Mathematische Zeitschrift manuscript No. (will be inserted by the editor)

Global curvature for rectifiable loops

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

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

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

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

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

Received: date / Revised version: date – \begin{matrix}c \\ ◯ \end{matrix} Springer-Verlag 2002

Abstract We study in detail the notion of global curvature defined on rectifiable closed curves, a concept which has been successfully applied in existence and regularity investigation regarding elastic self-contact problems in nonlinear elasticity. A bound on this purely geometric quantity serves as an excluded volume constraint to prevent selfintersections of slender elastic bodies modeled as elastic rods. Moreover, a finite global curvature characterizes simple closed curves, whose arc length parameterizations possess a Lipschitz continuous tangent field. The investigation of local and non-local properties of global curvature motivates, in particular, an extended definition of local curvature at any point of a rectifiable loop. Finally we show how a bound on global curvature can be used to define and control topological constraints such as a given knot type for closed loops or a prescribed linking number for closed framed curves, suitable to describe, e.g., supercoiling phenomena of biomolecules.

Mathematics Subject Classification (2000): 53A04, 57M25, 74K10, 74M15, 92C40

1 Introduction

An elastic string, rope or wire being deformed in space cannot penetrate itself. It is surprisingly difficult to provide a mathematically precise and analytically tractable formulation for this physically evident phenomenon of self-avoidance, also called the excluded volume constraint. While most investigations in elasticity neglect this effect, it becomes crucial to exclude interpenetration in order to describe the physically relevant solutions, for instance, when dealing with closed knotted configurations where self-contact is to be expected.

Send offprint requests to: Heiko von der Mosel

In many situations the deformed state of a slender ring-shaped elastic body can be idealized as the tubular neighbourhood of a closed centrecurve. The excluded volume constraint is transferred to the centreline mathematically as a bound on the global curvature. This is a nonlocal quantity whose inverse, the global radius of curvature, was introduced by Gonzalez and Maddocks [11] in the context of ideal shapes of knots. Since this notion is not restricted to smooth curves (as is the case, e.g., for the classical normal injectivity radius), a bound on the global curvature is accessible by variational methods. In addition, the global curvature carries important information about the geometrical object. For example, we will prove here that a bounded global curvature characterizes the set of simple rectifiable loops, whose arc length parameterizations have a Lipschitz continuous tangent, see Theorem 1. This furnishes a remarkable equivalence between a purely geometric concept and analytic properties of closed space curves. Moreover, the relation between the pointwise global curvature function to the classical local curvature for smooth curves motivates our geometric definition of generalized local curvature at arbitrary points on curves that are merely rectifiable. To account for sophisticated topological constraints occurring in applications we consider framed curves, which are curves associated with an orthonormal frame at each point. A bound on the global curvature ensures that the linking number between the centreline and a curve generated by the frame is well-defined. The linking number allows us to identify the mechanically relevant solutions, which is important, for instance, to account for supercoiling phenomena of biomolecules modeled as elastic rods. The geometrically exact condition of bounded global curvature turns out to be suitable for the direct methods in the calculus of variations to prove existence results for highly nonlinear problems, see [12]. Moreover, in contrast to alternative approaches, e.g., using repulsive potentials, where the necessary regularization can lead to serious mathematical and computational difficulties (see [9],[14],[18]), the geometrically exact condition of bounded global curvature allows us to derive the Euler-Lagrange equations including contact terms for minimizing elastic rods without shear and extension, see [16]. It should be emphasized that a rigorous derivation of variational equations in nonlinear elasticity taking into account self-contact has never been done before.

Crucial for all these applications is a thorough understanding of the purely geometric concept of global curvature. It is the objective of this paper to present a comprehensive investigation of this notion, both extending previous results in [12] and providing new and detailed analytical information about global curvature partly used in [16].

After defining the global curvature functions in Section 2 we present the central regularity result, Theorem 1. Part (ii) states that the tangent cone, as defined in geometric measure theory, reduces to a one-dimensional linear subspace at those points of a rectifiable loop, where the global curvature is finite. Part (iii) contains the remarkable fact that finite global curvature characterizes simple curves whose arc length parameterization possesses a Lipschitz continuous tangent field with Lipschitz constant equal to the global curvature. Geometrically, the global radius of curvature is the radius of the largest open ball that can be rotated tangentially about any point on the rectifiable loop without intersecting it, see part (iv) of Theorem 1.

Then the different cases of approaching the limit in the definition of the curvature functions are investigated, reflecting the local and nonlocal properties of global curvature. In particular, Theorem 2 (i) shows how the pointwise global curvature is related to the classical local curvature where the latter exists. This motivates the definition of the generalized local curvature at any point of a rectifiable loop, which turns out to be bounded from above by the approximate limes superior of |Γ′′| at that point (in the sense of geometric measure theory), and which equals |Γ′′| a.e., see Proposition 1. An alternative characterization of global curvature is given in Lemma 2. Specifically the geometric statement of Proposition 2 and the analytic information contained in Lemma 3 about the set of parameters, where global curvature is achieved, is essential for computing the structure of the contact term in the Euler-Lagrange equations in [16].

The excluded volume constraint in terms of global curvature is discussed in Section 3. Curves with finite global curvature K possess a tubular neighbourhood of uniform radius K^{−1}, where the next-point projection onto the curve is single-valued and continuous, Proposition 3. Also the reverse statement is true, which constitutes a purely geometric justification for adopting the notion of global curvature as an excluded volume constraint. Furthermore, global injectivity for deformations of unshearable elastic rods is equivalent to a finite bound on the global curvature of the centreline (Theorem 3). This means that a bound on the global curvature serves as an exact excluded volume constraint for unshearable rods also in the mechanical context.

Section 4 deals with functional properties of global curvature considered as a quantity defined on the space of rectifiable closed loops, and its role when dealing with topological constraints such as a given knot or link class. We prove in Lemma 4 that a bound on global curvature furnishes a closed condition with respect to uniform convergence, which is a sharpened version of [12, Lemma 4]. Proposition 4 states that isotopy classes, identifying the knot type for the centreline, are stable with respect to the C^{0}-topology on the set of loops with uniformly bounded global curvature. Thus the pulltight phenomenon of small knots in the limit, not detected by the uniform topology alone, is excluded by an additional uniform upper bound on the global curvature. The closedness of isotopy classes of rectifiable loops with uniformly bounded global curvature follows easily, see Theorem 4. In contrast to, and supplementing the upper semicontinuity of the global radius of curvature proved in [4], we show that global curvature is continuous with respect to C^{1,1}-convergence of the arc length parameterizations (Theorem 5). This turns out to be quite useful when dealing with perturbations of a minimizer of a mechanical variational problem in order to derive the Euler-Lagrange equations in [16]. The Gaussian linking number is invoked via an analytic formula (avoiding topological degree theory) to define link classes for framed curves, which enables us to treat the case where the curve of the assigned orthonormal frames is not closed in SO(3). With this notion we are able to distinguish the infinitely many components in the set of framed curves having the same boundary conditions, but where the number of rotations of the frame around the centreline differs. This distinction is necessary to select the correct solution in applications where supercoiling phenomena lead to a variety of configurations even for closed unknotted centrecurves. In Lemma 5 and Theorem 6 it is proved that a given link class constitutes a weakly closed condition and is stable with respect to small perturbations.

The proofs of all results are presented in Section 5.

Notation. We use x · y to denote the standard Euclidean inner product of x and y in ℝ^{3}, and | · | to denote the (intrinsic) distance between two points in ℝ^{3} or in some parameter set J ⊂ ℝ depending on the context. To denote the enclosed (smaller) angle between two non-zero vectors x and y in ℝ^{3} we use <)(x, y) ∈ [0, π]. The distance between a point x ∈ ℝ^{3} and a subset Σ ⊂ ℝ^{3} will be denoted by dist(x, Σ), the diameter of Σ will be denoted by diam(Σ). For any δ > 0 we define open neighbourhoods of x and Σ by

B_{δ}(x) = {y ∈ ℝ^{3} | |y − x| < δ} and B_{δ}(Σ) = {y ∈ ℝ^{3} | dist(y, Σ) < δ}.

The interior of a set Σ will be denoted by int Σ and its boundary by ∂Σ. The space of continuous functions on the closure of the interval I = (a, b) will be denoted by C^{0}(Ī), and C^{k,1}(Ī), k = 0, 1, 2, …, is the space of ktimes continuously differentiable functions whose k-th derivative is Lipschitz continuous on Ī. 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 that C^{k,1}(Ī) ≅ W^{k+1,∞}(I). The mean value integral of an integrable function f over a set E will be denoted by ∫ −_{E} f := |E|^{−1} ∫_{E} f.

2 Global curvature for rectifiable loops

We consider the set L of continuous and rectifiable closed curves γ : Ī → ℝ^{3}, with arc length parameterization Γ_{γ} : S_{L} → ℝ^{3}. Here I = (a, b) is an open interval, L = L(γ) := ∫_{I} |dγ| ≥ 0 denotes the length of γ, and 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). The intrinsic distance on S_{L} and also the Euclidean distance in ℝ^{3} will be denoted by | . |. In some instances we will need to use an ordering on the parameter set S_{L} to speak of inequalities between parameters s_{1}, s_{2} ∈ S_{L}, and to consider one-sided limits. For this we tacitly assume that s_{1}, s_{2} ∈ (0, L), where, for instance, s_{1} < s_{2} is welldefined. To treat parameters near the endpoints s = 0, s = L we simply shift the interval.

To simplify notation, we mostly omit the subscript γ and agree that Γ, Γ_{k} correspond to γ, γ_{k} and so on. According to [10, vol.II, p.255] this arc length parameterization Γ is Lipschitz continuous, i.e. of class C^{0,1}([0, L], ℝ^{3}). Note that, by Rademacher’s Theorem, Γ possesses a weak derivative Γ′ a.e. on [0, L] and C^{0,1}([0, L], ℝ^{3}) ≅ W^{1,∞}([0, L], ℝ^{3}).

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

ρ_{G}[γ](s) := \begin{cases}inf_{{σ,τ∈S_{L}\\{s\} \atop σ≠τ}} R(Γ (s), Γ (σ), Γ (τ)), & if L > 0, \\ 0, & if L = 0, \end{cases}  (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 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 and

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)

If two points coincide, however, say x = z or y = z, then there are many circles through the three points 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 Lemma 1 below we will see that ρ_{G}[γ](s) is always finite for closed curves. In the case of smooth curves that have no or only transversal crossings, our definition of ρ_{G} agrees with that in [11], but it is different for curves with double covered regions. In particular, at parameters s, where Γ is not injective, our definition leads to ρ_{G}[γ](s) = 0, whereas this is not necessarily the case according to the definition in [11], see also the discussion in [12].

The global radius of curvature of γ is defined as

R[γ] := inf_{s∈S_{L}} ρ_{G}[γ](s).  (3)

The global curvature of γ at s ∈ S_{L} is given by

κ_{G}[γ](s) := sup_{{σ,τ∈S_{L}\\{s\} \atop σ≠τ}} \frac{1}{R(Γ (s), Γ (σ), Γ (τ))} ,  (4)

which, of course, can take the value +∞. In analogy to R[γ] we define the global curvature of γ by

K[γ] := sup_{s∈S_{L}} κ_{G}[γ](s).  (5)

Notice that for the global radius of curvature R[γ] and for the global curvature K[γ] we can also write

R[γ] = inf_{{s,σ,τ∈S_{L} \atop s≠σ≠τ≠s}} R(Γ (s), Γ (σ), Γ (τ)),  (6)

K[γ] = sup_{{s,σ,τ∈S_{L} \atop s≠σ≠τ≠s}} \frac{1}{R(Γ (s), Γ (σ), Γ (τ))} .  (7)

We will see in Lemma 1 that R[γ] ∈ [0, ∞) and hence (by the identity

(8) below) K[γ] ∈ (0, ∞] for γ ∈ L. If in addition, γ ∈ L has an injective and smooth^{1} arc length parameterization, we infer from Theorem 1, part (iii), below that both R[γ] and K[γ] have values in (0, ∞).

As an immediate consequence of the definitions of κ_{G} and ρ_{G} we observe for γ ∈ L, that

κ_{G}[γ](s) = \frac{1}{ρ_{G}[γ](s)} ,  K[γ] = \frac{1}{R[γ]} ,  (8)

where we tacitly understand that the terms on the left-hand sides become infinite if the denominators on the right vanish.

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.

In part (iii) of the following central regularity result Theorem 1 it is shown that the condition of finite global curvature K[γ] identifies curves with a C^{1,1}-arc length parameterization without double points. On the other hand, however, there are simple curves γ ∈ L with infinite global curvature K[γ], e.g., curves with corner points.

Theorem 1. Let γ ∈ L with arc length parameterization Γ : S_{L} → ℝ^{3}.

(i) If Γ has a double point at the pair s, t ∈ S_{L} (s ≠ t), then ρ_{G}[γ](s) = ρ_{G}[γ](t) = 0.

(ii) At every point s ∈ S_{L} where ρ_{G}[γ](s) > 0, Γ possesses a geometric unit tangent T (s) satisfying

T (s) := lim_{{σ↓s \atop σ≠s}} \frac{Γ (σ) − Γ (s)}{|Γ (σ) − Γ (s)|} = − lim_{{τ↑s \atop τ≠s}} \frac{Γ (τ) − Γ (s)}{|Γ (τ) − Γ (s)|} .  (9)

In addition, at points s_{0} ∈ S_{L}, where Γ is differentiable, one has

Γ′(s_{0}) = T (s_{0}).  (10)

(iii) K[γ] < ∞ if and only if γ is simple and Γ ∈ C^{1,1}([0, L], ℝ^{3}) ≃ W^{2,∞}([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},  (11)

i.e., Γ′ has Lipschitz constant K[γ]. Thus (10) is true for all s_{0} ∈ S_{L}, if K[γ] < ∞.

