result for “pointwise” global curvature is shown which ensures that the length parameters of the ideal shape corresponding to “contact” form a closed set. On the other hand, the “pointwise” global curvature function is not continuous in general: we construct an example of a C^{1,1}-curve with discontinuous “pointwise” global curvature in the Appendix. The precise definition of ideal knots is given in Section 3, which also contains the main results about their characterization. In Section 4 we discuss the representation of C^{1,1}-curves as framed curves, which is an essential ingredient in the subsequent proof. Section 5 provides the proof of the main results, which is carried out in several steps. The present paper is a slightly extended and modified version of [14].

Notation. We use x · y to denote the standard Euclidean inner product of x and y in ℝ^{3}, and x∧y for their cross product. |·| denotes the (intrinsic) distance between two points in ℝ^{3} or in some parameter set J ⊂ ℝ depending on the context. The space of continuous functions on the closure of the interval I = (0, 1) will be denoted by C^{0}(Ī), and C^{k,1}(Ī), k = 0, 1, 2, …, is the space of k-times continuously differentiable functions whose k-th derivative is Lipschitz continuous on Ī. L^{p}(I), 1 ≤ p ≤ ∞, stands for the Lebesgue space of p-integrable functions. For Sobolev spaces of functions, whose weak derivatives up to order m are p-integrable, we use the standard notation W^{m,p}(I). Notice the isomorphism C^{k,1}(Ī) ≅ W^{k+1,∞}(I). For general Banach spaces X with dual space X^{∗} we denote the duality pairing on X^{∗} ×X by ⟨., .⟩_{X^{∗}×X}.

2. Global curvature

We consider the set L of continuous and rectifiable closed curves γ : Ī → ℝ^{3}, I = (0, 1), with arc length parameterization Γ_{γ} : S_{L} → ℝ^{3} and length L = L(γ) := ∫_{I} |dγ| ≥ 0. The parameter set S_{L} is the circle with perimeter L which corresponds to the interval [0, L] with identified endpoints, i.e., S_{L} ≅ ℝ/(L · ℤ). To simplify notation, we mostly omit the subscript γ and agree that Γ, Γ_{k} correspond to γ, γ_{k} and so on. It is well-known that this arc length parameterization Γ is Lipschitz continuous, i.e. of class C^{0,1}([0, L], ℝ^{3}) ≅ W^{1,∞}([0, L], ℝ^{3}).

For a closed curve γ ∈ L of positive length L the global curvature κ_{G}[γ](s) at the point s ∈ S_{L} is defined as

κ_{G}[γ](s) := sup_{{σ,τ∈S_{L}\\{s\} \atop σ≠τ}} \frac{1}{R(Γ (s), Γ (σ), Γ (τ))} ,  (1)

where R(x, y, z) ≥ 0 is the radius of the smallest circle containing the points x, y, z ∈ ℝ^{3}. For collinear but pairwise distinct points x, y, z we set R(x, y, z) to be infinite. When x, y and z are non-collinear (and thus distinct) there is a unique circle passing through them with radius

R(x, y, z) = \frac{|x − y|}{|2 sin[<)(x − z, y − z)]|} = \frac{|x − y|}{2 |\frac{x−z}{|x−z|} ∧ \frac{y−z}{|y−z|} |} ,  (2)

where <)(ξ, η) ∈ [0, π] denotes the enclosed (smaller) angle between two nonzero vectors ξ, η ∈ ℝ^{3}. If two points coincide, however, say x = z or y = z, then there are many circles through x, y and z, and we take R(x, y, z) to be the smallest possible radius namely the distance |x − y|/2. We should point out that with this choice the function R(x, y, z) fails to be continuous at points where at least two of the arguments x, y, z, coincide. Notice nevertheless that, by definition, R(x, y, z) is symmetric in its arguments. In [12] it was shown that κ_{G}[γ](s) is always positive for closed curves. In the case of smooth curves that have no or only transversal crossings, our definition of κ_{G} agrees with the inverse of the global radius of curvature defined in [6], but it is different for curves with double covered regions; see [12] and the discussion in [7]. We define the global curvature of γ by

K[γ] := sup_{s∈S_{L}} κ_{G}[γ](s).  (3)

The curve γ is said to be simple if its arc length parameterization Γ : S_{L} → ℝ^{3} is injective. Otherwise there exist s, t ∈ S_{L} (s ≠ t) for which Γ (s) = Γ (t). Any such pair will be called a double point of Γ.

Proposition 1. [12, Theorem 1 (iii)] Let γ ∈ L with arc length parameterization Γ : S_{L} → ℝ^{3}. Then K[γ] < ∞ if and only if γ is simple and Γ ∈ C^{1,1}([0, L], ℝ^{3}). In particular, if K[γ] is finite, then

|Γ′(s_{1}) − Γ′(s_{2})| ≤ K[γ]|s_{1} − s_{2}| ∀s_{1}, s_{2} ∈ S_{L},  (4)

i.e., Γ′ has Lipschitz constant K[γ].

Remark. The necessary condition for K[γ] to be finite implies that the second derivative Γ′′(s) of Γ exists for a.e. s ∈ S_{L}, since C^{1,1}([0, L], ℝ^{3}) is isomorphic to the Sobolev space W^{2,∞}([0, L], ℝ^{3}). Thus (4) actually implies

∥Γ′′∥_{L^{∞}} ≤ K[γ] .  (5)

The next result (cf. [12, Theorem 2 (ii)]) clarifies how the supremum in (1) is realized on a closed curve. Later this will lead to the definition of generalized local curvature for rectifiable loops that are not necessarily differentiable, and also to an alternative characterization of global curvature in terms of the radius of a circle uniquely determined by two points and one tangent.

Proposition 2. Let γ ∈ L with K[γ] < ∞, and let s ∈ S_{L}. Then there exists a sequence (σ_{j}, τ_{j}) → (σ, τ) in S_{L} × S_{L} with s ≠ σ_{j} ≠ τ_{j} ≠ s for all j ∈ ℕ satisfying

κ_{G}[γ](s) = lim_{j→∞} \frac{1}{R(Γ (s), Γ (σ_{j}), Γ (τ_{j}))} ,  (6)

such that either (a) s = σ = τ, or (b) s ≠ σ = τ.

Remarks.

1. The different options in Proposition 2 can occur simultaneously, as can, e.g., be seen for the planar circle. Here, at every point the local curvature and the global curvature coincide, i.e., the different limit cases hold simultaneously at every point. Another interesting example is that of a stadium curve, which consists of two parallel straight line segments of equal length and distance d connected by two planar half circles of radius d/2. These curves have constant local curvature κ = 0 along the line segments and κ = 2/d along the half circles, i.e., there are curvature jumps. The global curvature K[γ], however, is equal to 2/d and K[γ] = κ_{G}[γ](s) for all s ∈ S_{L}. Stadium curves appear also in the examples of ideal C^{1,1}-links considered in [3].

2. The previous example has shown that a jump in local curvature must not necessarily be accompanied by a discontinuity of κ_{G}[γ](·). Nevertheless κ_{G}[γ](·) may fail to be continuous in the case where K[γ] < ∞. Indeed, in the Appendix we construct an example of a closed curve γ of class C^{1,1} with a discontinuity for κ_{G}[γ](.).

3. The statement of Proposition 2 says, roughly speaking, that the supremum in the definition of κ_{G}[γ](s) cannot exclusively be achieved by two distinct parameters σ, τ ∈ S_{L}. Moreover, due to the symmetry of R(., ., .) a third feasible option (c), namely s = σ ≠ τ, does not occur exclusively, that is, without option (b) at the same time. Note that (6) is not true for every sequence (σ_{j}, τ_{j}) → (σ, τ) due to the lack of continuity of R at coinciding arguments.

