A GEOMETRIC CURVATURE DOUBLE INTEGRAL OF MENGER TYPE FOR SPACE CURVES

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1 Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 34, 009, 95 4 A GEOMETRIC CURVATURE DOUBLE INTEGRAL OF MENGER TYPE FOR SPACE CURVES Paweł Strzelecki, Marta Szumańska and Heiko von der Mosel Uniwersytet Warszawski, Instytut Matematyki ul. Banacha, PL Warsaw, Poland; pawelst@mimuw.edu.pl Politechnika Warszawska, Wydział Matematyki i Nauk Informacyjnych Plac Politechniki, PL Warsaw, Poland; M.Szumanska@mini.pw.edu.pl RWTH Aachen, Institut für Mathematik Templergraben 55, D-506 Aachen, Germany; heiko@instmath.rwth-aachen.de Abstract. We consider rectifiable closed space curves for which the energy I p () := inf z R p (x, y, z) dh (x) dh (y), p, is finite. Here, R(x, y, z) denotes the radius of the smallest circle passing through x, y, and z. It turns out that I p is a self-avoidance energy (curves of finite energy have no self-intersections). For p >, we study regularizing effects of I p : we prove that the arclength parametrization Γ of a curve with I p () < is everywhere differentiable, and its derivative, Γ, is Hölder continuous with exponent /p. Moreover, we obtain compactness results for classes of curves with uniformly bounded I p energy, and briefly discuss their variational applications.. Introduction In 930 Menger [Men30] introduced a metric version of curvature for so-called metric arcs, i.e., for subsets A X of a metric space (X, d) such that A is homeomorphic to the unit interval [0, ] R. For this purpose Menger defined the radius of curvature R(x, y, z) of three pairwise distinct points x, y, z A as (.) R(x, y, z) := d xy d yz d zx [dxy +d yz +d zx ][d xy +d yz d zx ][d xy d yz +d zx ][ d xy +d yz +d zx ], where d xy := d(x, y), etc. He then introduced (local) curvature κ(ξ) for any ξ A as (.) κ(ξ) := lim x,y,z ξ R(x, y, z). Menger s overall goal was a coordinate free description of metric continua to study their differential properties and to generalize differential geometric concepts to more general spaces; see, e.g., the treatise of Blumenthal and Menger [BlM70], in particular Chapter 0. In Euclidean 3-space R(x, y, z) as defined in (.) equals the classical circumcircle radius of the points x, y, z R 3, and even in that Euclidean setting some authors refer to the quotient /R(x, y, z) as the Menger curvature of the triple (x, y, z). 000 Mathematics Subject Classification: Primary 8A75, 49J45, 49Q0, 53A04, 57M5. Key words: Menger curvature, global curvature, self-avoidance energies, minimizing knots, imbedding theorems.

2 96 Paweł Strzelecki, Marta Szumańska and Heiko von der Mosel Motivated by quite different mathematical questions regarding curves in R 3 Gonzalez and Maddocks [GM99] had the ingenious idea to minimize R over pairs or triples of curve points leading to the global radius of curvature function (.3) ρ G [](x) := inf R(x, y, z), x y z x y,z and to (.4) [] := inf R(x, y, z), x y z x x,y,z which we refer to as the global radius of curvature of. Gonzalez and Maddocks used this quantity to characterize the thickness of the curve. Indeed, a positive lower bound on [] does not only control curvature (as (.) would suggest), but also prevents the curve from self-intersecting by establishing a uniform tubular neighbourhood as an excluded volume constraint; see, e.g., [GMSvdM0, Lemmata 3] for the precise statements. For C -smooth curves [] is equal to the normal injectivity radius as observed in [GM99]. But the global radius of curvature is well-defined for merely continuous and rectifiable curves as well. Moreover, it is analytically tractable, which paves the way towards variational calculus on embedded curves. In [GMSvdM0] various energy minimization problems for nonlinearly elastic curves and rods under topological constraints such as a prescribed knot class, or a given linking number on framings, could be solved (see also [CKS0] and [GL03]). Here, an inequality constraint involving on the class of competing curves guarantees the right topology of the respective minimizer. The dependence of on the underlying curve is not only highly nonlinear but also nonsmooth which turned out to be a challenge for the regularity investigations in [SvdM03b], [CFK + 04], and for a numerical treatment [CPR05], [ACPR05]. A relaxed variant of a possible self-avoidance energy involving Menger curvature as suggested by [GM99] and later by Banavar et al. [BGMM03] is (.5) M p () := R p (x, y, z) dh (x) dh (y) dh (z), p. In contrast to the common singular and therefore divergent repulsive potentials of the form (.6) x y dh (x) dh (y) p as discussed, e.g., in [O H9], [FHW94], [KS98a], [AS93], [O H03], the three-point interaction function R p (x, y, z) in M p requires no regularization: For a smooth embedded curve the integrand in (.5) tends to the p th power of local curvature at x as y, z x. Analytically not much is known about the M p -energy. If p = the functional is called the total Menger curvature and Melnikov [Mel95] had discovered its importance for complex analysis: M turned out to be a crucial quantity (defined on one-dimensional subsets of the complex plane) for the solution of the Vitushkin conjecture on the removability of singularities of bounded analytic functions; see, e.g., the surveys [Ma98], [Ma04], or the monograph [P0]. Moreover, one-dimensional Borel sets in R n with bounded M -energy are in fact countably -rectifiable in the sense of geometric measure theory, which was proved by Léger [Le99]. Later this result was generalized to the metric setting by Hahlomaa [Ha05a], [Ha05b]; see also