1 It suffices that the corresponding arc length parameterization Γ is of class C^{1,1}.

(iv) For θ > 0 let D_{θ}(z, z′) denote the open planar disk in ℝ^{3} of radius θ centred at z ∈ ℝ^{3} perpendicular to z′ ∈ ℝ^{3}\{0}. Let s ∈ S_{L} be given with ρ_{G}[γ](s) > 0 and geometric tangent T (s) as in (ii). Set

C(s, θ) = ∂D_{θ}(Γ (s), T (s)) and M(s, θ) = ⋃_{z∈C(s,θ)} B_{θ}(z).

Then

(a) Γ (S_{L}) ∩ M(s, ρ_{G}[γ](s)) = ∅.

(b) Suppose K[γ] < ∞, and let ϑ > 0 be a given constant. Then the following holds:

Γ (S_{L}) ∩ M(s, ϑ) = ∅ for all s ∈ S_{L} ⟺ K[γ] ≤ ϑ^{−1}.

Remarks. 1. Let Tan (γ(Ī), Γ (s)) denote the tangent cone of γ(Ī) at the point Γ (s) in the sense of Federer [7, Sec. 3.1.21]. Then (ii) implies that

Tan (γ(Ī), Γ (s)) = {λT (s) : λ ∈ ℝ}

for every s ∈ S_{L} with ρ_{G}[γ](s) > 0. In other words, the tangent cone reduces to a 1-dimensional linear subspace at points with positive global radius of curvature.

2. The necessary condition for K[γ] to be finite in part (iii), which was shown in [12, Lemma 2], 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 (11) actually implies

∥Γ′′∥_{L^{∞}} ≤ K[γ] = \frac{1}{R[γ]} .  (12)

3. Item (iv)(a) of the above result implies that an open ball of radius ρ_{G}[γ](s) placed tangent at the point Γ (s) may be rotated around the tangent vector Γ′(s) without intersecting the curve. On the other hand, if K[γ] > ϑ^{−1}, then there is a point on the curve about which a similar rotation of a ball of radius ϑ could not be effected without intersecting the curve, according to part (iv)(b). Thus R[γ] is the radius of the largest open ball that can be rotated tangentially about every point of a curve γ without intersecting it.

The technical and lengthy proof of Theorem 1 as well as all the other proofs are deferred to Section 5. In the following lemma we provide bounds on R[γ] and K[γ] in terms of simple geometric quantities of the curve γ.

Lemma 1. Let γ ∈ L with length L(γ) = L. Then

R[γ] ≤ ρ_{G}[γ](s) ≤ \frac{diam (γ(Ī))}{2} ≤ \frac{L}{4} for all s ∈ S_{L},  (13)

K[γ] ≥ κ_{G}[γ](s) ≥ \frac{2}{diam (γ(Ī))} ≥ \frac{4}{L} for all s ∈ S_{L}.  (14)

The next theorem clarifies if and how the infima and suprema in (1),(3),(4) and (5) are 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.

Theorem 2. (i) For every γ ∈ L with arc length parameterization Γ ∈ C^{1,1}([0, L], ℝ^{3}), there exists at least one parameter s ∈ S_{L}, such that either

R[γ] = ρ_{G}[γ](s) and K[γ] = κ_{G}[γ](s),  (15)

or there is a sequence (s_{j}, σ_{j}, τ_{j}) → (s, s, s) with s_{j} ≠ σ_{j} ≠ τ_{j} ≠ s_{j} for all j ∈ ℕ, such that

R[γ] = lim_{j→∞} R(Γ (s_{j}), Γ (σ_{j}), Γ (τ_{j})) and  (16)
K[γ] = lim_{j→∞} \frac{1}{R(Γ (s_{j}), Γ (σ_{j}), Γ (τ_{j}))}.  (17)

Moreover, if in the second alternative (16), (17), the parameter s ∈ S_{L} is a Lebesgue point of Γ′′, and if there is a constant c > 0, such that

max{|τ_{j} − s|, |σ_{j} − s|, |s_{j} − s|} ≤ c max{|τ_{j} − s_{j}|, |τ_{j} − σ_{j}|, |s_{j} − σ_{j}|} (18)

for all j ∈ ℕ, then

R[γ] = ρ_{G}[γ](s) = |Γ′′(s)|^{−1} and  (19)
K[γ] = κ_{G}[γ](s) = |Γ′′(s)|,  (20)

where Γ′′ is the precise representative in the sense of [6, p.46]. (ii) Let γ ∈ L with R[γ] > 0, 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

ρ_{G}[γ](s) = lim_{j→∞} R(Γ (s), Γ (σ_{j}), Γ (τ_{j})),  (21)
κ_{G}[γ](s) = lim_{j→∞} \frac{1}{R(Γ (s), Γ (σ_{j}), Γ (τ_{j}))} ,  (22)

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

Remarks. 1. The different options in Theorem 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 in (i) and (ii) hold simultaneously at every point due to the high degree of symmetry of the circle. 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., curvature jumps. The global curvature, however, is equal to 2/d and (15) holds everywhere along the curve. Stadium curves appear also in the examples of ideal C^{1,1}-links considered in [3] and [4].

2. If s = s_{j} for infinitely many j ∈ ℕ in (16),(17), then (15) holds true, too. Notice also that (18) holds (and therefore (19) and (20) as well) in the particular case, when s = s_{j} for all j ∈ ℕ, where s ∈ S_{L} is a Lebesgue point of Γ′′.

3. Statement (ii) says, roughly speaking, that the infimum in the definition of ρ_{G}[γ](s) as well as the supremum in κ_{G}[γ](s) cannot exclusively be achieved by two distinct parameters σ, τ ∈ S_{L}. Consequently, the infimum in (6) and the supremum in (7) cannot exclusively be achieved by three distinct parameters s, 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. Observe, however, that R might not be continuous at the limit point (s, σ, τ).

4. Notice that the assumption R[γ] > 0 is equivalent to demanding that γ possesses an injective arc length parameterization of class C^{1,1} according to Theorem 1, (iii).

In case (ii)(a) of Theorem 2 the value ρ_{G}[γ](s) expresses a local property of the curve at s ∈ S_{L}, which coincides with the classical local radius of curvature of γ ∈ L at s if γ is smooth.^{2} For curves γ ∈ L which are only continuous and rectifiable in general, this observation motivates the following definition of the generalized local radius of curvature ρ[γ](s) of γ ∈ L at s ∈ S_{L} as

ρ[γ](s) := lim inf_{{(τ_{j} ,σ_{j})→(s,s) \atop s≠τ_{j} ≠σ_{j} ≠s}} R(Γ (s), Γ (τ_{j}), Γ (σ_{j})).  (23)

Analogously, we define the generalized local curvature 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}))} .  (24)

Note that for γ ∈ L, both ρ[γ] and κ[γ] can take values in [0, ∞], and that

ρ[γ](s) ≥ ρ_{G}[γ](s) and κ[γ](s) ≤ κ_{G}[γ](s) for all s ∈ S_{L}. (25)

Proposition 1. 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) ≤ ap lim sup_{σ→s} |Γ′′(σ)| for all s ∈ S_{L}. (ii) κ[γ](s) = |Γ′′(s)| = ρ[γ](s)^{−1} for a.e. s ∈ S_{L}.

2 This can easily be verified by expanding Γ about s when calculating the term R(Γ (s), Γ (σ_{j}), Γ (τ_{j})) in (21) in case (a) for j large, see also Lemma 7 in Section 5.

Here, ap lim sup denotes the approximate limes superior as defined, e.g. in [6, p. 47]. Suppose we have the local bound |Γ′′(σ)| ≤ κ_{0} for a.e. σ in some open subinterval J ⊂ S_{L}, then, according to (i), κ[γ](σ) ≤ κ_{0} for all σ ∈ J. The essence of part (ii) is that for curves γ ∈ L with finite global curvature (hence γ simple and Γ ∈ W^{2,∞} by Theorem 1 (iii)), we can identify κ[γ] with |Γ′′| a.e. on [0, L], and from (12) we infer

∥κ[γ]∥_{L^{∞}} = ∥Γ′′∥_{L^{∞}} ≤ K[γ] for all γ ∈ L with K[γ] < ∞.  (26)

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

∥Γ′′∥_{L^{∞}} < K[γ].  (27)

Curves with this property are considered in Proposition 2 and Lemma 3 below and they play an essential role in [16], where the Euler-Lagrange equations for energy minimizing rods are derived.

For γ ∈ L with finite global curvature the alternative (b) in part (ii) of Theorem 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 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|} |} ,  (28)

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] < ∞, part (iii) of Theorem 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}, the radius r(Γ (s), Γ (σ), Γ′(σ)) is well defined, and we obtain the following identities for ρ_{G}, κ_{G}, R and K:

Lemma 2. Let γ ∈ L be such that K[γ] < ∞, then at least one of the following statements (A),(B) is true:

(A)

ρ_{G}[γ](s) = inf_{{σ∈S_{L} \atop σ≠s}} r(Γ (s), Γ (σ), Γ′(σ)) and

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

(B)

ρ_{G}[γ](s) = inf_{{σ∈S_{L} \atop σ≠s}} r(Γ (σ), Γ (s), Γ′(s)) = ρ[γ](s) and

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

If for s ∈ S_{L} part (ii)(b) of Theorem 2 holds, then alternative (A) above is true.

In addition,

R[γ] = inf_{{s,σ∈S_{L} \atop s≠σ}} r(Γ (s), Γ (σ), Γ′(σ)) ,  (29)

K[γ] = sup_{{s,σ∈S_{L} \atop s≠σ}} \frac{1}{r(Γ (s), Γ (σ), Γ′(σ))} .  (30)

Because of the representation of K[γ] as a supremum in (30) 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] : K[γ] = \frac{1}{r(Γ (s), Γ (σ), Γ′(σ))} \},  (31)

and one can show

Proposition 2. Let γ ∈ L be such that K[γ] < ∞. If (s, σ) ∈ A[γ], then s ≠ σ. If, in addition, K[γ] is locally not attained, then

|Γ (s) − Γ (σ)| = 2R[γ], and  (32)
Γ′(s) · (Γ (s) − Γ (σ)) = Γ′(σ) · (Γ (s) − Γ (σ)) = 0  (33)

for all (s, σ) ∈ A[γ].

Note that also Γ (s) ≠ Γ (σ) for (s, σ) ∈ A[γ], if K[γ] is finite. 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. In other words, for an ellipse, K[γ] is attained exclusively locally. On the other hand, if γ describes a circle, one has A[γ] = [0, L]×[0, L]\ diagonal.

For curves γ ∈ L such that K[γ] is locally not attained in the sense defined in (27), we have the following characterization of K[γ] as a maximum over pairs of parameters in a well defined compact subset of [0, L] × [0, L] away from the diagonal.

Lemma 3. Let γ ∈ L with K[γ] < ∞ such that K[γ] is locally not attained, and set

η(γ) := \frac{1 − R[γ] · ∥Γ′′∥_{L^{∞}}}{∥Γ′′∥_{L^{∞}}} ,  (34)
Q = Q[γ] := {(s, σ) ∈ [0, L] × [0, L] : L − η(γ) ≥ |s − σ| ≥ η(γ)}. (35)

Then (i) 0 < η(γ) < L/(2π), (ii) A[γ] ∩ Q ≠ ∅, i.e., K[γ] = max_{(s,σ)∈Q} (r(Γ (s), Γ (σ), Γ′(σ)))^{−1}, (iii) K[γ] > (r(Γ (s), Γ (σ), Γ′(σ)))^{−1} for all (s, σ) ∈ [0, L]^{2}\Q.

Note that the upper bound on η(γ) in (i) can be improved to L/(4π), if γ is non-trivially knotted by virtue of the Fáry-Milnor Theorem on the total curvature of knotted curves [13].

The significance of Lemma 3 is that r(Γ (s), Γ (σ), Γ′(σ)) is continuously differentiable on a small neighbourhood of Q, which is a crucial ingredient for the analysis in applications involving a bound on K as a constraint for variational problems, cf. [16].

3 Excluded volume constraint

Theorem 1, part (iv), establishes the fact that if a curve γ ∈ L has global curvature bounded by some constant θ^{−1}, then γ is restricted on how tightly it can bend locally, by the inequality κ[γ](s) ≤ K[γ] ≤ θ^{−1} for a.e. s ∈ I, and on how close it can come to self-intersection globally. The interplay of local and global effects is strongly connected to the excluded volume constraint, which says that some tube surrounding the curve γ as a centreline, does not intersect itself.

To make this physically intuitive condition mathematically precise, we recall that for curves γ ∈ L with K[γ] < ∞ we have a well-defined continuous tangent field Γ′. Thus we can speak of planar disk-shaped cross-sections of uniform size perpendicular to Γ′ along the curve. The excluded volume constraint then consists of the condition that two different cross-sections do not intersect, which locally translates into a bound on the (local) curvature, and globally into a restriction on how close different curve points can approach each other in space.

One mathematical notion to describe such a constraint would be the normal injectivity radius from differential geometry, which, however, requires a certain amount of smoothness of the centreline γ, compare, e.g. [5]. In [11] it is shown that for smooth simple curves γ the normal injectivity radius coincides with R[γ]. Proposition 3 below shows that the condition K[γ] ≤ θ^{−1} models in a geometrically exact way the excluded volume constraint for the tubular neighbourhood of γ ∈ L of uniform radius θ > 0. An alternative characterization of the excluded volume constraint involving global injectivity of a deformation mapping will be discussed afterwards.

The set B_{δ}(γ(Ī)) = B_{δ}(Γ (S_{L})) is said to be the tubular neighbourhood of γ of radius δ > 0, where γ ∈ L has arc length parameterization Γ : S_{L} → ℝ^{3}. We say that B_{δ}(Γ (S_{L})) is non-self-intersecting or regular if the closest-point projection map Π_{Γ} : B_{δ}(Γ (S_{L})) → Γ (S_{L}) is single-valued and continuous. That is to say, for any x ∈ B_{δ}(Γ (S_{L})) there is exactly one s(x) ∈ S_{L} such that Π_{Γ} (x) := Γ (s(x)) satisfies dist(x, Γ (S_{L})) = |Γ (s(x)) − x|, and Π_{Γ} (x) is a continuous function of x ∈ B_{δ}(Γ (S_{L})). The following result is contained in [12, Lemma 3].