In case (a) of Proposition 2 the value κ_{G}[γ](s) expresses a local property of the curve at s ∈ S_{L}, which coincides with the classical local curvature of γ ∈ L at s if γ is smooth.^{1} For curves γ ∈ L which are only continuous and

1 This can easily be verified by expanding Γ about s when calculating the term R(Γ (s), Γ (σ_{j}), Γ (τ_{j})) in (6) in case (a) for j large; see also [12, Lemma 7].

rectifiable in general, this observation motivates the following definition of the generalized local curvature κ[γ](s) of γ ∈ L at s ∈ S_{L} as

κ[γ](s) := lim sup_{{(τ_{j} ,σ_{j})→(s,s) \atop s≠τ_{j} ≠σ_{j} ≠s}} \frac{1}{R(Γ (s), Γ (τ_{j}), Γ (σ_{j}))} .  (7)

Note that for γ ∈ L the generalized local curvature κ[γ] can take values in [0, ∞], and that

κ[γ](s) ≤ κ_{G}[γ](s) for all s ∈ S_{L}.  (8)

For curves γ ∈ L with a C^{1,1}-arc length parameterization Γ we easily deduce from [12, Proposition 1] the following relation between the generalized local curvature and the second derivative of Γ :

Proposition 3. Let Γ : S_{L} → ℝ^{3} be the arc length parameterization of a simple curve γ ∈ L with Γ ∈ C^{1,1}([0, L], ℝ^{3}). Then

(i) κ[γ](s) ≤ ∥Γ′′∥_{L^{∞}} for all s ∈ S_{L}, (ii) κ[γ](s) = |Γ′′(s)| for a.e. s ∈ S_{L}.

From (ii) and (5) we infer

∥κ[γ]∥_{L^{∞}} = ∥Γ′′∥_{L^{∞}} ≤ K[γ] for all γ ∈ L with K[γ] < ∞.  (9)

In the light of this inequality, we say that, for curves γ with K[γ] < ∞, the global curvature K[γ] is not attained locally if and only if

∥Γ′′∥_{L^{∞}} < K[γ].  (10)

Curves with this property are considered in Proposition 4 below. They play an essential role in [13], where the Euler-Lagrange equations for energy minimizing elastic rods are derived. In fact, it seems that this property is needed for the variational techniques we apply in the proof of Theorem 1.

For γ ∈ L with finite global curvature the alternative (b) in part (ii) of Proposition 2 expresses a nonlocal property of the curve. It motivates a different characterization of K[γ] which is analytically more tractable. Let x, y, z ∈ ℝ^{3} be such that the vectors x − y and z are linearly independent. By P we denote the plane spanned by x − y and z. Then there is a unique circle contained in P through x and y, and tangent to z in the point y. We denote the radius of that circle by r(x, y, z) and set r(x, y, z) := ∞, if x ≠ y, and x − y and z are collinear. Using elementary geometric arguments r can be computed as

r(x, y, z) = \frac{|x − y|}{2 |\frac{x−y}{|x−y|} ∧ \frac{z}{|z|} |} ,  (11)

which shows that r(x, y, z) is continuous on the set of triples (x, y, z) with the property, that x − y and z are linearly independent. But it fails to be continuous at points, where, e.g., x and y coincide. Recall that for curves γ with K[γ] < ∞ Proposition 1 says, that the corresponding arc length parameterization Γ possesses a Lipschitz continuous unit tangent field Γ′ on [0, L]. Hence, for every pair (s, σ) ∈ S_{L}×S_{L}, s ≠ σ, the radius r(Γ (s), Γ (σ), Γ′(σ)) is well defined, and we obtain the following identities for κ_{G} and K (cf. [12, Lemma 2]): Lemma 1. Let γ ∈ L be such that K[γ] < ∞, then at least one of the following statements (A),(B) is true:

κ_{G}[γ](s) = sup_{{σ∈S_{L} \atop σ≠s}} \frac{1}{r(Γ (s), Γ (σ), Γ′(σ))} ,  (A)

κ_{G}[γ](s) = sup_{{σ∈S_{L} \atop σ≠s}} \frac{1}{r(Γ (σ), Γ (s), Γ′(s))} = κ[γ](s).  (B)

If for s ∈ S_{L} part (b) of Proposition 2 holds, then alternative (A) is true. In addition,

K[γ] = sup_{{s,σ∈S_{L} \atop s≠σ}} \frac{1}{r(Γ (s), Γ (σ), Γ′(σ))} .  (12)

Because of the representation of K[γ] as a supremum in (12) the following set A[γ], where the global curvature K[γ] is attained, is of particular interest.

A[γ] := \{(s, σ) ∈ [0, L] × [0, L], σ ≤ s : K[γ] = \frac{1}{r(Γ (s), Γ (σ), Γ′(σ))} \}. (13)

Note that Γ (s) ≠ Γ (σ) for (s, σ) ∈ A[γ], K[γ] < ∞, since r(·, ·, ·) is not defined otherwise. The set A[γ] can be empty, e.g., in the case when γ parameterizes a regular ellipse, where the local curvature κ[γ] is maximal and equal to K[γ] at exactly the two vertices. Here K[γ] is attained exclusively locally. On the other hand, if γ describes a circle, one has A[γ] = [0, L] × [0, L]\{(s, σ) : s = σ}.

We conclude this section with a continuity property of the global curvature κ_{G}[γ](.) on the set of parameters with sufficiently large values for κ_{G}[γ](s) (notice that this condition is not met in the example presented in the Appendix and mentioned above in Remark 2 following Proposition 2). Proposition 4. Let γ ∈ L satisfy (10) with K[γ] < ∞. Then κ_{G}[γ](.) : S_{L} → (0, ∞) is continuous on the set

S^{∗} := {s ∈ S_{L} : κ_{G}[γ](s) > ∥Γ′′∥_{L^{∞}}}.

As a consequence we obtain:

Corollary 1. Let γ ∈ L satisfy (10) with K[γ] < ∞. Then the set

S_{ρ}^{∗} := {s ∈ S_{L} : κ_{G}[γ](s) ≥ ∥Γ′′∥_{L^{∞}} + ρ}

is closed for each ρ ∈ (0, K[γ]−∥Γ′′∥_{L^{∞}}] and κ_{G}[γ](.) is Lipschitz continuous on S_{ρ}^{∗} with a Lipschitz constant depending on ρ.

The proofs can be found in Section 5.

3. Ideal knots

An ideal knot in the sense of [2], [9] and [11] is a closed non-self-intersecting tube of fixed radius θ > 0 and prescribed knot type with a centreline γ of minimal length. To be more precise, an ideal knot is a solution of the variational problem

L(γ) = ∫_{I} |γ′(σ)| dσ → Min!, γ ∈ W^{1,q}, q ∈ (1, ∞),  (14)

subject to the conditions

γ(0) = γ(1), K[γ] ≤ θ^{−1} and γ(Ī) ≃ γ̃(Ī).^{2}  (15)

Here θ > 0 is a constant representing the prescribed thickness of the curves in competition. In fact, the upper bound on the global curvature corresponds to a lower bound on the global radius of curvature (cf. [12]), and it provides by [7, Lemma 3] or [12, Proposition 3] a geometrically exact model for the self-contact or excluded-volume constraint imposed on the curve by a tubular neighbourhood of fixed radius θ. The fixed curve γ̃ ∈ W^{1,q} ∩ L represents the prescribed knot or isotopy type and is assumed to satisfy K[γ̃] ≤ θ^{−1}.

The existence of a curve of class W^{1,q}(I, ℝ^{3}) solving the minimization problem (14),(15) is established in [7, Theorem 4] (see also [3]) for each knot type γ̃, and the corresponding arc length parameterization Γ belongs to C^{1,1}([0, L], ℝ^{3}) where L denotes the length of the minimizing curve. Using the simple reparameterization γ(τ) := Γ (L · τ) for τ ∈ Ī = [0, 1] we obtain the same curve parameterized on [0, 1] with constant speed L. Obviously γ is again a solution of (14),(15) and γ ∈ C^{1,1}([0, 1], ℝ^{3}). The following theorem is the main result of the present paper providing a necessary condition for such a solution, from which we can extract information regarding the shape of an ideal knot.