Proposition 3. Consider γ ∈ L with K[γ] < ∞ and let Γ ∈ C^{1,1}(S_{L}, ℝ^{3}) denote its corresponding arc length parameterization. Let θ > 0 be a given constant and D_{θ}(z, z′) be the disk of radius θ centred at z, perpendicular to z′ ≠ 0. Then

(i) B_{θ}(Γ (S_{L})) is regular if and only if K[γ] ≤ θ^{−1},

(ii) Π_{Γ} has the property Π_{Γ}^{−1}(Γ (s_{0})) ∩ B_{θ}(Γ (S_{L})) = D_{θ}(Γ (s_{0}), Γ′(s_{0})) for every s_{0} ∈ S_{L}, if B_{θ}(Γ (S_{L})) is regular.

Items (i) and (ii) imply that the regularity of the tubular neighbourhood B_{θ}(Γ (S_{L})) is equivalent to the condition R[γ] ≥ θ, and that B_{θ}(Γ (S_{L})) is the envelope of disjoint disks D_{θ}(Γ (s_{0}), Γ′(s_{0})). Since each point x ∈ B_{θ}(Γ (S_{L})) is in a unique disk D_{θ}(Γ (s_{0}), Γ′(s_{0})) normal to the curve, we deduce that B_{θ}(Γ (S_{L})) has the structure of a uniform tube of radius θ centred on γ. Moreover, according to item (i), any tubular neighbourhood of radius larger than R[γ] would fail to have this structure, which strongly justifies the formulation of the excluded volume constraint as a prescribed upper bound on the global curvature K[γ]. In addition, continuity and compactness properties of K[.] on certain subsets of L, which are treated in Section 4, turn out to be valuable in proving various existence results involving elastic rods and ideal knots, see [12].

The excluded volume constraint for γ ∈ L with arc length parameterization Γ : S_{L} → ℝ^{3}, can be rephrased in terms of a deformation mapping p : Ω → ℝ^{3} defined as

p(s, ξ_{1}, ξ_{2}) = Γ (s) + ξ_{1}d_{1}(s) + ξ_{2}d_{2}(s) for (s, ξ_{1}, ξ_{2}) ∈ Ω,  (36)

where Ω is the open^{3} parameter set given by

Ω := { (s, ξ_{1}, ξ_{2}) ∈ ℝ^{3} | s ∈ S_{L}, ξ_{1}^{2} + ξ_{2}^{2} < θ^{2}}.

Here, Γ : S_{L} → ℝ^{3} describes the closed centreline of the deformed ringshaped body p(Ω), which might be considered as elastic rod, see [1]. d_{1}(s), d_{2}(s) are orthogonal unit vectors describing the orientation of the deformed crosssection at s ∈ S_{L}. We interpret s as length parameter and ξ_{1}, ξ_{2} as thickness parameters of the rod. With d_{3} := d_{1} ∧ d_{2} we get a right-handed orthonormal basis {d_{1}, d_{2}, d_{3}} at each s ∈ S_{L}, whose vectors are called directors, and which can be identified with an orthogonal matrix D = (d_{1}|d_{2}|d_{3}) ∈ SO(3) (the right-hand side denotes the matrix with columns d_{1}, d_{2}, d_{3}). We assume that D : S_{L} → SO(3) is continuous and that

Γ′(s) = d_{3}(s) for all s ∈ S_{L},  (37)

which models unshearability for the rod, i.e., cross sections remain orthogonal to the centreline. The pair (Γ, D), which we call framed curve, uniquely determines a deformed configuration of the rod.

3 Notice that in [16] we consider the closed parameter sets in accordance with the majority of works on elastic rods, see, e.g., [1].

A closed framed curve is given by the conditions

Γ (0) = Γ (L), d_{3}(0) = d_{3}(L),  (38)

which means that the centreline Γ is a closed curve and that the crosssections of the rod at s = 0 and s = L coincide.

We have seen above that the constraint K[γ] ≤ θ^{−1} prevents the tube B_{θ}(γ(Ī)) from self-intersecting. However, as a model for a physical object, it is the points of p(Ω) that are naturally identified with material points, and the excluded volume constraint should guarantee the global injectivity of the mapping p : Ω → ℝ^{3}. The following theorem, which was proved in [12, Lemma 7], shows that the condition K[γ] ≤ θ^{−1} is in fact equivalent to global injectivity of p.

Theorem 3. Consider a closed framed curve (Γ, D), where Γ is the arc length parameterization of γ ∈ L and where D ∈ C^{0}(S_{L}, SO(3)). Suppose that K[γ] < ∞ and that (37) holds. Then p : Ω → ℝ^{3} is globally injective if and only if K[γ] ≤ θ^{−1}.

Condition (37) allows us to identify the deformed rod p(Ω) with B_{θ}(Γ (S_{L})) and the result follows from the regularity of B_{θ}(Γ (S_{L})) as discussed above. When Γ′ is not parallel to d_{3}, however, the condition K[γ] ≤ θ^{−1} does not imply global injectivity of p and vice versa. Notice that p(Ω) itself is not a uniform tube of radius θ if Γ′ · d_{1} or Γ′ · d_{2} is non-zero. An alternative approach to obtain global injectivity also for this general case is given in [15].

4 Continuity properties and topological constraints

Here we examine the properties of global curvature considered as a functional on the space L of continuous rectifiable closed curves as well as its impact on the topological constraint of a given isotopy class on a subset of curves in L and of a given link class on closed framed curves.

Since one is interested in compactness properties for sequences of loops in L, compare [12], we are going to analyze the behaviour of the function K[.] on sequences in L that converge uniformly. Our first result in that direction states that the set of curves γ ∈ L with bounded length L(γ) ≤ L_{0} < ∞, satisfying K[γ] ≤ θ^{−1} < ∞ is closed with respect to uniform convergence.

Lemma 4. Let L_{0}, θ > 0 be fixed. Then the set

F_{1} := {γ ∈ L : L(γ) ≤ L_{0} and K[γ] ≤ θ^{−1}}

is closed in C^{0}(Ī, ℝ^{3}), i.e., with respect to uniform convergence.

In order to deal with minimization problems on isotopy (or knot) classes of closed curves, one has to investigate under which circumstances the isotopy relation is conserved. This is important for compactness arguments in the existence theory, see [12], but also for the variational aspects when deriving the Euler-Lagrange equations, see [16]. For that we recall the definition of isotopy:

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.

Proposition 4. Let γ ∈ L satisfy K[γ] ≤ θ^{−1} for some fixed constant θ > 0. Then there exists ϵ = ϵ(γ, θ) > 0, such that for all curves γ̃ ∈ L with K[γ̃] ≤ θ^{−1} and ∥γ − γ̃∥_{C^{0}} ≤ ϵ, one has γ(Ī) ≃ γ̃(Ī).

The statement of the lemma is no longer true if one removes the assumptions on the global curvature, small knotted regions might pull tight in the uniform topology. As an immediate consequence of Proposition 4 we get

Theorem 4. Let L_{0}, θ > 0 be fixed and K ∈ L be a given simple closed curve. Then the set

F_{2} := {γ ∈ L : L(γ) ≤ L_{0}, γ(Ī) ≃ K, K[γ] ≤ θ^{−1}}

is closed in C^{0}(Ī, ℝ^{3}), i.e., with respect to uniform convergence.

The next theorem provides an important continuity property of K[.] on some subset of L which is also an essential ingredient for our variational approach in [16].

Theorem 5. Let L^{∗} ⊂ L be the set of curves γ ∈ L of fixed length L(γ) = L > 0, with the property that the corresponding arc length parameterization Γ is of class C^{1,1}([0, L], ℝ^{3}). Then K[.] (and hence R[.]) is continuous on L^{∗} with respect to convergence of the corresponding arc length parameterizations in C^{1,1}([0, L], ℝ^{3}).

Cantarella et al. [4] have pointed out that R[.] is merely upper semicontinuous with respect to uniform convergence on the space of Lipschitz curves, and that R[.] can actually jump upwards in a limit even for C^{1}-convergence. For instance, consider a circle which is approximated in C^{1} by a sequence of curves Γ_{n} such that ∥Γ′′_{n}∥_{L^{∞}} → ∞.

A closed framed curve satisfying (37) consists of a base curve Γ and an associated frame field D, both of which can be subjected to topological restrictions. We have discussed above how to prescribe knot classes for the base curve incorporating the concept of isotopy. Moreover, if we fix the rotation between the terminal frames D(0) and D(L) there are still infinitely many topologically distinct components in the space of closed framed curves within the same isotopy class for the base curve, which may be seen as follows.

If one prescribes the knot type for the centreline and if one glues together the terminal frames at s = 0 and s = L such that d_{3}(0) = d_{3}(L) and, for simplicity, d_{1}(0) = d_{1}(L) hold, one has not entirely fixed the topological type of the framed curve. Indeed, every full rotation of the pair d_{1}(L), d_{2}(L) within the cross section respects the boundary conditions, but changes the linking number between the centreline and the curve Γ (.)+(θ/2)d_{1}(.), which is a topological invariant. Since such a change of topological type is accompanied by an (often drastic) change of the equilibrium configuration for an elastic rod, we need to prescribe the linking number in order to identify particular solutions, see also the discussion in [2]. The approach in [12] using the concept of homotopies in SO(3) distinguishes only two topologically different classes, since the fundamental group of SO(3) is ℤ_{2}.

One way to determine the link between two disjoint closed (but not necessarily simple) curves is to compute the Gaussian linking number, which is usually defined in terms of the topological degree, see, e.g., [17, p. 402]. For a pair of absolutely continuous disjoint curves, however, there is an analytically more convenient formula, which we adopt as definition for the linking number. For γ_{1}, γ_{2} ∈ L ∩ W^{1,1}(I, ℝ^{3}) with γ_{1}(Ī) ∩ γ_{2}(Ī) = ∅, the linking number l(γ_{1}, γ_{2}) is given by

l(γ_{1}, γ_{2}) := \frac{1}{4π} ∫_{I} ∫_{I} \frac{γ_{1}(s) − γ_{2}(t)}{|γ_{1}(s) − γ_{2}(t)|^{3}} · [γ′_{1}(s) ∧ γ′_{2}(t)] ds dt.  (39)

One can show that l(γ_{1}, γ_{2}) is integer-valued and stable with respect to smooth perturbations preserving the non-intersection property. Before we use the linking number to identify topological classes of closed framed curves we present some general properties.

Lemma 5. (i) Let γ_{1,n}, γ_{2,n} ∈ L ∩ W^{1,q}(I, ℝ^{3}), q > 1, n ∈ ℕ, with γ_{1,n} ⇀ γ_{1}, γ_{2,n} ⇀ γ_{2} in W^{1,q}(I, ℝ^{3}), and let

dist (γ_{1,n}(Ī), γ_{2,n}(Ī)) ≥ c for all n ∈ ℕ,  (40)
l(γ_{1,n}, γ_{2,n}) = l_{0} for all n ∈ ℕ,  (41)

where c > 0, l_{0} ∈ ℤ are given constants. Then l(γ_{1}, γ_{2}) = l_{0}.

(ii) Let γ_{1}, γ_{2} ∈ L ∩ W^{1,q}(I, ℝ^{3}), q > 1, such that

γ_{1}(Ī) ∩ γ_{2}(Ī) = ∅.  (42)

Then there is a constant ϵ = ϵ(γ_{1}, γ_{2}) > 0, such that β_{1}(Ī) ∩ β_{2}(Ī) = ∅, and l(γ_{1}, γ_{2}) = l(β_{1}, β_{2}) for all β_{1}, β_{2} ∈ L ∩ W^{1,q}(I, ℝ^{3}) with ∥γ_{i} − β_{i}∥_{W^{1,q}} ≤ ϵ, i = 1, 2.

For a closed framed curve respecting (37) we want to consider the linking number of the curves Γ (.) and Γ (.)+(θ/2)d_{1}(.). The problem here is that the second curve might not be closed and that the two curves might intersect each other. The first problem can be solved by closing the curve Γ (.) + (θ/2)d_{1}(.) up in a unique way, namely by

β^{D}(s) := \begin{cases}Γ (s) + \frac{θ}{2}d_{1}(s) for s ∈ [0, L], \\ Γ (L) + \frac{θ}{2}[cos(ϕ_{D}(s − L))d_{1}(L) + sin(ϕ_{D}(s − L))d_{2}(L)] \\ for s ∈ [L, L + 1], \end{cases}

where ϕ_{D} ∈ [0, 2π) is the angle between d_{1}(0) and d_{1}(L), such that ϕ_{D} − π has the same sign as (d_{1}(0) ∧ d_{1}(L)) · d_{3}(0). For technical reasons we identify Γ with its trivial extension onto [0, L + 1] according to Γ (s) := Γ (L) for s ∈ [L, L + 1]. Notice that Γ, β^{D} ∈ W^{1,q}([0, L + 1], ℝ^{3}), 1 ≤ q ≤ ∞, if Γ ∈ W^{1,q}([0, L], ℝ^{3}), and that Γ and β^{D} are closed. Demanding the global curvature bound K[Γ] ≤ θ^{−1} we ensure that Γ ([0, L + 1]) ∩ β^{D}([0, L + 1]) = ∅ by Theorem 3. Thus the linking number of a closed framed curve (Γ, D) satisfying (37), K[Γ] ≤ θ^{−1}, Γ ∈ W^{1,1}([0, L], ℝ^{3}) and D ∈ W^{1,1}([0, L], ℝ^{3×3}), is well-defined by

l(Γ, D) := l(Γ, β^{D}).  (43)

Lemma 5 now readily implies the following properties of l(Γ, D), which are needed to derive existence results along the lines of [12] and to verify the Euler-Lagrange equations as necessary minimality conditions as carried out in [16].

Theorem 6. (i) Let Γ_{n} ⇀ Γ in W^{1,q}([0, L], ℝ^{3}), q > 1, D_{n} ⇀ D in W^{1,p}([0, L], ℝ^{3×3}), p > 1, for a sequence (Γ_{n}, D_{n}) of closed framed curves respecting (37), K[Γ_{n}] ≤ θ^{−1}, and

l(Γ_{n}, D_{n}) = l(Γ_{1}, D_{1}) for all n ∈ ℕ.  (44)

Then l(Γ, D) is well-defined and l(Γ, D) = l(Γ_{1}, D_{1}). (ii) Let (Γ, D) ∈ W^{1,q}([0, L], ℝ^{3}) × W^{1,p}([0, L], ℝ^{3×3}), p, q > 1, be a closed

framed curve satisfying (37) and K[Γ] ≤ θ^{−1}. Then there is ϵ > 0, such that l(Γ̃, D̃) is well-defined and

l(Γ, D) = l(Γ̃, D̃)  (45)

for all closed framed curves (Γ̃, D̃) ∈ W^{1,q}([0, L], ℝ^{3})×W^{1,p}([0, L], ℝ^{3×3}) satisfying

∥Γ − Γ̃∥_{W^{1,q}} ≤ ϵ, ∥D − D̃∥_{W^{1,p}} ≤ ϵ.  (46)

5 Proofs

We will first prove two technical lemmas, which will be quite useful in later proofs, too. The first one concerns the approximate limes superior and inferior of integrable bounded functions as defined, e.g., in [6, p.47].

Lemma 6. Let f ∈ L^{∞}(Ω) for some open set Ω ⊂ ℝ^{n} and let x ∈ Ω. Suppose E_{r} ⊂ B_{r}(x) such that there is a constant c with

|B_{r}(x)| ≤ c|E_{r}| for all r > 0.  (47)

Then

ap lim inf_{z→x} f(z) ≤ lim inf_{r→0} ∫−_{E_{r}} f(y) dy

≤ lim sup_{r→0} ∫−_{E_{r}} f(y) dy ≤ ap lim sup_{z→x} f(z).  (48)

If x is a Lebesgue point of f, then (48) holds with equality everywhere, and the limes superior and limes inferior may be replaced by the limit.

Proof. The approximate limes superior of f as z → x ∈ Ω is defined as

a := ap lim sup_{z→x} f(z) := inf\{s : lim_{r↓0} \frac{|B_{r}(x) ∩ \{f > s\}|}{|B_{r}(x)|} = 0\}.

We may assume that 0 ≤ a < ∞ (otherwise consider f + ∥f∥_{L^{∞}}). Thus for any ā > a,

∫−_{E_{r}} f(y)dy = \frac{1}{|E_{r}|}[ ∫_{E_{r}∩\{f>ā\}} f(y) dy + ∫_{E_{r}∩\{f≤ā\}} f(y) dy] =: I + II. (49)

Then we estimate

I = \frac{1}{|E_{r}|} ∫_{E_{r}∩\{f>ā\}} f(y) dy ≤ ∥f∥_{L^{∞}} · \frac{|E_{r} ∩ \{f > ā\}|}{|E_{r}|}  (50)
≤_{(47)} c∥f∥_{L^{∞}} · \frac{|B_{r} ∩ \{f > ā\}|}{|B_{r}|} = o(1) as r → 0,

as well as

II = \frac{1}{|E_{r}|} ∫_{E_{r}∩\{f≤ā\}} f(y) dy ≤ ā · \frac{|E_{r} ∩ \{f ≤ ā\}|}{|E_{r}|} ≤ ā.  (51)

Inserting (50) and (51) into (49) gives the right inequality in (48). One proceeds analogously for the left inequality. The last statement follows from the well-known fact that an integrable function is approximate continuous at all its Lebesgue points, in which case equality holds everywhere in (48), see [6, p.48]. ⊓ ⊔

The second lemma gives an explicit representation of the circumcircle radius R of three different points on a curve γ whose arc length parameterization Γ possesses weak second derivatives.

Lemma 7. Let γ ∈ L be simple with arc length parameterization Γ ∈ W^{2,1}([0, L], ℝ^{3}). For s_{1}, s_{2}, s_{3} ∈ S_{L} with s_{1} < s_{2} < s_{3}, one has

R(Γ (s_{1}), Γ (s_{2}), Γ (s_{3})) = \frac{∏3_{{i,k=1 \atop i<k}} C(s_{i}, s_{k})}{2|A(s_{1}, s_{2}, s_{3}) + (s_{2} − s_{1})B(s_{1}, s_{2}, s_{3})|}, (52)

where

C(s_{i}, s_{k}) := |Γ′(s_{k}) + ∫−_{s_{k}}^{s_{i}} ∫_{s_{k}}^{t} Γ′′(ω) dω dt|, i ≠ k,  (53)

A(s_{1}, s_{2}, s_{3}) := Γ′(s_{2}) ∧ ∫_{0}^{1} t ∫−_{s_{2}−t(s_{2}−s_{1})}^{s_{2}+t(s_{3}−s_{2})} Γ′′(ω) dω dt,  (54)

B(s_{1}, s_{2}, s_{3}) := ∫_{0}^{1} t ∫−_{s_{2}}^{s_{2}−t(s_{2}−s_{1})} Γ′′(ω) dω dt  (55)

∧ ∫_{0}^{1} t ∫−_{s_{2}−t(s_{2}−s_{1})}^{s_{2}+t(s_{3}−s_{2})} Γ′′(ω) dω dt.

If s ∈ S_{L} is a Lebesgue point of Γ′′, and if s_{j} < σ_{j} < τ_{j}, with (s_{j}, σ_{j}, τ_{j}) → (s, s, s) as j → ∞, such that

r_{j} := max{|s − s_{j}|, |s − τ_{j}|} ≤ c(τ_{j} − s_{j}),  (56)

where c > 0 is a constant independent of j, then

lim_{j→∞} R(Γ (s_{j}), Γ (σ_{j}), Γ (τ_{j})) = |Γ′′(s)|^{−1}.  (57)

If |Γ′′(s)| = 0, then (57) means that R(Γ (s_{j}), Γ (σ_{j}), Γ (τ_{j})) → ∞ as j → ∞.

Proof. Let us assume that Γ (s_{i}), i = 1, 2, 3, are not collinear. (They are distinct, since γ is simple and s_{i} ≠ s_{k} for i ≠ k.) Without loss of generality we assume that |s_{1} − s_{2}| ≤ |s_{2} − s_{3}|, (otherwise we choose a different substitution to derive (60) below). We consider the expansions

Γ (s_{i}) − Γ (s_{k}) = ∫_{s_{k}}^{s_{i}}(Γ′(τ) − Γ′(s_{k})) dτ + (s_{i} − s_{k})Γ′(s_{k})

= (s_{i} − s_{k})[Γ′(s_{k}) + ∫−_{s_{k}}^{s_{i}} ∫_{s_{k}}^{t} Γ′′(ω) dω dt],  (58)

Γ (s_{i}) − Γ (s_{k}) = ∫_{0}^{1} Γ′(s_{k} + t(s_{i} − s_{k})) dt(s_{i} − s_{k})  (59)

= (s_{i} − s_{k}) ∫_{0}^{1} [Γ′(s_{k}) + t(s_{i} − s_{k}) ∫_{0}^{1} Γ′′(s_{k} + σt(s_{i} − s_{k})) dσ] dt

= Γ′(s_{k})(s_{i} − s_{k}) + (s_{i} − s_{k})^{2} ∫_{0}^{1} t ∫_{0}^{1} Γ′′(s_{k} + σt(s_{i} − s_{k})) dσ dt.

In particular, substituting ρ(σ) := σ(s_{1} − s_{2})/(s_{3} − s_{2}), we deduce

Γ (s_{1}) − Γ (s_{2}) = Γ′(s_{2})(s_{1} − s_{2})  (60)

+(s_{1} − s_{2})(s_{3} − s_{2}) ∫_{0}^{1} t ∫0^{\frac{s_{1}−s_{2}}{s_{3}−s_{2}}} Γ′′(s_{2} + tρ(s_{3} − s_{2})) dρ dt.

Using (59) and (60) we compute

|(Γ (s_{1}) − Γ (s_{2})) ∧ (Γ (s_{3}) − Γ (s_{2}))|

= |[Γ′(s_{2})(s_{1} − s_{2}) + (s_{1} − s_{2})(s_{3} − s_{2}) ∫_{0}^{1} t ∫0^{\frac{s_{1}−s_{2}}{s_{3}−s_{2}}} Γ′′(s_{2} + tρ(s_{3} − s_{2})) dρ dt]

∧[Γ′(s_{2})(s_{3} − s_{2}) + (s_{3} − s_{2})^{2} ∫_{0}^{1} t ∫_{0}^{1} Γ′′(s_{2} + tρ(s_{3} − s_{2})) dρ dt]|

= |Γ′(s_{2})(s_{3} − s_{2})^{2}(s_{1} − s_{2}) ∧ ∫_{0}^{1} t ∫ 1_{\frac{s_{1}−s_{2}}{s_{3}−s_{2}}} Γ′′(s_{2} + tρ(s_{3} − s_{2})) dρ dt

+(s_{1} − s_{2})(s_{3} − s_{2})^{3} ∫_{0}^{1} t ∫0^{\frac{s_{1}−s_{2}}{s_{3}−s_{2}}} Γ′′(s_{2} + tρ(s_{3} − s_{2})) dρ dt

∧ ∫_{0}^{1} t ∫ 1_{\frac{s_{1}−s_{2}}{s_{3}−s_{2}}} Γ′′(s_{2} + tσ(s_{3} − s_{2})) dσ dt|

With a suitable substitution we get

|(Γ (s_{1}) − Γ (s_{2})) ∧ (Γ (s_{3}) − Γ (s_{2}))|

= |Γ′(s_{2})(s_{3} − s_{2})(s_{1} − s_{2}) ∧ ∫_{0}^{1} ∫_{s_{2}−t(s_{2}−s_{1})}^{s_{2}+t(s_{3}−s_{2})} Γ′′(ω) dω dt

+(s_{1} − s_{2})(s_{3} − s_{2}) ∫_{0}^{1} ∫_{s_{2}}^{s_{2}−t(s_{2}−s_{1})} Γ′′(ω) dω dt

∧ ∫_{0}^{1} ∫_{s_{2}−t(s_{2}−s_{1})}^{s_{2}+t(s_{3}−s_{2})} Γ′′(ω) dω dt|

= |s_{3} − s_{2}||s_{1} − s_{2}||s_{3} − s_{1}||Γ′(s_{2}) ∧ ∫_{0}^{1} t ∫−_{s_{2}−t(s_{2}−s_{1})}^{s_{2}+t(s_{3}−s_{2})} Γ′′(ω) dω dt  (61)

+(s_{2} − s_{1}) ∫_{0}^{1} t ∫−_{s_{2}}^{s_{2}−t(s_{2}−s_{1})} Γ′′(ω) dω dt ∧ ∫_{0}^{1} t ∫−_{s_{2}−t(s_{2}−s_{1})}^{s_{2}+t(s_{3}−s_{2})} Γ′′(ω) dω dt| .

Inserting (61) and (58) in (2) one verifies (52).

If Γ (s_{i}), i = 1, 2, 3, are collinear, then the right-hand side of (2) has vanishing denominator, which is then also true for the right-hand side in (52). That corresponds to an infinite value for R(Γ (s_{1}), Γ (s_{2}), Γ (s_{3})) in its definition.

To prove (57) we apply (48) in Lemma 6 to the mean value expression in the term A(s_{j}, σ_{j}, τ_{j}) as defined in (54) of Lemma 7. We set x = s, f := (Γ^{k})′′, k = 1, 2, 3, replace E_{r} in Lemma 6 by the set

E_{j}^{t} := [σ_{j} − t(σ_{j} − s_{j}), σ_{j} + t(τ_{j} − σ_{j})]

and B_{r} by the B_{r_{j}}(s). For given ϵ > 0 we choose δ := ϵ∥Γ′′∥_{L^{∞}}^{−1}/2. Notice that (56) implies |B_{r_{j}}| ≤ c|E_{j}^{t}|/δ, i.e., condition (47) in Lemma 6 for each t ≥ δ. According to the last statement of Lemma 6 we get

lim_{j→∞} ∫−_{E_{j}^{t}} Γ′′(ω) dω = Γ′′(s) for all t ∈ [δ, 1].

Since Γ′′ ∈ L^{∞}([0, L], ℝ^{3}) we may apply the Dominated Convergence Theorem to conclude

lim_{j→∞} ∫_{δ}^{1} t ∫−_{E_{j}^{t}} Γ′′(ω) dωdt = Γ′′(s) [\frac{1 − δ^{2}}{2} ] .

Hence there is j_{0} such that for all j ≥ j_{0}

∫_{0}^{1} t ∫−_{E_{j}^{t}} Γ′′(ω) dωdt = Γ′′(s) [\frac{1 − δ^{2}}{2} ] + ϵ

by our choice of δ. Since δ → 0 as ϵ → 0 we have shown

lim_{j→∞} ∫_{0}^{1} t ∫−_{E_{j}^{t}} Γ′′(ω) dωdt = Γ′′(s)/2,

hence lim_{j→∞} A(s_{j}, σ_{j}, τ_{j}) = Γ′(s) ∧ Γ′′(s)/2.

Since B(s_{j}, σ_{j}, τ_{j}) is bounded by 4^{−1}∥Γ′′∥_{L^{∞}}^{2}, for all j ∈ ℕ, and the numerator in (52) for s_{1} = s_{j}, s_{2} = σ_{j}, s_{3} = τ_{j}, converges to 1 as j → ∞, the claim follows from the fact that |Γ′(σ)| = 1, i.e., Γ′(σ) ⊥ Γ′′(σ) for all σ ∈ S_{L}. ⊓ ⊔

Proof of Theorem 1. Part (i) was proved in [12, Lemma 1].