Theorem 1. Let γ ∈ C^{1,1}([0, 1], ℝ^{3}) be a solution of problem (14),(15) parameterized with constant speed such that the global curvature K[γ] is not attained locally. Then K[γ] = θ^{−1}, and for all s ∈ [0, 1] there exists a (nonnegative) Radon measure μ_{s} on [0, 1] supported on

I_{s} := {τ ∈ [0, 1] : (s, τ) ∈ A[γ] or (τ, s) ∈ A[γ]}

(i.e., μ_{s} is the zero measure if I_{s} = ∅) such that

γ′′(s) = ∫_{0}^{1}(γ(τ) − γ(s)) dμ_{s}(τ) for a.e. s ∈ [0, 1].  (16)

2 Two continuous closed curves K_{1}, K_{2} ⊂ ℝ^{3} are isotopic, denoted as K_{1} ≃ K_{2}, if there are open neighbourhoods N_{1} of K_{1}, N_{2} of K_{2}, and a continuous mapping Φ : N_{1} × [0, 1] → ℝ^{3} such that Φ(N_{1}, τ) is homeomorphic to N_{1} for all τ ∈ [0, 1], Φ(x, 0) = x for all x ∈ N_{1}, Φ(N_{1}, 1) = N_{2}, and Φ(K_{1}, 1) = K_{2}. Roughly speaking, two curves are in the same isotopy class if one can be continuously deformed onto the other.

Remarks.

1. The condition that K[γ] be not attained locally excludes the circle of radius θ, for which K[γ] is attained locally and globally at all points. This circle is the obvious ideal knot in the trivial isotopy class of unknots, as one can easily deduce from [7, Lemma 3 (i)].

2. A simple scaling argument (cf. Section 5) shows that K[γ] = θ^{−1} for any ideal knot, i.e., the bound on the global curvature K[γ] is automatically attained for an ideal configuration.

3. Notice that

γ′′(s) ∈ \overline{conv} \{ ρ (γ(τ) − γ(s))| ρ ≥ 0, τ ∈ I_{s}\} for a.e. s ∈ [0, 1]

by (16) and the definition of the integral. (Here, \overline{conv} A denotes the closed convex hull of the set A.)

4. The solutions of (14),(15) cannot expected to be unique, i.e., there might be local minimizers that are not global ones. In fact, Cantarella et al. constructed concrete examples of nonunique minimizing links [3]. In the light of this mountain pass arguments suggest the existence of certain (mechanically unstable) “saddle points” (critical points with respect to the constraints) which satisfy equation (16). Thus it seems reasonable to extend the definition of ideal knots to stationary points of the variational problem (14),(15), i.e., to solutions of (16).

Let us define the set of contact parameters

I_{c} := {s ∈ [0, 1] : I_{s} ≠ ∅}.

Then the following consequences of the previous theorem provide fundamental properties characterizing ideal knots.

Corollary 2. Let the assumptions of the previous theorem be satisfied. Then:

(i) I_{c} is closed. (ii) If κ_{G}[γ](s) < θ^{−1} (i.e., s ∉ I_{c}), then there is δ(s) > 0 such that

γ′′(τ) = 0 for a.e. τ ∈ [0, 1] with |τ − s| < δ(s), i.e., γ is straight on a neighbourhood of s.

(iii) Let I^{∗} ⊂ [0, 1] be an open interval and let I_{c}∩I^{∗} have (Lebesgue) measure

zero. Then γ is straight on I^{∗}. (iv) Let γ be curved on an open interval Ĩ ⊂ [0, 1], i.e., γ′′ ≠ 0 a.e. on Ĩ,

then Ĩ ⊂ I_{c} and κ_{G}[γ](s) = θ^{−1} on Ĩ.

Remarks.

1. If s_{0} is an isolated point in I_{c} (or also an accumulation point of isolated points s_{n} in I_{c}), then γ has to be straight on a neighbourhood of s_{0} by (iii).

2. Notice that the previous results can be transferred to ideal links, i.e., ideal configurations composed of several closed knotted and linked curves having minimal length under a uniform upper bound on the global curvature.

4. Framed curves

In this section we provide a special representation of curves γ ∈ C^{1,1}([0, 1], ℝ^{3}) having constant speed parameterization as a key ingredient for the proof of Theorem 1. This allows us to apply the methods developed in [13].

We consider pairs (γ, D), called framed curves, where γ : Ī → ℝ^{3} is a curve of class W^{2,∞}(I, ℝ^{3}) parameterized by constant speed equipped with a frame field D : Ī → SO(3) of class W^{1,∞}(I, ℝ^{3}). Notice that D(s) = (d_{1}(s)|d_{2}(s)|d_{3}(s)) consists of three orthonormal column-vectors d_{i}(s) (i = 1, 2, 3) for each s ∈ Ī = [0, 1]. Thus a framed curve can be considered as a curve γ with an orthonormal frame attached to each point γ(s). Let us mention that our smoothness assumptions for (γ, D) constitute a proper subset of more general framed curves treated, e.g., in [7] and [12]. We call (γ, D) a closed framed curve if

γ(0) = γ(1) and d_{3}(0) = d_{3}(1).  (17)

Proposition 5 below states that a framed curve (γ, D) ∈ W^{2,∞}(I, ℝ^{3}) × W^{1,∞}(I, ℝ^{3×3}) may be uniquely determined from shape and placement variables

w = (u, v, γ_{0}, D_{0}) ∈ X_{0} := L^{∞}(I, ℝ^{3}) × (0, ∞) × ℝ^{3} × SO(3)

with u = (u^{1}, u^{2}, u^{3}) via the equations

d′_{k}(s) = [ ∑_{i=1}^{3} u^{i}(s)d_{i}(s)] ∧ d_{k}(s) for a.e. s ∈ I, k = 1, 2, 3,
γ′(s) = vd_{3}(s) for a.e. s ∈ I,
γ(0) = γ_{0}, D(0) = D_{0}.

(18)

Notice that X_{0} is a proper subset of the corresponding Banach space

X := L^{∞}(I, ℝ^{3}) × ℝ × ℝ^{3} × ℝ^{3×3} .

The triples (u^{1}, u^{2}, u^{3}) and (0, 0, v) may be identified as the coordinates, in the moving frame {d_{i}}, of the Darboux vector for the frame field D(s) and the tangent vector γ′(s), respectively. Notice that u ∈ L^{∞}(I, ℝ^{3}) and v ∈ (0, ∞) describe the shape of a framed curve, whereas γ_{0} and D_{0} describe its spatial placement. The constant v > 0 in particular is the speed of the parameterization of γ.

The following result is a special case of [7, Lemma 6].

Proposition 5. To each framed curve (γ, D) ∈ W^{2,∞} × W^{1,∞} we can associate a unique w = w(γ, D) ∈ X_{0} determined by (18). Conversely, to each w ∈ X_{0} we can associate a unique framed curve (γ, D) = (γ[w], D[w]) ∈ W^{2,∞} × W^{1,∞} such that (18) holds.

As already mentioned our aim is to apply the variational techniques for framed curves developed in [13] in the proof of our main results. Here the key observation is that for each curve γ ∈ W^{2,∞} having a constant speed parameterization we can find a frame field D ∈ W^{1,∞} such that (γ, D) is a framed curve.

Proposition 6. For each curve γ ∈ W^{2,∞}(I, ℝ^{3}) with |γ′(σ)| = v > 0 there are u^{1}, u^{2} ∈ L^{∞}(I), γ_{0} ∈ ℝ^{3}, and D_{0} ∈ SO(3), such that for any u^{3} ∈ L^{∞}(I) equations (18) provide a frame field D : Ī → SO(3) of class W^{1,∞}(I, ℝ^{3×3}) such that (γ, D) is a framed curve.

The proof is deferred to Section 5.

Remarks.

1. The preceding proposition allows us to represent any curve of class W^{2,∞}(I, ℝ^{3}) by an element w ∈ X_{0}. Thus, instead of considering perturbations of an ideal knot in W^{2,∞} we can study corresponding variations in X_{0}. This has the essential advantage that the freedom of different parameterizations of the competing curves is removed.

2. Below we assign the length of γ to framed curves (γ, D) as some kind of energy. For a mechanical interpretation one may view these objects as unshearable elastic strings which are able to develop tensile stresses but without resistance towards bending and torsion.