A careful inspection of the proof of [12, Lemma 2] shows that the local information ρ := ρ_{G}[γ](s) > 0 is sufficient to carry over steps 1–8 of that proof to prove (ii) and the necessity in part (iii). For the first step (and similarly for step 5) one merely needs to take the connected arc A_{1} := Γ ([s, σ_{1}]) with |σ_{1} − s| so small that diam A_{1} < 2ρ, |P_{1} − P_{0}| < ρ/2 and Γ (τ) ≠ Γ (σ_{1}) for all τ ∈ (s, σ_{1}). This is possible, since otherwise we could find a sequence τ_{i} → s with Γ (τ_{i}) = Γ (σ_{1}), which would imply that Γ (s) = Γ (σ_{1}). But Γ (s) is not a double point by virtue of part (i), because ρ > 0.

To prove the converse in (iii) we argue as follows. If γ is simple and Γ ∈ C^{1,1}([0, L], ℝ^{3}), we take a minimal sequence of triples of distinct parameters s_{j}, σ_{j}, τ_{j} ∈ S_{L} for R[γ] using the characterization (6). Taking a subsequence we may assume that (s_{j}, σ_{j}, τ_{j}) → (s, σ, τ) ∈ S_{L} × S_{L} × S_{L}, since S_{L} is compact, and we are going to look at the different possible limit cases, where we will repeatedly use (8).

Case I. If s ≠ σ ≠ τ ≠ s then also Γ (s) ≠ Γ (σ) ≠ Γ (τ) ≠ Γ (s), since Γ is simple by assumption. This means that according to the representation (2)

R[γ] = lim_{j→∞} R(Γ (s_{j}), Γ (σ_{j}), Γ (τ_{j})) = \frac{|Γ (s) − Γ (σ)|}{|\frac{Γ (s)−Γ (τ)}{|Γ (s)−Γ (τ)|} ∧ \frac{Γ (σ)−Γ (τ)}{|Γ (σ)−Γ (τ)|} |} (62)

unless Γ (s_{j}), Γ (σ_{j}) and Γ (τ_{j}) are collinear for infinitely many j ∈ ℕ, in which case R[γ] = ∞, i.e., K[γ] = 0. (In fact, the latter cannot occur for closed curves γ ∈ L according to Lemma 1.) If Γ (s_{j}), Γ (σ_{j}) and Γ (τ_{j}) are not collinear, the denominator on the right of (62) is positive and bounded from above by 2, whereas the numerator does not vanish, since γ is assumed to be simple, hence R[γ] > 0, i.e., K[γ] < ∞.

Case II. If two but not all of the three parameters s, σ, τ coincide, say s = σ ≠ τ, then we can use (2) again for R(Γ (s_{j}), Γ (σ_{j}), Γ (τ_{j})). Before going to the limit we select a subsequence such that the difference s_{j} − σ_{j} has the same sign for all j. Since Γ ∈ C^{1,1}([0, L], ℝ^{3}) we may use Taylor’s expansion to prove

lim_{j→∞} \frac{Γ (s_{j}) − Γ (σ_{j})}{|Γ (s_{j}) − Γ (σ_{j})|} = ±Γ′(s).  (63)

Consequently, using the symmetry of R(., ., .),

R[γ] = lim_{j→∞} R(Γ (s_{j}), Γ (σ_{j}), Γ (τ_{j})) = lim_{j→∞} R(Γ (τ_{j}), Γ (s_{j}), Γ (σ_{j})) (64)
= lim_{j→∞} \frac{|Γ (τ_{j}) − Γ (s_{j})|}{2|\frac{Γ (τ_{j})−Γ (σ_{j})}{|Γ (τ_{j})−Γ (σ_{j})|} ∧ \frac{Γ (s_{j})−Γ (σ_{j})}{|Γ (s_{j})−Γ (σ_{j})|} |} = \frac{|Γ (τ) − Γ (s)|}{2|\frac{Γ (τ)−Γ (s)}{|Γ (τ)−Γ (s)|} ∧ Γ′(s)|} ,

where the denominator is bounded from above by 2, and the numerator is different from 0 as before, since γ is simple, hence R[γ] > 0 and K[γ] < ∞ in this case.

Case III. If s = σ = τ, we apply Lemma 7. Notice that in our situation Γ′′ ∈ L^{∞}([0, L], ℝ^{3}), and ∥Γ′′∥_{L^{∞}} > 0, (otherwise Γ′ ≡ const. and γ could not be closed). So

|A(s_{1}, s_{2}, s_{3})| ≤ ∥Γ′′∥_{L^{∞}}/2,  (65)
|B(s_{1}, s_{2}, s_{3})| ≤ ∥Γ′′∥_{L^{∞}}^{2}/4,  (66)
|C(s_{i}, s_{k})| ≥ 1 − |s_{i} − s_{k}|, for i ≠ k.  (67)

By symmetry of R(., ., .) we may assume that s_{j} < τ_{j} < σ_{j} for all j ∈ ℕ, and by (52),(65)–(67) we obtain

R(Γ (s_{j}), Γ (σ_{j}), Γ (τ_{j})) ≥ \frac{1 + o(1)}{∥Γ′′∥_{L^{∞}} · (1 + o(1))} as j → ∞.  (68)

This means that

R[γ] = lim_{j→∞} R(Γ (s_{j}), Γ (σ_{j}), Γ (τ_{j})) ≥ ∥Γ′′∥_{L^{∞}}^{−1} > 0,  (69)

i.e., K[γ] < ∞ in this case, which concludes the proof of sufficiency in part (iii).

Part (iv) (b) was shown in [12, Lemma 3]. To prove (iv) (a), we argue as follows. Set ρ := ρ_{G}[γ](s) > 0 and let s_{n} → s, P_{n} := Γ (s_{n}), P_{0} := Γ (s) and C_{n} := { z ∈ ℝ^{3} | |z − P_{0}| = |z − P_{n}| = ρ }. Notice that C_{n} is the circle of radius ρ_{n} := \sqrt{ρ^{2} − |P_{n} − P_{0}|^{2}/4} centred at y_{n} := (P_{n} + P_{0})/2 and perpendicular to the unit vector (P_{n} − P_{0})/|P_{n} − P_{0}|. We claim that

dist_{H}(C_{n}, C(s, ρ)) → 0 as n → ∞,  (70)

where C(s, ρ) is the circle defined in the statement of the theorem (for θ = ρ), and dist_{H}(A, B) denotes the Hausdorff distance [7, p. 183] between two subsets A, B of ℝ^{3}. To establish this result, we note first that ρ_{n} → ρ and y_{n} → P_{0}. Moreover, by part (ii), the only possible limits for (P_{n} − P_{0})/|P_{n} − P_{0}| as n → ∞ are ±T (s). Thus C_{n} converges to a circle of radius ρ with centre P_{0} in the plane perpendicular to T (s). Since these properties completely characterize C(s, ρ) our claim in (70) follows.

To establish (iv)(a), we consider the sets Ξ_{n} := ⋃_{z∈C_{n}} B_{ρ}(z). We assume for contradiction that there is a point P̄ ∈ Γ (S_{L}) ∩ M(s, ρ), which implies dist(P̄, C(s, ρ)) < ρ. For n ∈ ℕ sufficiently large, we deduce from (70) that dist(P̄, C_{n}) < ρ, which implies P̄ ∈ Ξ_{n}, and moreover we have |P̄ − P_{0}| > |P_{n} − P_{0}|. These observations lead to the result P̄ ∈ Ξ_{n}\\overline{l_{n}}, where l_{n} := ⋂_{z∈C_{n}} B_{ρ}(z). By exactly the same arguments as in the proof of [12, Lemma 2] we arrive at a contradiction to the definition of ρ = ρ_{G}[γ](s), so claim (iv)(a) must be true. ⊓ ⊔

Proof of Lemma 1. Only the second inequality in (13) requires a proof.

(14) follows immediately from (8). If ρ_{G}[γ](s) = 0 the statement is trivial, so assume that ρ := ρ_{G}[γ](s) > 0, which in particular means that the curve has positive length. Notice that we did not assume that R[γ] > 0, hence we may not have a tangent of Γ at s. However, the geometric tangent T (s) exists according to part (ii) of Theorem 1, and we may apply part (iv) (a) of the same theorem to find that the curve Γ does not intersect M(s, ρ). On the other hand, by (9), Γ intersects the plane E through Γ (s) perpendicular to T (s) transversally. Because Γ (s) is not a double point (ρ_{G}[γ](s) > 0), Γ must intersect this plane in at least one different point Γ (σ) ≠ Γ (s), since γ is closed. Then Γ (σ) ∈ E\M(s, ρ), i.e., |Γ (s) − Γ (σ)| ≥ 2ρ, which proves the lemma. ⊓ ⊔

Proof of Theorem 2. It suffices to prove the first relation in (15) and the identities (16),(19) and (21), since the remaining identities follow from

(i) If R[γ] = 0, then we know by Theorem 1, part (iii), that γ is not simple, i.e., there exist t ≠ s such that Γ (s) = Γ (t). But then, by Theorem 1, part (i), ρ_{G}[γ](s) = ρ_{G}[γ](t) = 0, thus (i) is trivially true in this case.

If R[γ] > 0, we argue as follows. Taking a minimizing sequence of triples of distinct parameters s_{j}, σ_{j}, τ_{j} ∈ S_{L} with R(Γ (s_{j}), Γ (σ_{j}), Γ (τ_{j})) → R[γ], which is possible according to (6), we may assume that (s_{j}, σ_{j}, τ_{j}) → (s, σ, τ) as j → ∞ by compactness. Note that R[γ] is bounded by Lemma 1 and that γ is simple, by Theorem 1 (iii), hence we may assume that Γ (s_{j}), Γ (σ_{j}) and Γ (τ_{j}) are distinct and non-collinear for all j ∈ ℕ.

As in the proof of part (iii) of Theorem 1 we distinguish three different cases.

Case I. If s ≠ σ ≠ τ then we can use (2) to obtain

R[γ] = lim_{j→∞} R(Γ (s_{j}), Γ (σ_{j}), Γ (τ_{j}))
= R(Γ (s), Γ (σ), Γ (τ)) ≥ ρ_{G}[γ](s) ≥ R[γ],  (71)

hence equality holds everywhere, which proves (15) in this case. Observe that also Γ (s), Γ (σ), Γ (τ) are distinct and non-collinear here.

Case II. If two but not all of the three parameters coincide, say s = σ ≠ τ we can repeat the calculation that led to (64) in the proof of Theorem 1, part (iii) to obtain

R[γ] = lim_{j→∞} R(Γ (s_{j}), Γ (σ_{j}), Γ (τ_{j})) = \frac{|Γ (τ) − Γ (s)|}{2|\frac{Γ (τ)−Γ (s)}{|Γ (τ)−Γ (s)|} ∧ Γ′(s)|} . (72)

Notice that Γ is differentiable everywhere, since K[γ] < ∞ now. According to Theorem 1 we may insert in (72)

Γ′(s) = T (s) = lim_{{t_{k}↓s \atop t_{k}≠s}} \frac{Γ (t_{k}) − Γ (s)}{|Γ (t_{k}) − Γ (s)|}

to obtain by the definition of ρ_{G}[γ](τ)

R[γ] = lim_{{t_{k}↓s \atop t_{k}≠s}} \frac{|Γ (τ) − Γ (s)|}{2|\frac{Γ (τ)−Γ (s)}{|Γ (τ)−Γ (s)|} ∧ \frac{Γ (t_{k})−Γ (s)}{|Γ (t_{k})−Γ (s)|} |}

= lim_{{t_{k}↓s \atop t_{k}≠s}} R(Γ (τ), Γ (s), Γ (t_{k})) ≥ ρ_{G}[γ](τ) ≥ R[γ],

which implies equality, i.e.,(15).

Case III. If s = σ = τ we have (16), which finishes the proof of the first statement in (i).

In order to apply (57) of Lemma 7 in the case when s ∈ S_{L} is a Lebesgue point of Γ′′ we may assume a fixed ordering, say s_{j} < σ_{j} < τ_{j}, by taking subsequences, because we know from (16) that the limit as j → ∞ exists. Condition (18) then implies (56), hence we infer from (57) and (16) that

R[γ] = |Γ′′(s)|^{−1}.  (73)

As pointed out in the first remark following Theorem 2, any sequence of the form (s, σ_{j}, τ_{j}) → (s, s, s) respects (18) as well. Applying (73) to a minimizing sequence (s, σ_{j}, τ_{j}) with lim_{j→∞} R(Γ (s), Γ (σ_{j}), Γ (τ_{j})) = ρ_{G}[γ](s) leads to (19).

(ii) Fix s ∈ S_{L} and take a minimal sequence of pairs (u_{k}, t_{k}) with s ≠ u_{k} ≠ t_{k} ≠ s realizing the infimum in the definition of ρ := ρ_{G}[γ](s), i.e., with

lim_{k→∞} R(Γ (s), Γ (u_{k}), Γ (t_{k})) = ρ_{G}[γ](s).  (74)

As before by taking a subsequence we can assume that (u_{k}, t_{k}) → (σ, τ) ∈ S_{L} × S_{L}. We are going to distinguish between three different cases.

Case I. s = σ = τ, this is case (a) of the statement of the theorem and we are done.

Case II. s ≠ σ ≠ τ. In this case we will show, that γ intersects or touches non-transversally the unique sphere of radius ρ, which contains the circumcircle of Γ (s), Γ (σ) and Γ (τ) as a great circle. (Note that Γ (s), Γ (σ) and Γ (τ) are distinct and non-collinear, since γ is simple (R[γ] > 0) and because R[γ] is bounded by Lemma 1.) In other words, if y ∈ ℝ^{3} is the centre of that sphere, then we claim that

[(Γ (s) − y) · Γ′(s)][(Γ (σ) − y) · Γ′(σ)][(Γ (τ) − y) · Γ′(τ)] = 0.  (75)

In fact, if (75) were false, we could find a slightly (but strictly) smaller sphere within the previous one and with center y_{1} ∈ [y, Γ (s)], which touches the original sphere in Γ (s) and still contains two other distinct points Γ (σ_{1}), Γ (τ_{1}) close to Γ (σ) and Γ (τ), respectively. Intersecting this smaller sphere with the plane through Γ (s), Γ (σ_{1}) and Γ (τ_{1}) gives a circle with radius strictly smaller than ρ contradicting the definition of ρ = ρ_{G}[γ](s), which proves our claim. (Notice that also Γ (s), Γ (σ_{1}) and Γ (τ_{1}) are non-collinear, hence span a plane for Γ (σ_{1}) and Γ (τ_{1}) sufficiently close to Γ (σ) and Γ (τ), respectively.)

Now by (75) we know that γ is tangential to the original sphere centred at y ∈ ℝ^{3} with radius ρ in at least one of the points Γ (s), Γ (τ), Γ (σ), say in Γ (τ). Intersecting this sphere with the plane P spanned by the (noncollinear) vectors Γ (s) − Γ (τ) and Γ′(τ) we obtain a circle C with radius r(C), which satisfies

ρ ≥ r(C) = \frac{|Γ (s) − Γ (τ)|}{2|\frac{Γ (s)−Γ (τ)}{|Γ (s)−Γ (τ)|} ∧ Γ′(τ)|} = lim_{j→∞} R(Γ (s), Γ (τ), Γ (σ_{j})) ≥ ρ, (76)

for a sequence σ_{j} → τ as j → ∞. This follows from (28) and the calculation that led to (64) in the proof of Theorem 1. Hence we have equality everywhere in (76), which proves that in this situation we have (21) (b) with the sequence (σ_{j}, τ_{j}) → (τ, τ) as j → ∞, where τ_{j} := τ for all j ∈ ℕ.

Case III. If two but not all of the three parameters s, σ, τ coincide we have two subcases.

IIIa If τ ≠ s = σ (the case σ ≠ τ = s can be treated analogously) one can go to the limit in (74) as in the calculation that led to (64) to get

ρ_{G}[γ](s) = lim_{k→∞} R(Γ (s), Γ (u_{k}), Γ (t_{k})) = \frac{|Γ (τ) − Γ (s)|}{2|\frac{Γ (τ)−Γ (s)}{|Γ (τ)−Γ (s)|} ∧ Γ′(s)|}.

By (28), the expression on the right equals the radius of the unique circle C through Γ (s) and Γ (τ) tangent to Γ in Γ (s). If ∂B_{ρ}(y) is the unique sphere containing C as a great circle then B_{ρ}(y) ⊂ M(s, ρ) as defined in Theorem 1, part (iv), which implies that Γ must also be tangential to ∂B_{ρ}(y) in the point Γ (τ), i.e., Γ′(τ)·(Γ (τ)−y) = 0. Now we look at the circle C^{∗} obtained by intersecting ∂B_{ρ}(y) with the plane spanned by the vectors Γ (s) − Γ (τ) and Γ′(τ). Then we get for the radius r(C^{∗}) of C^{∗}

ρ ≥ r(C^{∗}) = lim_{l→∞} R(Γ (s), Γ (τ), Γ (v_{l}))
≥ ρ_{G}[γ](s) = ρ,  (77)

for a sequence v_{l} → τ as l → ∞. Hence we have equality everywhere, which implies that we have (21) with (b) with the sequence (v_{l}, τ) converging to (τ, τ) as l → ∞.

IIIb If s ≠ τ = σ we are immediately in case (b) of the statement in part (ii) of the theorem, and we are done, which finishes the proof. ⊓ ⊔

Proof of Proposition 1. (i) We take a sequence (τ_{j}, σ_{j}) → (s, s), s ≠ τ_{j} ≠ σ_{j} ≠ s, realizing the limes superior in the definition of κ[γ](s), see (24). If s < τ_{j} < σ_{j} for infinitely many j ∈ ℕ we apply (48) of Lemma 6 to the mean value expression in the term A(s, τ_{j}, σ_{j}) as defined in (54) of Lemma 7. For that set x = s, replace E_{r} in Lemma 6 by the set

E_{j}^{t} := [τ_{j} − t(τ_{j} − s), τ_{j} + t(σ_{j} − τ_{j})],

and B_{r} by B_{j} := B_{σ_{j}−s}(s) and set f := |Γ′′|. Then for given ϵ > 0 we choose

δ := ϵ∥Γ ∥_{L^{∞}}^{−1}/2.  (78)

Notice that according to Lemma 6 we can find j_{0} sufficiently large, such that for all j ≥ j_{0} and t ∈ [δ, 1]

∫−_{τ_{j}−t(τ_{j}−s)}^{τ_{j}+t(σ_{j}−τ_{j})} |Γ′′(ω)| dω ≤ ap lim sup_{z→s} |Γ′′(z)| + ϵ/2,  (79)

Combining (78) and (79) we obtain

| ∫_{0}^{1} t ∫−_{E_{j}^{t}} Γ′′(ω) dωdt| ≤ ap lim sup_{z→s} |Γ′′(z)| + ϵ

for all j ≥ j_{0}. Consequently,

κ[γ](s) = lim_{j→∞} \frac{1}{R(Γ (s), Γ (τ_{j}), Γ (σ_{j}))}
=_{(52)} lim_{j→∞} \frac{2|A(s, τ_{j}, σ_{j}) + (τ_{j} − s)B(s, τ_{j}, σ_{j})|}{C(s, τ_{j})C(s, σ_{j})C(τ_{j}, σ_{j})}
≤ lim sup_{j→∞} 2 ∫_{0}^{1} ∫−_{E_{t}^{j}} |Γ′′(ω)| dω dt ≤ ap lim sup_{z→s} |Γ′′(z)|.

Here we have used the fact that |B(s, τ_{j}, σ_{j})| ≤ ∥Γ′′∥_{L^{∞}}^{2}/4 and that C(s, τ_{j}), C(s, σ_{j}) and C(τ_{j}, σ_{j}) converge to 1 as j → ∞.

If τ_{j} < s < σ_{j} for infinitely many j ∈ ℕ, use A(τ_{j}, s, σ_{j}) and B(τ_{j}, s, σ_{j}) instead; all the other possible orderings of an infinite number of the parameters s, σ_{j}, τ_{j} can be treated similarly.

(ii) One simply needs to recall that a.e. point s ∈ S_{L} is a Lebesgue point of Γ′′, since Γ ∈ C^{1,1} ≃ W^{2,∞}. This means that we can use (57) in Lemma 7 for s_{j} = s for all j ∈ ℕ, to replace the limes superior in the definition of κ[γ](s), (24), by the limit as j → ∞ to obtain (ii). Notice that for the sequence (τ_{j}, σ_{j}) → (s, s) with s ≠ τ_{j} ≠ σ_{j} ≠ s realizing the limes superior we may assume a fixed ordering (τ_{j} < σ_{j} < s, or σ_{j} < s < τ_{j}, etc. for all j ∈ ℕ) by taking suitable subsequences. Thus Lemma 7 is applicable, because (56) is automatically satisfied if s = s_{j} for all j ∈ ℕ, as pointed out in the first remark following Theorem 2. ⊓ ⊔

Proof of Lemma 2. Notice first that it suffices to prove the statements for ρ_{G} and R, the relations for κ_{G}, and K follow from (8).

It is easy to see that the right-hand side of (A) (and (B)) is bounded from above by diam (γ(Ī))/2, since we may consider the continuously differentiable function, (recall that the assumption K[γ] < ∞ leads to a Lipschitz continuous tangent field Γ′ by Theorem 1), f(σ) := |Γ (s)−Γ (σ)|^{2}, bounded by ( diam (γ(Ī)))^{2}. Here s ∈ S_{L} is fixed. Notice that f attains its maximum on S_{L} by compactness. Let τ ∈ S_{L} be such that f(τ) is maximal. Then Γ (S_{L}) is tangent to the sphere of radius δ := |Γ (s) − Γ (τ)|/2 centred at m := (Γ (s) + Γ (τ))/2 in the point Γ (τ), since f′(τ) = 0. Any great circle through Γ (s) and Γ (τ) and tangent to Γ′(τ) has radius δ ≤ diam (γ(Ī))/2, hence also the infimum of such radii must have that upper bound.

Taking a minimal sequence {σ_{j}} ⊂ S_{L} for the right-hand side of (A), i.e., with r(Γ (s), Γ (σ_{j}), Γ′(σ_{j})) tending to the infimum in (A), we find for a given ϵ > 0 an index j_{0} such that for all j ≥ j_{0}

r(Γ (s), Γ (σ_{j}), Γ′(σ_{j})) ≤ inf_{{σ∈S_{L} \atop σ≠s}} r(Γ (s), Γ (σ), Γ′(σ)) + ϵ/2.  (80)

Since the right-hand side of (A) is finite, the same is true for the left-hand side of (80) for all j ≥ j_{0}, which means that the vectors Γ (s) − Γ (σ_{j}) and Γ′(σ_{j}) are non-collinear for j ≥ j_{0}. Hence we can apply (28). On the other hand, approximating the left-hand side of (80) for j = j_{0} as we did to derive (64), we find for the above ϵ some t ∈ S_{L} close to σ_{j_{0}} such that

R(Γ (s), Γ (σ_{j_{0}}), Γ (t)) ≤ inf_{{σ∈S_{L} \atop σ≠s}} r(Γ (s), Γ (σ), Γ′(σ)) + ϵ,

which immediately gives

ρ_{G}[γ](s) ≤ inf_{{σ∈S_{L} \atop σ≠s}} r(Γ (s), Γ (σ), Γ′(σ)).  (81)

Analogously one obtains

ρ_{G}[γ](s) ≤ inf_{{σ∈S_{L} \atop σ≠s}} r(Γ (σ), Γ (s), Γ′(s)).  (82)

For the reverse inequalities we first note that according to (ii) of Theorem 2 we can represent ρ_{G}[γ](s) as limit as in (21) with two possible cases. In case (ii)(b) of that theorem we are done, since then by the same calculation that led to (64),

ρ_{G}[γ](s) = r(Γ (s), Γ (τ), Γ′(τ)) ≥ inf_{{σ∈S_{L} \atop σ≠s}} r(Γ (s), Γ (σ), Γ′(σ)).  (83)

In case (ii)(a) there is a sequence (σ_{j}, τ_{j}) → (s, s), such that

ρ_{G}[γ](s) = lim_{j→∞} R(Γ (s), Γ (σ_{j}), Γ (τ_{j})) ≥ ρ[γ](s) ≥_{(25)} ρ_{G}[γ](s),  (84)

by definition of ρ[γ], see (23), hence equality holds.

Remaining in case (ii)(a), we renumber the minimal sequence (σ_{j}, τ_{j}) → (s, s) appropriately, so that we can assume that for each j ∈ ℕ

θ_{j} := R(Γ (s), Γ (σ_{j}), Γ (τ_{j})) ≤ ρ_{G}[γ](s) + \frac{1}{j} .  (85)

Then for fixed j ∈ ℕ, and θ_{j}^{∗} := inf_{{t∈S_{L} \atop t≠s,σ_{j}}} R(Γ (s), Γ (σ_{j}), Γ (t)) ≤ θ_{j}, we can

choose a minimal sequence t_{k} converging w.l.o.g. to some τ_{j}^{∗}, such that

θ_{j}^{∗} = lim_{k→∞} R(Γ (s), Γ (σ_{j}), Γ (t_{k})),  (86)

where s ≠ σ_{j} ≠ t_{k} ≠ s for all k ∈ ℕ. There are three cases in this situation:

Case I. If τ_{j}^{∗} = σ_{j} (≠ s), then with the same calculation that led to (64) before, we obtain

θ_{j}^{∗} = r(Γ (s), Γ (σ_{j}), Γ′(σ_{j})) ≥ inf_{{σ∈S_{L} \atop σ≠s}} r(Γ (s), Γ (σ), Γ′(σ)),  (87)

and by (85)–(87) we get as j → ∞

ρ_{G}[γ](s) ≥ inf_{{σ∈S_{L} \atop σ≠s}} r(Γ (s), Γ (σ), Γ′(σ)).  (88)

Case II. If τ_{j}^{∗} = s (≠ σ_{j}), then we obtain in the same way as in Case I

ρ_{G}[γ](s) ≥ inf_{{σ∈S_{L} \atop σ≠s}} r(Γ (σ), Γ (s), Γ′(s)).  (89)

Case III. If τ_{j}^{∗} ≠ s ≠ σ_{j} ≠ τ_{j}^{∗}, then we can go to the limit k → ∞ in formula (2) for R(., ., .) to obtain θ_{j}^{∗} = R(Γ (s), Γ (σ_{j}), Γ (τ_{j}^{∗})). Let ∂B_{θ_{j}^{∗}}(y_{j}) be the unique sphere of radius θ_{j}^{∗} containing the unique circumcircle of Γ (s), Γ (σ_{j}), Γ (τ_{j}^{∗}) as a great circle. Then we claim that

Γ′(τ_{j}^{∗}) ⊥ Γ (τ_{j}^{∗}) − y_{j}.  (90)

This is in fact true, since otherwise we could find Γ (τ̃) with τ̃ close to τ_{j}^{∗}, such that

R(Γ (s), Γ (σ_{j}), Γ (τ̃)) < θ_{j}^{∗},  (91)

which would contradict the definition of θ_{j}^{∗} in (86) as infimum over such radii. (91) holds for the following reason: Consider the triangle △ with vortices Γ (s), Γ (σ_{j}), Γ (τ_{j}^{∗}). Recall that we are in case (ii)(a) of Theorem 2 now, which means that by taking j sufficiently large from the beginning on we may assume that |Γ (s) − Γ (σ_{j})| < R[γ]/2, i.e., that the angle α of △ in Γ (τ_{j}^{∗}) is either in (0, π/2) or in (π/2, π). In either case, considering the perturbed triangle ˜△ with vortices Γ (s), Γ (σ_{j}), Γ (τ̃) for Γ (τ̃) ∈ B_{θ_{j}^{∗}}(y_{j}) with τ̃ close to τ_{j}^{∗}, leads to (91). In fact, intersecting the plane spanned by Γ (s), Γ (σ_{j}), Γ (τ̃) with the ∂B_{θ_{j}^{∗}}(y_{j}) gives a circle ∂D_{ϑ} with radius ϑ ≤ θ_{j}^{∗}, and with Γ (τ̃) contained in the open disk D_{ϑ}. (Notice that, since θ_{j}^{∗} is finite, the points Γ (s), Γ (σ_{j}), Γ (τ_{j}^{∗}) are non-collinear, which remains true if we replace Γ (τ_{j}^{∗}) by Γ (τ̃).) Also the angle α̃ of ˜△ in Γ (τ̃) is in the same interval as α (either (0, π/2) or (π/2, π)), for |τ̃ − τ_{j}^{∗}| sufficiently small. But then

R(Γ (s), Γ (σ_{j}), Γ (τ̃)) < R(Γ (s), Γ (σ_{j}), ξ) = ϑ ≤ θ_{j}^{∗}

for any comparison point ξ ∈ ∂D_{ϑ}, as one easily checks with (2). Thus we have shown (91) and hence also (90).