The previous investigations enable us to reformulate the variational problem stated in Section 3 in terms of w ∈ X_{0}. This way we restrict our attention to constant speed parameterizations on the fixed parameter interval Ī = [0, 1] representing curves in W^{2,∞}. To each element w = (u, v, γ_{0}, D_{0}) in X_{0} = L^{∞}(I, ℝ^{3}) × (0, ∞) × ℝ^{3} × SO(3) we assign the “energy”

E(w) := ∫_{0}^{1} vdσ = v = L(γ).  (19)

Then we seek minimizers of the problem

E(w) → Min! , w ∈ X_{0},  (20)
γ[w](1) = γ_{0},  (21)
d_{3}[w](1) = d_{3}[w](0),  (22)
K[w] ≤ θ^{−1},  (23)
γ[w](Ī) ≃ γ̃(Ī).  (24)

Here and from now on we use the short notation K[w], A[w] etc. for K[γ[w]], A[γ[w]], etc. Note that (22) is automatically satisfied for a constant speed ideal knot, since γ′ is continuous and γ′ = vd_{3}. Consequently, if γ is a (reparameterized) solution of (14), (15) and w = w[γ] ∈ X_{0} is the element assigned by means of Proposition 6 (with, e.g., u^{3} ≡ 0), then w solves the variational problem (20)–(24).

5. Proofs

5.1. Proofs for Section 2

Proof of Proposition 4. Let s ∈ S^{∗}. According to Proposition 2 we can find a sequence (σ_{j}, τ_{j}) → (σ, τ) in S_{L} × S_{L} with s ≠ σ_{j} ≠ τ_{j} ≠ s for all j ∈ ℕ with

κ_{G}[γ](s) = lim_{j→∞} \frac{1}{R(Γ (s), Γ (σ_{j}), Γ (τ_{j}))} ,

such that either (a) s = σ = τ, or (b) s ≠ σ = τ.

Case (a) is not possible since by (7), Proposition 3 (i), and by our assumption that s ∈ S^{∗} we would obtain

κ_{G}[γ](s) ≤_{(7)} κ[γ](s) ≤_{Prop.3(i)} ∥Γ′′∥_{L^{∞}} <_{Def. of S^{∗}} κ_{G}[γ](s),

which is absurd. Hence we are in the situation (b) which allows us to go to the limit j → ∞ in the formula (2), since γ is simple and Γ ∈ C^{1,1}([0, L], ℝ^{3}) by Proposition 1 and K[γ] < ∞. We obtain

κ_{G}[γ](s) = \frac{1}{r(Γ (s), Γ (τ), Γ′(τ))} ,  (25)

where the function r(., ., .) is given by (11). By continuity of r(x, y, z) in the vicinity of non-collinear vectors x − y and z we can find for

ε ∈ (0, κ_{G}[γ](s) − ∥Γ′′∥_{L^{∞}})

some number δ = δ(s) > 0 such that for all s′, τ′ ∈ S_{L} with |s − s′| ≤ δ and |τ − τ′| ≤ δ

\frac{1}{r(Γ (s′), Γ (τ′), Γ′(τ′))} ≥ \frac{1}{r(Γ (s), Γ (τ), Γ′(τ))} − ε/2.

Consequently, by (1), and for |τ′′ − τ′| sufficiently small by (2) and (11)

κ_{G}[γ](s′) ≥_{(1)} \frac{1}{R(Γ (s′), Γ (τ′), Γ (τ′′))} ≥ \frac{1}{r(Γ (s′), Γ (τ′), Γ′(τ′))} − ε/2
≥ \frac{1}{r(Γ (s), Γ (τ), Γ′(τ))} − ε = κ_{G}[γ](s) − ε > ∥Γ′′∥_{L^{∞}}.

Hence s′ ∈ S^{∗} as well, and as before we obtain

κ_{G}[γ](s′) = \frac{1}{r(Γ (s′), Γ (τ̄), Γ′(τ̄))}  (26)

for some τ̄ ≠ s′. Therefore, by Lemma 1

κ_{G}[γ](s) ≥_{Lemma 1} \frac{1}{r(Γ (s), Γ (τ̄), Γ′(τ̄))}
= \frac{1}{r(Γ (s′), Γ (τ̄), Γ′(τ̄))} + \frac{1}{r(Γ (s), Γ (τ̄), Γ′(τ̄))}

−\frac{1}{r(Γ (s′), Γ (τ̄), Γ′(τ̄))}

= κ_{G}[γ](s′) + \frac{1}{r(Γ (s), Γ (τ̄), Γ′(τ̄))} − \frac{1}{r(Γ (s′), Γ (τ̄), Γ′(τ̄))} .

Hence we arrive at

κ_{G}[γ](s′) − κ_{G}[γ](s) ≤ \frac{1}{r(Γ (s′), Γ (τ̄), Γ′(τ̄))} − \frac{1}{r(Γ (s), Γ (τ̄), Γ′(τ̄))},

and exchanging the roles of s and s′ in the previous considerations also

κ_{G}[γ](s) − κ_{G}[γ](s′) ≤ \frac{1}{r(Γ (s), Γ (τ), Γ′(τ))} − \frac{1}{r(Γ (s′), Γ (τ), Γ′(τ))}.

Since the right-hand side of each inequality (for fixed τ, and τ̄) tends to zero as s′ → s we have proven the claim. ⊓ ⊔

Proof of Corollary 1. Let s_{i} → s_{0} as i → ∞, where s_{i} ∈ S_{ρ}^{∗} for all i ∈ ℕ. Then, since s_{i} ∈ S^{∗} for all i ∈ ℕ, where S^{∗} is defined in Proposition 4, we obtain as in (25)

κ_{G}[γ](s_{i}) = \frac{1}{r(Γ (s_{i}), Γ (τ_{i}), Γ′(τ_{i}))},  (27)

where s_{i} ≠ τ_{i} for all i ∈ ℕ. A straightforward argument involving Taylor’s expansion implies that for all i ∈ ℕ

|s_{i} − τ_{i}| > η := (1 − \frac{∥Γ′′∥_{L^{∞}}}{∥Γ′′∥_{L^{∞}} + ρ}) /∥Γ′′∥_{L^{∞}} ,  (28)

since ∥Γ′′∥_{L^{∞}} + ρ ≤ κ_{G}[γ](s_{i}) (see [12, Lemma 8] for details). Without loss of generality we can assume that τ_{i} → τ_{0} ∈ S_{L}. Hence |s_{0} − τ_{0}| > η and Γ (s_{0}) ≠ Γ (τ_{0}). Using the continuity of the function r^{−1}(., ., .) (cf. (11)) we obtain for all i sufficiently large

\frac{1}{r(Γ (s_{0}), Γ (τ_{i}), Γ′(τ_{i}))} ≥ \frac{1}{r(Γ (s_{i}), Γ (τ_{i}), Γ′(τ_{i}))} − \frac{ρ}{4}.  (29)

Using (1) one deduces for τ′_{i} ∈ [0, 1] with |τ′_{i} − τ_{i}| sufficiently small by (2) and (11)

κ_{G}[γ](s_{0}) ≥ \frac{1}{R(Γ (s_{0}), Γ (τ_{i}), Γ (τ′_{i}))} ≥ \frac{1}{r(Γ (s_{0}), Γ (τ_{i}), Γ′(τ_{i}))} − \frac{ρ}{4}. (30)

Thus, by (27)–(30)

κ_{G}[γ](s_{0}) ≥ κ_{G}[γ](s_{i}) − ρ/2 ≥ ∥Γ′′∥_{L^{∞}} + ρ/2,

since s_{i} ∈ S_{ρ}^{∗}. Hence s_{0} ∈ S^{∗} defined in Proposition 4, i.e. s_{i} → s_{0} in S^{∗}. Now we can use the continuity of κ_{G}[γ](.) on S^{∗} to obtain

κ_{G}[γ](s_{0}) = lim_{i→∞} κ_{G}[γ](s_{i}) ≥ ∥Γ′′∥_{L^{∞}} + ρ ,

hence s_{0} ∈ S_{ρ}^{∗}. Applying estimate (28) to (s, τ) in (25) in the previous proof we deduce the Lipschitz continuity of κ_{G}[γ](.) on S_{ρ}^{∗} as claimed. ⊓ ⊔

5.2. Proofs for Section 3

We claim to prove Theorem 1. Similarly as in [13] we proceed in several steps first modifying and reducing the variational problem and then deriving variational formulas for the quantities involved. Throughout this section we assume that w = (u, v, γ_{0}, D_{0}) ∈ X_{0} is a solution of the variational problem (20)–(24) such that the global curvature K[w] is not attained locally.

Modified variational problem. We follow the approach in [13, Section 5] to linearize the space X_{0} by using the correspondence of small variations of the identity in SO(3) with small variations in ℝ^{3} near the origin by means of the rotation vector. To be more precise, small perturbations of D_{0} ∈ SO(3) have the form D_{0}U(α^{△}) with a continuous and invertible mapping U : B_{δ_{1}}(0) ⊂ ℝ^{3} → SO(3) for sufficiently small δ_{1} > 0, where U(0) = Id ∈ SO(3). We can identify each slightly perturbed configuration

(u+ u^{△}, v+ v^{△}, γ_{0}+ γ^{△}_{0}, D_{0} D^{△}) ∈ X_{0}

with an element

w^{△}= (u^{△}, v^{△}, γ^{△}_{0}, α^{△}) ∈ L^{∞}(I, ℝ^{3}) × ℝ × ℝ^{3} × ℝ^{3} =: Y  (31)

by D^{△}= U(α^{△}). With the norm

∥ w^{△} ∥_{Y} := ∥ u^{△} ∥_{L^{∞}} + | v^{△} | + | γ^{△}_{0} | + | α^{△} |  (32)

for w^{△}= (u^{△}, v^{△}, γ^{△}_{0}, α^{△}) ∈ Y we obtain (Y, ∥.∥_{Y}) as a Banach space, whereas the original set X_{0} is not a linear space.

For notational convenience we introduce the modified energy function

Ĕ(w^{△}) := E(v+ v^{△}) for w^{△}∈ B_{δ}(0) ⊂ Y,  (33)

where B_{δ}(0) is a small neighbourhood of 0 ∈ Y with δ > 0 not fixed yet but sufficiently small. Analogously, we define γ̆[w^{△}], D̆[w^{△}], K̆[w^{△}], etc. Note that γ̆[0] = γ[w], D̆[0] = D[w], etc.

Now we consider the modified variational problem

Ĕ(w^{△}) ⟶ Min!, w^{△}∈ Y,  (34)
γ̆[w^{△}](1) = γ_{0}+ γ^{△}_{0},  (35)
d̆_{3}[w^{△}](1) = d̆_{3}[w^{△}](0),  (36)
K̆[w^{△}] ≤ θ^{−1},  (37)
γ̆[w^{△}](Ī) ≃ γ̃(Ī).  (38)

Since w ∈ X_{0} was assumed to be a solution of (20)–(24),

w^{△}= 0 is a local minimizer of (34)–(38).  (39)

Reduction of the modified problem. Some of the constraints of the modified variational problem are redundant and need to be reformulated to obtain a clear-cut situation. Furthermore we will replace K[γ] by its square to obtain better regularity properties for the side condition (37).

We recall from [13, Lemma 7] that small variations

w^{△}_{1}:= (u^{△}, 0, γ^{△}_{0}, α^{△}) ∈ Y

satisfying (35) do not violate the constraint of a given knot class. Variations of the form w^{△}_{2}:= (0, v^{△}, 0, 0) ∈ Y on the other hand, correspond to merely changing the length of the curve within the same knot class. Since general variations w^{△}∈ Y may be decomposed as w^{△} = w^{△}_{1} + w^{△}_{2} we conclude that the topological condition of a prescribed knot class can be removed without affecting the fact that w^{△} = 0 is a local minimizer of the modified variational problem:

Lemma 2. Let w = (u, v, γ_{0}, D_{0}) ∈ X_{0} be a solution of the variational problem (20)–(24). Then γ̆[w^{△}] ≃ γ[w] for all ∥ w^{△} ∥_{Y} sufficiently small.

In order to replace (37) by an equivalent condition we introduce the functions

P [w^{△}](s, σ) := (γ̆[w^{△}](s), γ̆[w^{△}](σ), γ̆′[w^{△}](σ)),  (40)
H(x, y, t) := \frac{4|(x − y) ∧ t|^{2}}{|x − y|^{4}|t|^{2}} , for x, y, t ∈ ℝ^{3}, x ≠ y, t ≠ 0, (41)

and note that according to (11),(12) we may write

K̆[w^{△}]^{2} = sup_{{s,σ∈S_{L} \atop s≠σ}} H(P [w^{△}](s, σ)).  (42)

Now we can replace (37) with

g(w^{△}) := K̆[w^{△}]^{2} − θ^{−2} ≤ 0.  (43)

To remove the redundancy in the boundary condition (36) we are going to replace the three scalar conditions (36) by just two scalar equations; see (47), (48) below. (Note that d_{3}[w^{△}](s) is always a unit vector for each w^{△}∈ Y, thus it has only two degrees of freedom.)

This way we get the reduced variational problem

Ĕ(w^{△}) → Min! , w^{△}∈ Y,  (44)
g(w^{△}) ≤ 0,  (45)
g_{0}(w^{△}) := γ̆[w^{△}](1) − (γ_{0}+ γ^{△}_{0}) = 0,  (46)
g_{1}(w^{△}) := d̆_{3}[w^{△}](1) · (D_{0}U(α^{△}))_{1} = 0,  (47)
g_{2}(w^{△}) := d̆_{3}[w^{△}](1) · (D_{0}U(α^{△}))_{2} = 0,  (48)

where, for M ∈ ℝ^{3×3}, we denote the k-th column vector by (M)_{k}, k = 1, 2, 3. Lemma 3. The reduced variational problem (44)–(48) has a local minimizer

at w^{△}= 0.

Proof. Taking into account the previous arguments one merely needs to show that (47), (48) imply (36) for ∥ w^{△} ∥_{Y} sufficiently small. Note that d̆_{3}[w^{△}](1) is collinear with d̆_{3}[w^{△}](0) by (47), (48). [7, Lemma 8] implies that d̆_{3}[w^{△}](s) → d̆_{3}[0](s) for all s ∈ [0, 1] as ∥w^{△} ∥_{Y} → 0. By

d̆_{3}[0](1) = d_{3}[w](1) = d_{3}[w](0) = d̆_{3}[0](0)

(cf. (22)) we conclude that d̆_{3}[w^{△}](1) = d̆_{3}[w^{△}](0) for ∥w^{△} ∥_{Y} sufficiently small. ⊓ ⊔

Computation of derivatives. The mapping

(w^{△}, s) ↦ (γ̆[w^{△}](s), D̆[w^{△}](s))

from B_{δ}(0) × [0, 1] into ℝ^{3} × ℝ^{3×3} is of class C^{1}(B_{δ}(0) × [0, L], ℝ^{3} × ℝ^{3×3}) for some sufficiently small δ > 0 (depending on w), which can be shown as in [13, Lemma 9]. In [13, Lemma 16] we gave an explicit formula for the Fréchet derivative of the mapping w^{△} ↦ d̆_{k}[w^{△}](s), k = 1, 2, 3, at the origin 0 ∈ Y, which we denote by ∂_{w}d̆_{k}[0](s). (Note that d̆_{k}[w^{△}](s) is independent of variations v^{△} which are not considered in [13].) In fact, we have

∂_{w}d̆_{k}[0](s) w^{△} = z(s) ∧ d_{k}[w](s), k = 1, 2, 3, s ∈ [0, 1].  (49)

Here z = z[u^{△}, α^{△}] is a special characterization of elements u^{△}∈ L^{∞}([0, 1], ℝ^{3}) and α^{△}∈ ℝ^{3} by the uniquely assigned function

z(s) = z(0) + ∫_{0}^{s} ∑_{i=1}^{3} u^{△} ^{i}(τ)d_{i}[w](τ) dτ  (50)

with

z(0) ∧ d_{k}[w](0) = (D_{0}U′(0) α^{△})_{k}, k = 1, 2, 3,  (51)

where U′ denotes the derivative of U with respect to α at 0 ∈ ℝ^{3}, which is invertible as pointed out in [13, Appendix]. Note that z ∈ W^{1,∞}([0, 1], ℝ^{3}). In particular,

z(0) = 0 for w^{△}= (u^{△}, v^{△}, γ^{△}_{0}, 0) ∈ Y.  (52)

Thus, by (18),

∂_{w}γ̆[0](s) w^{△} = γ^{△}_{0} +v ∫_{0}^{s} z(τ) ∧ d_{3}[w](τ) dτ+ v^{△} ∫_{0}^{s} d_{3}[w](τ) dτ, (53)

for all s ∈ [0, 1], w^{△}= (u^{△}, v^{△}, γ^{△}_{0}, α^{△}) ∈ Y.