Now we intersect the plane P_{j} spanned by the two vectors Γ (s) − Γ (τ_{j}^{∗}) and Γ′(τ_{j}^{∗}) with ∂B_{θ_{j}^{∗}}(y_{j}) to obtain a circle C_{j}^{∗} through Γ (s) and Γ (τ_{j}^{∗}) and tangent to Γ′(τ_{j}^{∗}) with radius r(C_{j}^{∗}) ≤ θ_{j}^{∗}. On the other hand, we may write

θ_{j}^{∗} ≥ r(C_{j}^{∗}) = r(Γ (s), Γ (τ_{j}^{∗}), Γ′(τ_{j}^{∗})) ≥ inf_{{σ∈S_{L} \atop σ≠s}} r(Γ (s), Γ (σ), Γ′(σ)),

which together with (85)–(86) gives the desired inequality

ρ_{G}[γ](s) ≥ inf_{{σ∈S_{L} \atop σ≠s}} r(Γ (s), Γ (σ), Γ′(σ)).  (92)

Summarizing (81), (82), (83), (88), (89), (92) leads to the first equality in (A) and (B), respectively. The second equality in (B) was shown in (84).

Notice that ρ_{G}[γ](s) = ρ[γ](s) may or may not hold in alternative (A). Taking the infimum over s ∈ S_{L} on both sides of the equations (A) and (B) gives (29). ⊓ ⊔

Proof of Proposition 2. By Lemma 1 we have K[γ] > 0. Hence the expression r(Γ (s), Γ (σ), Γ′(σ)) is finite for all (s, σ) ∈ A[γ], therefore s ≠ σ by definition of r(., ., .).

Let (s, σ) ∈ A[γ]. If (32) is true, then Γ (s), Γ (σ) lie diametrically on the unique circle C through these points, which is tangential to Γ′(σ) in Γ (σ), with radius r(Γ (s), Γ (σ), Γ′(σ)) = K[γ]^{−1} = R[γ]. Theorem 1, part (iv) (b) gives M(σ, R[γ]) ∩ Γ (S_{L}) = ∅, which means that in the points Γ (s) and Γ (σ), the curve Γ is tangential to the unique sphere of radius R[γ] that contains C as a great circle. Now (33) follows.

It remains to show (32). We start with a technical lemma.

Lemma 8. Let ∥Γ′′∥_{L^{∞}} ≤ l, and assume that Γ (s), Γ (t) ∈ C_{R} where s ≠ t, and where C_{R} is a circle of radius R < l^{−1} tangential to Γ′(t). Then

|s − t| > \frac{1 − Rl}{l} .  (93)

Proof. Without loss of generality we may assume s > t, and choose the coordinate system such that Γ (t) = 0, Γ′(t) = (1, 0, 0), and that the center of C_{R} is the point (0, 0, R). Then we have the expansions

Γ^{3}(s) = ∫_{t}^{s} Γ^{3}′(τ) dτ = ∫_{t}^{s}(Γ^{3}′(τ) − Γ^{3}′(t)) dτ,
Γ^{1}(s) = Γ^{1}′(t)(s − t) + ∫_{t}^{s}(Γ^{1}′(τ) − Γ^{1}′(t)) dτ,
∫_{t}^{s}(Γ^{j}′(τ) − Γ^{j}′(t)) dτ = ∫_{t}^{s} ∫_{t}^{τ} Γ^{j}′′(σ) dσdτ,

which immediately gives the estimates

|Γ^{3}(s)| ≤ l|s − t|^{2}/2,  |Γ^{1}(s)| ≥ |s − t| − l|s − t|^{2}/2.  (94)

If the right-hand side of (94) is non-positive then |s − t| ≥ 2/l > (1 − Rl)/l, and we are done. If on the other hand, the right-hand side of (94) is positive, we estimate for Γ (s) ∈ C_{R}

R^{2} ≥ (Γ^{1}(s))^{2} + (Γ^{3}(s) − R)^{2}
≥ [1 − Rl − l|s − t|]|s − t|^{2} + R^{2} + l^{2}|s − t|^{4}/4 > R^{2},

unless the term in brackets is negative, which means |s − t| > (1 − Rl)/l. ⊓ ⊔

To prove (32) we assume the contrary, i.e., that for (s, t) ∈ A[γ], P := Γ (s), Q := Γ (t), θ := R[γ], 2ρ := |P − Q| < 2θ. Then m := (P + Q)/2 ∈ B_{θ}(Γ (S_{L})), where B_{θ}(Γ (S_{L})) is regular according to Proposition 3, which is proved independently. Hence we can look at the continuous projection Π_{Γ} and conclude that Π_{Γ} (m) ≠ P, Q, since |P − m| = |Q − m| = ρ, P ≠ Q, since s ≠ t, but the projection Π_{Γ} (m) is a unique point on Γ. Hence there is a unique S ≠ P, Q, S ∈ Γ (S_{L}) ∩ B_{ρ}(m) with Π_{Γ} (m) = S. Let ∂B_{θ}(y) be the sphere of radius θ containing the unique circle through Γ (s), Γ (t), tangent to Γ′(t) in Γ (t), as a great circle. Then by Theorem 1, part (iv) (b), we know that S ∉ B_{θ}(y) ⊂ M(t, θ).

Case I. Let S ∉ \overline{B_{θ}(y)}. The angle of the triangle △(P, Q, S) in S lies in (π/2, π), since S ∈ B_{ρ}(m). Consequently,

R(Q, P, S) = \frac{|Q − P |}{2| sin <)(Q − S, P − S)|}
< \frac{|Q − P |}{2| sin <)(Q − S^{∗}, P − S^{∗})|} = R(P, Q, S^{∗}) ≤ θ,  (95)

for any S^{∗} ∈ ∂B_{θ}(y) ∩ E^{∗}, S^{∗} ≠ P, Q, where E^{∗} is the plane spanned by P, Q, S. (95) contradicts the definition of θ = R[γ].

Case II. If S ∈ ∂B_{θ}(y), then we claim that we find S̃ ∈ Γ (S_{L}) arbitrarily close to S with S̃ ∉ \overline{B_{θ}(y)} and apply the argument as in Case I to S̃ instead of S to arrive at a contradiction. To show that S̃ exists, we assume not, i.e, we assume that for S = Γ (σ) we have Γ (B_{δ}(σ)) ⊂ ∂B_{θ}(y), for some small δ > 0. For (τ_{j}, s_{j}) → (σ, σ) one gets R(Γ (σ), Γ (τ_{j}), Γ (s_{j})) ≤ θ, for j sufficiently large. Hence κ[γ](σ) ≥ θ^{−1} by (24). This is also true for any σ̄ ∈ B_{δ}(σ), which means that ∥Γ′′∥_{L^{∞}} ≥ θ^{−1} = K[γ], contradicting the assumption that K[γ] is locally not attained. ⊓ ⊔

Proof of Lemma 3. (i). By the classical theorem of Fenchel [8] we have ∫_{0}^{L} |Γ′′(t)| dt ≥ 2π, which readily implies ∥Γ′′∥_{L^{∞}} ≥ 2π/L, hence

η(γ) ≤ \frac{1 − R[γ] · (2π/L)}{(2π/L)} < \frac{L}{2π} .

On the other hand, η(γ) > 0, since K[γ] is locally not attained, see (27).

(ii)&(iii). We take a maximizing sequence {s_{j}}_{j∈ℕ} ⊂ S_{L} for K[γ] in the definition (5). In other words, for every ϵ > 0, we find j_{0} such that for all j ≥ j_{0},

K[γ] ≤ κ_{G}[γ](s_{j}) + ϵ.  (96)

Take 0 < ϵ < K[γ] − ∥Γ′′∥_{L^{∞}}, which is possible by (27). According to Theorem 2 (ii) we have two possible cases (a) and (b) for κ_{G}[γ](s_{j}) for fixed j ≥ j_{0}. In case (a) there is a sequence (σ_{k}, τ_{k}) → (s_{j}, s_{j}) with

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

by (22). We use Proposition 1 to conclude that

κ_{G}[γ](s_{j}) ≤ lim sup_{{(t_{l},u_{l})→(s_{j} ,s_{j}) \atop s_{j} ≠t_{l}≠u_{l}≠s_{j}}} \frac{1}{R(Γ (s_{j}), Γ (t_{l}), Γ (u_{l}))} = κ[γ](s_{j}) ≤ ∥Γ′′∥_{L^{∞}}.

Hence by choice of ϵ we get K[γ] ≤ ∥Γ′′∥_{L^{∞}} + ϵ < K[γ], which is absurd, hence Case (a) cannot occur.

If on the other hand, Case (b) holds, then we know by the calculation that led to (64) that

κ_{G}[γ](s_{j}) = \frac{1}{r(Γ (s_{j}), Γ (τ_{j}), Γ′(τ_{j}))} =: r_{j}^{−1}

for some τ_{j} ≠ s_{j}. Now apply Lemma 8 for l = ∥Γ′′∥_{L^{∞}}, R = r_{j}, which is applicable since by choice of ϵ, one has r_{j}^{−1} ≥ K[γ] − ϵ > ∥Γ′′∥_{L^{∞}} = l. We get |s_{j} − τ_{j}| ≥ η(γ) for all j ≥ j_{0}. Then we can go to the limit for a subsequence (s_{j}, τ_{j}) → (s, τ) in (96) to get

K[γ] = \frac{1}{r(Γ (s), Γ (τ), Γ′(τ))},  (97)

where |s − τ| ≥ η(γ).

This in fact furnishes the proof of (iii), since assuming (97) we can apply Lemma 8 directly to get |s − τ| ≥ η(γ). Recall that for s, τ ∈ S_{L}, the term |s − τ| denotes the intrinsic distance on S_{L}, whence the other statement, |s − τ| ≤ L − η(γ) for s, τ ∈ [0, L] is automatically proved as well.

(ii) It remains to show that we can restrict our attention to the set Q as stated in the Lemma. In fact, if

R[γ] = r(Γ (s), Γ (σ), Γ′(σ)),  (98)

for some (s, σ) ∉ Q (but with |s−σ| ≥ η(γ) as we have just shown), then we will show that the reversed pair (σ, s) ∈ Q also satisfies (98), which shows that statement (ii) is true. Indeed, we know by Theorem 1, part (iv) (b), that for θ := R[γ]

M(σ, θ) ∩ Γ (S_{L}) = ∅.  (99)

If ∂B_{θ}(y) is the sphere that contains the circle through Γ (s), Γ (σ) tangent to Γ′(σ) as a great circle, then (99) implies, that (Γ (s)−y)·Γ′(s) = 0. Thus we can also look at the circle through Γ (s), Γ (σ) tangent to Γ′(s) with radius ρ ≤ θ. But (29) actually implies ρ = θ, hence R[γ] = r(Γ (σ), Γ (s), Γ′(s)). ⊓ ⊔

Proof of Lemma 4. Let {γ_{i}} ⊂ F_{1} be a sequence with

∥γ_{i} − γ∥_{C^{0}} → 0 as i → ∞.  (100)

Since L(γ_{i}) ≤ L_{0} for all i ∈ ℕ, we know that γ ∈ BV (I, ℝ^{3}) ∩ L with length L(γ) ≤ L_{0}. The remaining part of the proof is identical to the proof (in terms of R[.] rather than K[.]) of [12, Lemma 4]. One simply needs to check that the uniform convergence (100) suffices to carry out the arguments in [12, pp. 29,30]. ⊓ ⊔

Proof of Proposition 4. Assuming the contrary we can find a sequence γ_{j} → γ in C^{0}(Ī, ℝ^{3}), with K[γ_{j}] ≤ θ^{−1} for all j ∈ ℕ, i.e., by (8), R[γ_{j}] ≥ θ for all j ∈ ℕ, such that γ ≄ γ_{j} for all j ∈ ℕ. This contradicts the proof of [12, Lemma 5], where under the above conditions an isotopy mapping is constructed explicitly to get γ_{j} ≃ γ for sufficiently large j ∈ ℕ. ⊓ ⊔

Proof of Theorem 4. By virtue of Lemma 4 it suffices to prove that for {γ_{i}}_{i∈ℕ} ⊂ F_{2} with

∥γ_{i} − γ∥_{C^{0}} → 0 as i → ∞,  (101)

one has γ(Ī) ≃ K. This was shown in [12, Lemma 5] but follows here as a simple corollary of the stronger result Proposition 4. In fact, for i sufficiently large we have ∥γ−γ_{i}∥_{C^{0}} ≤ ϵ, for a given ϵ > 0, because of (101), and Lemma 4 implies K[γ] ≤ θ^{−1}, which is also needed in order to apply Proposition 4. ⊓ ⊔

Proof of Theorem 5. If R[γ] = 0, then by Theorem 1, part (iii), γ must have a double point, i.e., Γ (s) = Γ (t) for some s ≠ t in S_{L}. Then any sequence {γ_{i}} ⊂ L of fixed length L(γ_{i}) = L with ∥Γ_{i} − Γ ∥_{C^{0}} satisfies R[γ_{i}] → 0 as i → ∞. This is true, because assuming the contrary would lead to an immediate contradiction to Lemma 4. Now assume R[γ] > 0 for some γ ∈ L with length L(γ) = L, hence γ is simple in particular. We take a sequence of curves γ_{l} with arc length parameterizations Γ_{l} converging to Γ in C^{1,1}([0, L], ℝ^{3}), where Γ is the arc length parameterization of γ. We proceed in several steps.