The energy functional Ĕ : B_{δ}(0) ⊂ Y ⟶ ℝ is obviously differentiable satisfying

Ĕ′(0) w^{△} = ∫_{0}^{1} v^{△} dτ = v^{△} for w^{△}∈ Y.  (54)

Similarly as in [13, Lemma 11] one verifies that the constraint functions g_{0}, g_{1}, and g_{2} are differentiable on B_{δ}(0) ⊂ Y for δ > 0 sufficiently small, and we obtain

g′_{0}(0) w^{△} = vz(0) ∧ ∫_{0}^{1} d_{3}[w](t) dt+ v^{△} ∫_{0}^{1} d_{3}[w](t) dt
+v ∫_{0}^{1} z′(t) ∧ ∫_{t}^{1} d_{3}[w](τ) dτ dt
=_{(21)} v ∫_{0}^{1} z′(t) ∧ ∫_{t}^{1} d_{3}[w](τ) dτ dt,  (55)
g′_{1}(0) w^{△} = ∫_{0}^{1} z′(t) · d_{02} dt,  (56)
g′_{2}(0) w^{△} = − ∫_{0}^{1} z′(t) · d_{01} dt,  (57)

where z ∈ W^{1,∞}([0, 1], ℝ^{3}) is given by (50), (51).

Notice that g is not smooth, but one can show (cf. [13, Lemma 14]) that g is Lipschitz continuous on B_{δ}(0) ⊂ Y for some δ > 0 sufficiently small. Thus we can compute its generalized gradient in the sense of Clarke [4]. As a preliminary tool we show

Lemma 4. Let w be a minimizing configuration for (20)–(24) such that K[w] is not attained locally. Then there are constants δ > 0 and η̃ ∈ (0, L/2π) (both depending on the minimizer w) such that

g(w^{△}) = max_{(s,σ)∈Q̃} H(P [w^{△}](s, σ)) − θ^{−2} for all w^{△}∈ B_{δ}(0) ⊂ Y,  (58)

where

Q̃ := {(s, σ) ∈ [0, 1] × [0, 1] : 1 − η̃ ≥ s − σ ≥ η̃}.  (59)

In particular, A[γ̆[w^{△}]] ⊂ Q̃ for all w^{△}∈ B_{δ}(0), where A[γ̆[w^{△}]] is the set defined in (13).

Proof. It suffices to prove the continuity of the mapping K̆[.] near the origin of Y, i.e.,

K̆[w^{△}] → K̆[0] = K[w] as ∥ w^{△} ∥_{Y} → 0.  (60)

Then we can argue exactly as in the proof of [13, Lemma 13] to prove the claim.

For the proof of (60) we notice that according to (40)–(42) we have

K̆[w^{△}] = v(v+ v^{△})^{−1}K̆[w^{△}_{0}],  (61)

where

w^{△}_{0}:= (u^{△}, 0, γ^{△}_{0}, α^{△}),  (62)

since d̆_{3}[w^{△}] = d̆_{3}[w^{△}_{0}] by (18). Variations of the special form (62) keep the length fixed according to (18), and [13, Lemma 2] implies that K̆[.] is continuous with respect to these particular variations. Hence we conclude with (61) that

|K̆[w^{△}] − K̆[0]| ≤ |K̆[w^{△}] − K̆[w^{△}_{0}]| + |K̆[w^{△}_{0}] − K̆[0]|
→ 0 as w^{△}→ 0,

because w^{△}→ 0 implies w^{△}_{0}→ 0 and v^{△}→ 0. ⊓ ⊔

Due to the characterization (58) of g we can apply the nonsmooth chain rule proved in [13, Appendix] to get an explicit formula for the generalized gradient ∂g(0).

Lemma 5. Let w be a minimizer of (20)–(24) such that K[w] is not attained locally. Then the function g as defined in (43) is Lipschitz continuous on B_{δ}(0) ⊂ Y for some small δ > 0 depending on w, i.e., the generalized gradient ∂g(0) ⊂ X^{∗} exists and is nonempty. Furthermore, for any g^{∗} ∈ ∂g(0) there is a Radon measure μ^{∗} on [0, 1] × [0, 1] with nonempty support on A[w] (see (13)), such that

⟨g^{∗}, w^{△}⟩_{Y^{∗}×Y} = − ∫_{0}^{1} z′(t) · v ∫_{t}^{1} d_{3}(τ) ∧ f_{c}^{∗}(τ) dτ dt
− z(0) · v ∫_{0}^{1} d_{3}(t) ∧ f_{c}^{∗}(t) dt  (63)

− v^{△} ∫_{0}^{1} d_{3}[w] · f_{c}^{∗}(t) dt,

where

f_{c}^{∗}(τ) := ∫_{Q_{τ}} (γ[w](s) − γ[w](σ)) dμ^{∗}(s, σ),  (64)
Q_{τ} := {(s, σ) ∈ [0, 1] × [0, 1] : σ ≤ τ ≤ s} for τ ∈ [0, 1].  (65)

Proof. We can proceed as in the proof of [13, Lemma 14], the only difference being the evaluation of the Fréchet derivative P_{w}[0](s, σ) of the function P defined in (40). In contrast to [13, (131)] we obtain by (49), (53)

P_{w}[0](s, σ) w^{△} = ( \begin{matrix}γ^{△}_{0} +v ∫_{0}^{s} z(t) ∧ d_{3}[w](t) dt+ v^{△} ∫_{0}^{s} d_{3}[w](t) dt \\ γ^{△}_{0} +v ∫_{0}^{σ} z(t) ∧ d_{3}[w](t) dt+ v^{△} ∫_{0}^{σ} d_{3}[w](t) dt \\ z(σ) ∧ d_{3}[w](σ) \end{matrix} ) .

Thus for any g^{∗} ∈ ∂g(0) there is a probability Radon measure μ̄ on Q̃ (cf. (59)) supported on A[w] such that

⟨g^{∗}, w^{△}⟩_{Y^{∗}×Y} = −K[w]^{4} ∫Q̃(γ[w](s) − γ[w](σ)) ·

(v ∫_{σ}^{s} z(t) ∧ d_{3}[w](t) dt+ v^{△} ∫_{σ}^{s} d_{3}[w](t) dt) dμ̄(s, σ)

for all w^{△}∈ Y. After extending the measure μ̄ from Q̃ to the triangle

Q̄ := {(s, σ) ∈ [0, 1] × [0, 1] : s ≥ σ} ⊃ Q̃

by zero (again denoted by μ̄), we use Fubini’s Theorem and integrate by parts to obtain for μ^{∗} := K[w]^{4}μ̄ and f_{c}^{∗} as in (64)

⟨g^{∗}, w^{△}⟩_{Y^{∗}×Y} = − ∫_{0}^{1} z′(t) · v ∫_{t}^{1} d_{3}[w](τ) ∧ f_{c}^{∗}(τ) dτ dt
−z(0) · v ∫_{0}^{1} d_{3}[w](t) ∧ f_{c}^{∗}(t) dt  (66)

− v^{△} ∫_{0}^{1} d_{3}[w](t) · f_{c}^{∗}(t) dt. ⊓ ⊔

Lagrange multiplier rule. By Lemma 3 we know that w^{△}= 0 is a local minimizer for the reduced variational problem (44)–(48). We are now in the position to apply the nonsmooth Lagrange multiplier rule of Clarke (cf. [4, Chapter 6]) to this variational problem, since the energy function Ĕ and the constraint functions g, g_{0}, g_{i}, i = 1, 2, are Lipschitz continuous near 0 ∈ Y. Hence there exist multipliers λ_{E}, λ ≥ 0, λ_{0} ∈ ℝ^{3}, λ_{1}, λ_{2} ∈ ℝ, not all zero, such that

0 ∈ λ_{E}Ĕ′(0) + λ∂g(0) + λ_{0} · g′_{0}(0) + ∑_{i=1}^{2} λ_{i}g′_{i}(0)  (67)

with

λg(0) = 0.  (68)

In other words, there exists g^{∗} ∈ ∂g(0) ⊂ Y^{∗}, such that

0 = ⟨λ_{E}Ĕ′(0) + λ_{0} · g′_{0}(0) + ∑_{i=1}^{2} λ_{i}g′_{i}(0), w^{△}⟩_{Y^{∗}×Y} + λ⟨g^{∗}, w^{△}⟩_{Y^{∗}×Y} (69)

for all w^{△}∈ Y.

Inserting the expressions (54)–(57) and (63) into (69) and setting m_{0} := λ_{1}d_{02} − λ_{2}d_{01} we thus arrive at

0 = λ_{E} v^{△} +v ∫_{0}^{1} z′(t) · (∫_{t}^{1} d_{3}[w](τ) ∧ λ_{0} dτ) dt
+ ∫_{0}^{1} z′(t) · m_{0} dt − λv ∫_{0}^{1} z′(t) · ∫_{t}^{1} d_{3}[w](τ) ∧ f_{c}^{∗}(τ) dτ dt (70)

−λ v^{△} ∫_{0}^{1} d_{3}[w](t) · f_{c}^{∗}(t) dt − λz(0) · v ∫_{0}^{1} d_{3}[w](t) ∧ f_{c}^{∗}(t) dt

for all w^{△}∈ Y. Notice that the equation is independent of the variations γ^{△}_{0} .

First, taking special variations of the form w^{△}= (0, v^{△}, 0, 0) ∈ Y we obtain from (61) that K̆[w^{△}] = v(v+ v^{△})^{−1}K̆[0] = v(v+ v^{△})^{−1}K[w]. Assume that K[w] < θ^{−1}. Then, for ∥ w^{△} ∥_{Y} sufficiently small, one has by the continuity property (60) that K̆[w^{△}] < θ^{−1}, and w^{△} satisfies all side conditions (45)–(48). For v^{△}< 0, however, Ĕ(w^{△}) = v+ v^{△}< v = Ĕ(0) = E(w) according to (19). Thus w = 0 could not have been a local minimizer. Hence K[w] = K[γ] = θ^{−1}.

Next we show that λ ≠ 0 in (70). Indeed, λ = 0 in (70) implies

0 = λ_{E} v^{△} +v ∫_{0}^{1} z′(t) · (∫_{t}^{1} d_{3}[w](τ) ∧ λ_{0} dτ) dt + ∫_{0}^{1} z′(t) · m_{0} dt.

The choice v^{△}≠ 0 and u^{△}= 0 giving z′ ≡ 0 by (50) leads to λ_{E} = 0. Arbitrary variations u^{△}∈ L^{∞}(I, ℝ^{3}) correspond to arbitrary z′ ∈ L^{∞}(I, ℝ^{3}) by (50). Hence we can apply the Fundamental Lemma of the calculus of variations to obtain

0 = v ∫_{t}^{1} d_{3}[w](τ) ∧ λ_{0} dτ + m_{0}  (71)

for a.e. t ∈ [0, 1], and then by continuity for all t ∈ [0, 1]. Differentiation with respect to t gives

d_{3}[w](t) ∧ λ_{0} = 0 for all t ∈ [0, 1],  (72)

again by continuity of d_{3}[w] ∈ C^{0,1}([0, 1], ℝ^{3}). Inserting (72) into (71) we arrive at m_{0} = 0, i.e., λ_{1} = λ_{2} = 0 by definition of m_{0}. If λ_{0} = 0 then all Lagrange multipliers in (67) would vanish contradicting the Lagrange multiplier rule, hence λ_{0} ≠ 0. But then by (72) the vectors d_{3}[w](t) = γ′[w](t) and λ_{0} would be collinear for all t ∈ [0, 1] contradicting the fact that γ[w] is a closed curve. Hence λ ≠ 0, i.e., λ > 0 (since λ ≥ 0 by the Lagrange multiplier rule).

Evaluating (70) for v^{△}= 0 and z(0) = 0, and setting f_{c}(t) := λvf_{c}^{∗}(t) for t ∈ [0, 1] and f_{0} := vλ_{0} we obtain

0 = ∫_{0}^{1} z′(t) · (∫_{t}^{1} d_{3}[w](τ) ∧ [f_{0} − f_{c}(τ)] dτ + m_{0}) dt,

which implies by the Fundamental Lemma of the calculus of variations

0 = ∫_{t}^{1} d_{3}[w](τ) ∧ [f_{0} − f_{c}(τ)] dτ + m_{0}  (73)

for all t ∈ [0, 1] by continuity. Differentiating this with respect to t we obtain

d_{3}[w](t) ∧ (f_{0} − f_{c}(t)) = 0 for a.e. t ∈ [0, 1],

which by (73) leads to m_{0} = 0. Hence we can write

f_{c}(t) = b(t)d_{3}[w](t) + f_{0} for a.e. t ∈ [0, 1]  (74)

for some function b. Notice that [13, Corollary 1] applies to f_{c}, which is a constant multiple of f_{c}^{∗} defined in (64). Hence f_{c} · d_{3}[w] ∈ W^{1,∞}([0, 1]) and, by (74), b ∈ W^{1,∞}([0, 1]) and f_{c} ∈ W^{1,∞}([0, 1], ℝ^{3}). Taking derivatives in (74) we get

f′_{c}(t) = b′(t)d_{3}[w](t) + b(t)d′_{3}[w](t) for a.e. t ∈ [0, 1].

Since d′_{3}[w] · d_{3}[w] = 0 on [0, 1] and, by [13, Corollary 1], f′_{c} · d_{3}[w] = 0 a.e. on [0, 1], there is a constant b_{0} ∈ ℝ with

b(t) = b_{0} for all t ∈ [0, 1].

Consequently,

f′_{c}(t) = b_{0}d′_{3}[w](t) for a.e. t ∈ [0, 1].  (75)

If b_{0} = 0, then f_{c}(t) = f_{0} for all t ∈ [0, 1] by (74). With [13, Lemma 15] we conclude that

λvμ^{∗} = λvK[w]^{4}μ̄ = 0.

Hence μ̄ = 0 because λ > 0. This contradicts the fact that μ̄ is a probability Radon measure according to the proof of Lemma 5. Therefore b_{0} ≠ 0. By the identity γ′[w](t) = vd_{3}[w](t) on account of (18) we can rewrite (75) as

f′_{c}(t) = (b_{0}/v)γ′′[w](t) for a.e. t ∈ [0, 1].  (76)

To determine the sign of the constant b_{0} we take variations

w^{△} = (0, v^{△}, 0, 0) ∈ Y with v^{△}≠ 0

in (70) to obtain

0 = λ_{E} − λ ∫_{0}^{1} d_{3}[w](t) · f_{c}^{∗}(t) dt
= λ_{E} − v^{−1} ∫_{0}^{1} d_{3}[w](t) · [f_{c}(t) − f_{0}] dt
=_{(74)} λ_{E} − b_{0}/v,

where we used that ∫_{0}^{1} d_{3}[w](t) · f_{0} dt = 0 by (18) and (21). Consequently, λ_{E} ≠ 0, i.e., λ_{E} > 0 by the Lagrange multiplier rule, hence b_{0} > 0 as well. (The condition λ_{E} > 0 expresses normality in the Lagrange multiplier rule.)

Recall that f_{c} = λvf_{c}^{∗} with f_{c}^{∗} according to (64) and λ, v > 0. Now we can argue as in the proof of [13, Corollary 1 (iii)], which rests on [1, Theorem 2.28]. Hence there are (nonnegative) Radon measures μ_{s} for s ∈ [0, 1] such that

supp μ_{s} ⊂ I_{s} (i.e., μ_{s} = 0 if I_{s} = ∅.)

and

γ′′(s) = \frac{v}{b_{0}} f′_{c}(s) = ∫_{0}^{1}(γ(τ) − γ(s)) dμ_{s}(τ) for a.e. s ∈ [0, 1].

This verifies Theorem 1. ⊓ ⊔

Proof of Corollary 2. (i) We can rewrite the set I_{c} as

I_{c} = {s ∈ [0, 1] : κ_{G}[γ](s) = θ^{−1}}  (77)

by definition of A[γ] in (13), together with Lemma 1 and by the fact that K[γ] = θ^{−1} according to Theorem 1. Thus (i) follows from Corollary 1.