1. We claim that there is l_{0}, ρ > 0, such that

|Γ_{l}(s) − Γ_{l}(t)| ≥ \frac{|s − t|}{2} for all l ≥ l_{0}, |s − t| < ρ.  (102)

In fact, expanding Γ_{l} about t ∈ S_{L}, we obtain

|Γ_{l}(s) − Γ_{l}(t)| ≥ |s − t| − | ∫_{t}^{s}(Γ′_{l}(σ) − Γ′_{l}(t)) dσ|

≥ |s − t| − | ∫_{t}^{s}(Γ′(σ) − Γ′(t)) dσ| − 2∥Γ′_{l} − Γ′∥_{C^{0}}|s − t|.

Choosing first l_{0} so large that ∥Γ_{l} − Γ ∥_{C^{1}} < 1/8 for all l ≥ l_{0}, and then ρ > 0 (depending on the Lipschitz constant K[γ] of Γ′) so small that

|Γ′(t_{1}) − Γ′(t_{2})| ≤ 1/4 for all |t_{1} − t_{2}| < ρ,

we arrive at (102). For later purposes we take ρ so small that

ρ ≤ \frac{1}{4∥Γ′′∥_{L^{∞}}} .  (103)

2. We claim that there are constants d > 0, l_{1} ≥ l_{0} such that for all l ≥ l_{1} and all s, σ ∈ S_{L} with |s − σ| ≥ ρ one has

|Γ_{l}(s) − Γ_{l}(σ)| ≥ d.  (104)

It suffices to prove the claim for the fixed simple limit curve γ, (with some constant d = d_{γ}), since then the claim follows from the simple estimate

|Γ_{l}(s) − Γ_{l}(σ)| ≥ d_{γ} − 2∥Γ_{l} − Γ ∥_{C^{0}} ≥ d_{γ}/2

for all |s − σ| ≥ ρ and for all l ≥ l_{1}, with l_{1} ≥ l_{0} sufficiently large. Assuming that (104) is not true for Γ, one finds by compactness (s_{k}, σ_{k}) → (s, σ) with |s − σ| ≥ ρ, as k → ∞, with |Γ (s_{k}) − Γ (σ_{k})| ≤ k^{−1}. Hence by continuity of Γ we infer Γ (s) = Γ (σ), i.e., R[γ] = 0, which is excluded here.

3. Combining (102) and (104) we get for l sufficiently large, that γ_{l} is simple, which by Theorem 1, part (iii) implies that R[γ_{l}] > 0 for sufficiently large l ∈ ℕ.

4. The assumption that

lim_{l→∞} K[γ_{l}] < K[γ],  (105)

will lead to a contradiction. Indeed, then by (26) we have ∥Γ′′_{l}∥_{L^{∞}} ≤ K[γ]−ϵ for some fixed ϵ > 0 and l sufficiently large. But due to the convergence ∥Γ − Γ_{l}∥_{C^{1,1}} → 0 as l → ∞, we then get ∥Γ′′∥_{L^{∞}} ≤ K[γ] − ϵ. This means that K[γ] is locally not attained, see (27), which implies by Lemma 3, part (ii), that

K[γ] = \frac{1}{r(Γ (s), Γ (σ), Γ′(σ))}  (106)

for some s, σ ∈ S_{L} with |s − σ| ≥ η(γ), where

η(γ) = \frac{1 − R[γ] · ∥Γ′′∥_{L^{∞}}}{∥Γ′′∥_{L^{∞}}} = \frac{1 − R[γ] · (K[γ] − ϵ)}{K[γ] − ϵ} ≥ ϵR[γ]^{2} > 0.

But the right-hand side in (106) may be approximated as

K[γ_{l}] ≥_{(30)} \frac{1}{r(Γ_{l}(s), Γ_{l}(σ), Γ′_{l}(σ))} ⟶ \frac{1}{r(Γ (s), Γ (σ), Γ′(σ))} = K[γ]

contradicting (105).

5. Assuming on the other hand

lim_{l→∞} K[γ_{l}] > K[γ],  (107)

we find ϵ > 0 and l_{0} ∈ ℕ such that for all l ≥ l_{0} (by (26) applied to Γ),

K[γ_{l}] ≥ K[γ] + ϵ ≥ ∥Γ′′∥_{L^{∞}} + ϵ.  (108)

This together with ∥Γ_{l} − Γ ∥_{C^{1,1}} → 0 as l → ∞ gives us

K[γ_{l}] ≥ ∥Γ′′_{l}∥_{L^{∞}} + ϵ/2  (109)

for all l sufficiently large. Hence for all l sufficiently large we know that K[γ_{l}] is locally not attained. Moreover R[γ_{l}] is positive, hence K[γ_{l}] finite, and by Lemma 3 we conclude that for each such l there exist s_{l}, σ_{l} ∈ S_{L} with |s_{l} − σ_{l}| ≥ η(γ_{l}) > 0, such that

K[γ_{l}] = \frac{1}{r(Γ_{l}(s_{l}), Γ_{l}(σ_{l}), Γ′_{l}(σ_{l}))} .  (110)

We claim that there is a uniform constant c > 0, such that for all l sufficiently large

η(γ_{l}) ≥ c.  (111)

Indeed, assuming η(γ_{l}) → 0 as l → ∞, we get

R[γ_{l}]∥Γ′′_{l}∥_{L^{∞}} → 1,  (112)

since ∥Γ′′_{l}∥_{L^{∞}} → ∥Γ′′∥_{L^{∞}} ∈ (0, ∞). (112) together with (8) implies K[γ_{l}] → ∥Γ′′∥_{L^{∞}} contradicting (108), which proves the claim.

By compactness we may assume that s_{l} → s and σ_{l} → σ in S_{L} for l → ∞, with |s − σ| ≥ c > 0, by (111), which means that we can go to the limit in (110) to obtain

lim_{l→∞} K[γ_{l}] = lim_{l→∞} \frac{1}{r(Γ_{l}(s_{l}), Γ_{l}(σ_{l}), Γ′_{l}(σ_{l}))} = \frac{1}{r(Γ (s), Γ (σ), Γ′(σ))} ≤ K[γ],

where we used (30) applied to γ to obtain the last inequality. But this contradicts (107), hence if lim_{l→∞} K[γ_{l}] exists it must be equal to K[γ].

6. We actually proved with the previous argument that the sequence {K[γ_{l}]}_{l∈ℕ} ⊂ ℝ must be bounded, since otherwise we deduce a contradiction exactly as we have just done. Hence there exists a convergent subsequence l_{k}, k ∈ ℕ, with lim_{k→∞} K[γ_{l_{k}}] = K[γ]. By the subsequence principle, the whole sequence must actually converge to K[γ], since we have proved that if a limit exists it must equal K[γ]. This proves (sequential) continuity. ⊓ ⊔

Proof of Lemma 5. (i) The weak convergence implies

γ_{1,n} → γ_{1} and γ_{2,n} → γ_{2} in C^{0}(Ī, ℝ^{3})  (113)

as n → ∞. Hence we can assume that γ_{1}, γ_{2} ∈ L ∩ W^{1,q}(I, ℝ^{3}) and by (40)

dist (γ_{1}(Ī), γ_{2}(Ī)) ≥ c.  (114)

Thus l(γ_{1}, γ_{2}) is well-defined, and we can estimate

4π|l(γ_{1,n}, γ_{2,n}) − l(γ_{1}, γ_{2})| ≤

| ∫_{I} ∫_{I} (\frac{γ_{1,n}(s) − γ_{2,n}(t)}{|γ_{1,n}(s) − γ_{2,n}(t)|^{3}} − \frac{γ_{1}(s) − γ_{2}(t)}{|γ_{1}(s) − γ_{2}(t)|^{3}} ) · [γ′_{1,n}(s) ∧ γ′_{2,n}(t)] ds dt|

+| ∫_{I} ∫_{I} \frac{γ_{1}(s) − γ_{2}(t)}{|γ_{1}(s) − γ_{2}(t)|^{3}} · [(γ′_{1,n}(s) − γ′_{1}(s)) ∧ γ′_{2,n}(t)] ds dt|  (115)

+| ∫_{I} ∫_{I} \frac{γ_{1}(s) − γ_{2}(t)}{|γ_{1}(s) − γ_{2}(t)|^{3}} · [γ′_{1}(s) ∧ (γ′_{2,n}(t) − γ_{2}(t)] ds dt| =: I_{n} + II_{n} + III_{n}.

We estimate the three terms on the right separately.

I_{n} = | ∫_{I} γ′_{2,n}(t) · [ ∫_{I} (\frac{γ_{1,n}(s) − γ_{2,n}(t)}{|γ_{1,n}(s) − γ_{2,n}(t)|^{3}} − \frac{γ_{1}(s) − γ_{2}(t)}{|γ_{1}(s) − γ_{2}(t)|^{3}} ) ∧ γ′_{1,n}(s) ds] dt|

≤ sup_{s,t∈I} |\frac{γ_{1,n}(s) − γ_{2,n}(t)}{|γ_{1,n}(s) − γ_{2,n}(t)|^{3}} − \frac{γ_{1}(s) − γ_{2}(t)}{|γ_{1}(s) − γ_{2}(t)|^{3}} |∥γ′_{1,n}∥_{L^{1}}∥γ′_{2,n}∥_{L^{1}},  (116)

hence I_{n} → 0 as n → ∞, by (113),(114) and the fact that the W^{1,q}-norms of γ_{1,n} and γ_{2,n} are uniformly bounded, because of the weak convergence .

II_{n} = | ∫_{I} γ′_{2,n}(t) · [ ∫_{I} \frac{γ_{1}(s) − γ_{2}(t)}{|γ_{1}(s) − γ_{2}(t)|^{3}} ∧ (γ′_{1,n}(s) − γ′_{1}(s)) ds] dt|
=: | ∫_{I} γ′_{2,n}(t) · ξ_{n}(t) dt|,  (117)

where we have denoted the inner integral by ξ_{n}(t). By the weak convergence γ′_{1,n} ⇀ γ′_{1} in L^{q}(I, ℝ^{3}), one finds |ξ_{n}(t)| → 0 as n → ∞ for all t ∈ I, hence also |ξ_{n}(t)|^{q^{∗}} → 0 as n → ∞ for all t ∈ I, where q^{−1} + q^{∗}^{−1} = 1. Furthermore, using (114),

|ξ_{n}(t)|^{q^{∗}} ≤ \frac{|b − a|}{c^{2q^{∗}}}(∥γ′_{1,n}∥_{L^{q}}^{q^{∗}} + ∥γ′_{1}∥_{L^{q}}^{q^{∗}}) ≤ C̃ < ∞;  (118)

here C̃ does not depend on n. Thus by Lebesgue’s Dominated Convergence Theorem we conclude that ξ_{n} → 0 in L^{q^{∗}}. This together with the weak convergence γ_{2,n} ⇀ γ_{2} in W^{1,q}(I, ℝ^{3}) leads to II_{n} → 0 as n → ∞.

The third term III_{n} has the same structure as II_{n}.

(ii) The embedding W^{1,q} ↪ C^{0,1−(1/q)} implies that for ∥γ_{i} − β_{i}∥_{W^{1,q}} sufficiently small one has by (42)

β_{1}(Ī) ∩ β_{2}(Ī) = ∅.  (119)

Hence the linking number l(β_{1}, β_{2}) is well-defined. Going back to (116) and (117) with γ_{i,n} replaced by β_{i}, i = 1, 2, one observes that the terms I_{n}, II_{n} (and analogously III_{n}) can be made arbitrarily small by choosing ϵ sufficiently small. Notice that by (42) and (119) we can estimate the denominators on the right-hand side in (116) and (117) from below by some constant and that the norms ∥β′_{i}∥_{L^{1}} are bounded by, say 2∥γ′_{i}∥_{L^{1}} for i = 1, 2, for ϵ sufficiently small. ⊓ ⊔

Proof of Theorem 6. To apply Lemma 5 for γ_{1,n} := Γ_{n}, γ_{2,n} := Γ_{n} + (θ/2)d_{1,n}, γ_{1} := Γ and γ_{2} := Γ + (θ/2)d_{1}, we merely need to verify that hypothesis (40) holds. In fact, part (iv)(b) of Theorem 1 implies that

dist (Γ_{n}([0, L]), (Γ_{n} + (θ/2)d_{1,n})([0, L])) ≥ θ/2 for all n ∈ ℕ,

since K[Γ_{n}] ≤ θ^{−1} and because (37) was assumed to hold for all n ∈ ℕ. Thus the first part of Lemma 5 implies (i), and part (ii) follows immediately from the second part of that lemma. ⊓ ⊔

Acknowledgements Both authors were supported by the Max-Planck Institute for Mathematics in the Sciences in Leipzig and the Sonderforschungsbereich 256 at the University of Bonn. The second author enjoyed the hospitality of the Forschungsinstitut für Mathematik at the ETH Zürich during the spring term 2001.

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