(ii) If κ_{G}[γ](s) < θ^{−1} we find δ = δ(s) such that κ_{G}[γ](τ) < θ^{−1} for all τ ∈ [0, 1] with |τ − s| < δ. Otherwise we could find a sequence s_{i} → s with κ_{G}[γ](s_{i}) = θ^{−1} for all i ∈ ℕ, but then κ_{G}[γ](s) = θ^{−1} according to assertion (i). Thus τ ∉ I_{c} for |s − τ| < δ. Hence I_{τ} = ∅ and μ_{τ} = 0 for all |τ − s| < δ. Therefore, by (16), γ′′(τ) = 0 for a.e. |τ − s| < δ, which implies the assertion.

(iii) Obviously μ_{s} = 0 for a.e. s ∈ I^{∗}. Thus γ′′(s) = 0 for a.e. s ∈ I^{∗}, i.e., γ is straight on I^{∗} since γ ∈ W^{2,∞}(I, ℝ^{3}).

(iv) Since γ′′ ≠ 0 a.e. on Ĩ, μ_{s} ≠ 0 for a.e. s ∈ Ĩ. Thus there is a set I_{0} ⊂ Ĩ of (Lebesgue) measure zero such that Ĩ \ I_{0} ∈ I_{c}. Since the closure of Ĩ\ I_{0} relative to Ĩ is Ĩ and since I_{c} is closed, we readily get Ĩ ⊂ I_{c}, i.e., κ_{G}[γ](.) ≡ θ^{−1} on Ĩ by (77). ⊓ ⊔ 5.3. Proofs for Section 4

Proof of Proposition 6. Given a curve γ ∈ W^{2,∞}(I, ℝ^{3}) with |γ′(σ)| = const. = v > 0 we set d_{3} := γ′/|γ′| which is of class W^{1,∞}(I, ℝ^{3}) and has unit length. To obtain a complete frame field D = (d_{1}|d_{2}|d_{3}) : Ī → SO(3) of class W^{1,∞}(I, ℝ^{3}) we observe that the length of the curve d_{3}(Ī) on the unit sphere S^{2} ⊂ ℝ^{3} is bounded since γ ∈ W^{2,∞}(I, ℝ^{3}). Hence the two-dimensional Hausdorff-measure H^{2}(γ′(Ī)) of the image curve γ′(Ī) of tangents is zero. Thus we find a fixed vector d̃ ∈ S^{2}\d_{3}(Ī) and define d_{1} := d̃∧d_{3} and d_{2} := d_{3}∧d_{1} on Ī. Now we apply the first part of Proposition 5 to the framed curve (γ, D) ∈ W^{2,∞}(I, ℝ^{3}) × W^{1,∞}(I, ℝ^{3×3}) and obtain u^{i} ∈ L^{∞}(I), i = 1, 2, 3, and γ_{0} := γ(0), D_{0} := D(0), such that (18) holds. Finally we note that we can change the function u^{3} ∈ L^{∞}(I) arbitrarily without changing the differential equation and the initial condition for d_{3} in (18). Thus, by the second part of Proposition 5, we obtain a framed curve (γ, D) for any u^{3} ∈ L^{∞}(I). ⊓ ⊔

Appendix. A C^{1,1}-curve with discontinuous global curvature function

We are going to construct a planar closed curve γ of class C^{1,1} with γ(0) = 0 and with oscillations accumulating at the origin, such that in any neighbourhood of the origin there are points γ(s_{k}), s_{k} → 0 as k → ∞, with local curvature κ[γ](s_{k}) = 9/2, which implies κ_{G}[γ](s_{k}) ≥ 9/2 for all k ∈ ℕ by (8). On the other hand, we will see that there are two open disks D_{1}, D_{2} ⊂ ℝ^{2} of radius 1/2 with \overline{D}_{1} ∩ \overline{D}_{2} = \{0\}, such that γ ∩ (D_{1} ∪ D_{2}) = ∅ (see Fig. 1). We will show that this implies κ_{G}[γ](0) ≤ 2, and hence a discontinuity of κ_{G}[γ](.) at s = 0.

Let σ_{k} := 2^{−k}, k = 0, 1, 2, …, and let γ_{k}(σ) := (σ, h_{k}(σ)) be a curved arc given as a graph of the function h_{k} : [σ_{k+1}, σ_{k}] → ℝ defined by

h_{k}(σ) := \begin{cases}(−1)^{k+1}σ_{k+1}(σ − σ_{k+1}) & on [σ_{k+1}, σ_{k}^{∗}], \\ (−1)^{k}(σ − σ_{k})(σ_{k} + 2^{−1}κ_{0}(σ − σ_{k})) & on [σ_{k}^{∗}, σ_{k}], \end{cases} (78)

where σ_{k}^{∗} := 2σ_{k}/3 and κ_{0} := 9/2. One easily checks that h_{k} is of class W^{2,∞}([σ_{k+1}, σ_{k}]) with

h′′_{k}(σ) = \begin{cases}0 & on (σ_{k+1}, σ_{k}^{∗}), \\ (−1)^{k}κ_{0} & on (σ_{k}^{∗}, σ_{k}). \end{cases}  (79)

Hence the local curvature κ[γ_{k}] satisfies

κ[γ_{k}](σ) = \begin{cases}0 & for σ ∈ (σ_{k+1}, σ_{k}^{∗}), \\ \frac{(−1)^{k}κ_{0}}{(1+|h′_{k}(σ)|^{2})^{3/2}} & for σ ∈ (σ_{k}^{∗}, σ_{k}); \end{cases}  (80)

hence κ[γ_{k}](.) ≤ κ_{0} a.e. on [σ_{k+1}, σ_{k}]. Glueing together the arcs γ_{k}, k = 0, 1, 2, …, we obtain a curve γ^{∗} ∈ W^{2,∞}((0, 1], ℝ^{3}), since h′_{k}(σ_{k}−) = h′_{k}(σ_{k}+) for k = 1, 2, …. Setting γ^{∗}(0) := 0 and reflecting across the origin we obtain a curve γ^{∗∗} ∈ W^{2,∞}([−1, 1], ℝ^{3}), which can be closed up to obtain a closed curve γ ∈ W^{2,∞} by the larger arc of the circle ∂B_{1}((0, 1)) with the endpoints (1, 0) and (−1, 0), see Figure 1.

Fig. 1. The curve γ.

By (78)–(80) we have for σ_{k}^{0} := (7/9)σ_{k} ∈ [σ_{k}^{∗}, σ_{k}], k = 0, 1, 2, …, that h′_{k}(σ_{k}^{0}) = 0; hence

κ[γ_{k}](σ_{k}^{0}) = κ_{0} for all k = 0, 1, 2, ….  (81)

On the other hand, |h_{k}(σ)| ≤ σ_{k+1}(σ − σ_{k+1}) ≤ σ^{2} − σ_{k+1}^{2} < σ^{2} for all σ ∈ [σ_{k+1}, σ_{k}], which implies in particular that

γ([−1, 1]) lies in between the parabolas ±σ^{2}.  (82)

Reparameterizing γ by arc length we obtain Γ ∈ W^{2,∞}(S_{L}, ℝ^{3}) on account of (80), with |Γ′| ≡ 1 on S_{L}, where L = L(γ) denotes the length of γ. Without loss of generality we may assume that Γ (0) = 0; hence Γ′(0) is horizontal according to (82). We claim that

κ_{G}[γ](0) ≤ 2.  (83)

In fact, we can apply Lemma 1 for s = 0 where we have to distinguish between the two cases (A) and (B).

Case (A). One can use formula (11) to establish 1/2 as an explicit lower bound for the radius r(Γ (0), Γ (σ), Γ′(σ)) of any circle through Γ (0) which is tangential to Γ at some other point Γ (σ). Therefore (83) holds in this case.

Case (B). Any circle tangential to Γ in Γ (0) = 0 which intersects Γ in at least one more point Γ (σ), has radius r(Γ (σ), Γ (0), Γ′(0)) ≥ 1/2 by (82), which implies (83), and therefore our claim is proved.

Now (81) implies that there is a sequence of arc length parameters s_{k}^{0} :=

∫_{0}^{σ_{k}^{0}} |γ′(σ)| dσ ∈ S_{L} with s_{k}^{0} → 0 as k → ∞ and, by (8), with

κ_{G}[γ](s_{k}^{0}) ≥ κ_{0} = 9/2 > 2 ≥ κ_{G}[γ](0).

Hence κ_{G}[γ](.) is not continuous at s = 0.

Acknowledgements. We would like to thank the Sonderforschungsbereich 611 at the University of Bonn for supporting our research.

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