3 A geometric curvature double integral of Menger type for space curves 97 recent work of Schul [Schu06]. For Borel sets of fractal dimensions p/, 0 < p <, we refer to the work of Lin and Mattila [LM00]. But we are not aware of any existence or regularity results for M p -minimizing curves. The motivation for our recent investigations in [StvdM07] and for the present work is two-fold: Firstly, we would like to study a whole range of possible energies in between (and including) / involving a triple infimization (see (.4)) and the triple integral M p defined in (.5). Secondly, we aim at complementing the measure-theoretic achievements of Léger for M with a calculus of variation approach in the class of embeddings. In a variational context M p, or intermediate versions interpolating between / and M p, would on the one hand serve as a cost-function to obtain optimally embedded curves as minimizers. Or, on the other hand, these energies could appear in sideconditions to guarantee embeddedness for variational problems with different cost functions such as bending or torsional energies. As a first relaxation of (.4) we analyzed in [StvdM07] the semi-soft selfavoidance energy ( ) /p U p () := ρ G [](x) dh (x) p (.7) ( = inf y,z R(x, y, z) p dh (x) ) /p, p. Notice that U () = / []. One central result of [StvdM07] is the characterization of closed curves with U p () <, p >, as those embedded curves that have their arclength parametrization in the Sobolev class W,p. This generalized the corresponding characterization of C, -embeddings in [SvdM03a]. Moreover, quantitative estimates lead to compactness theorems and to existence theorems for embedded curves (or rods) under topological constraints analogous to the variational applications in [GMSvdM0] mentioned above. In the present paper we replace one more infimization in (.4) by an integration to get a further relaxation (.8) I p () := inf z R p (x, y, z) dh (x) dh (y), p. It turns out that I p behaves similar to a repulsive potential of the form (.6) on large parts of a generic curve away from the diagonal {(x, x) }. Nevertheless, no regularization near the diagonal is necessary for I p since the integrand tends to the p th power of local curvature for x = y as z x, provided that is embedded and sufficiently smooth. We are going to prove in Section that I p is in fact a self-avoidance energy for p. To be more precise, rectifiable closed curves : S R 3 (of length L) with I p () < have an injective arclength parametrization Γ: S L = R/LZ R 3 ; see Proposition.. The regularizing effect of the I p -energy is studied in Section 3: If I p () < and p >, then the arclength parametrization possesses a Hölder continuous tangent Γ C 0,α (S L, R 3 ) for α = (p )/(p+4) (Corollary 3.). Technically this is proved by establishing a uniform cone flatness, which basically means that locally near any curve point the curve is contained in a cone with arbitrarily small cone angle. The resulting Recall that C, = W,.

4 98 Paweł Strzelecki, Marta Szumańska and Heiko von der Mosel Hölder norm is solely controlled by the energy level I p (). This instantaneously gives compactness results and, with an additional geometric argument, convergence of I p - equibounded sequences to a simple limit curve (Corollary 3.3). Existence theorems for topologically constrained variational problems involving I p as cost-function or in side conditions are then immediate consequences; see, e.g., Theorem 3.4. Using the C,α - regularity we can use measure theoretic arguments to improve the Hölder exponent of Γ up to (/p), p >. This regularity theorem is vaguely reminiscent of the Morrey Sobolev embedding theorem: Menger curvature is related to local curvature, i.e., to Γ, and the domain is two-dimensional, which leads to the expected Hölder exponent (/p). The authors are convinced that the results in this paper can be generalized to curves in R n for n > 3. However, to keep the geometric ideas simple and transparent, we stick to the case of curves in three-dimensional space. The proofs of the present paper are based on a mixture of geometric and measuretheoretic arguments. A substantial part of this discussion carries over to the triple integral M p for p > 3 (see (.5)). However, to this end one has to change numerous technical details, not only in the proofs, but also in the assumptions. Moreover unlike in Proposition. in this paper arclength parametrizations of curves of finite M p energy do not have to be injective; this part of the analysis is significantly more complicated. This would make a joint presentation of the results rather lengthy, with long, sometimes even awkward, statements of theorems designed to cover various cases. Therefore, a detailed account on the M p -energy will be published elsewhere [SStvdM07], which together with the present work brings the variational calculus closer to the purely measure-theoretic setting of Léger s work in [Le99]. However, the situation for higher-dimensional analogues of geometric curvature energies such as (.7), (.8) or (.5) for surfaces, remains largely open. Acknowledgement. The first author was partially supported by KBN grant no PO 3A The second author was partially supported by EC FP6 Marie Curie ToK programme SPADE, MTKD-CT and Polish MNiSW SPB-M. The third author was partially supported by the Deutsche Forschungsgemeinschaft. We would also like to thank the referee for his/her careful work and valuable suggestions.. Injectivity In view of applications in the calculus of variations we prefer to work with geometric quantities defined in terms of parametrized curves. We assume that Γ C 0, (S L, R 3 ), where S L = R/LZ, is the arclength parametrization of a rectifiable closed curve : S R 3 with H ((S )) = L > 0. (The reader should bear in mind that throughout the whole paper, whenever capital Γ is used, we mean the arclength parametrization of a curve.) Fixing two distinct arclength parameters s, t S L we can define an intermediate global radius of curvature function as follows. Let (.) ϱ[γ](s, t) ϱ(s, t) := inf R(Γ(s), Γ(t), Γ(τ)), τ S L \{s,t} See however, our investigation [StvdM05], [StvdM06] on a generalization of the global radius of curvature (.4) to two-dimensional surfaces in Euclidean n-space.

5 A geometric curvature double integral of Menger type for space curves 99 where R(x, y, z) is the uniquely defined radius of the smallest circle containing the points x, y, z R 3. For points x, y, z that are not collinear R(x, y, z) equals the circumcircle radius which may be expressed as R(x, y, z) := x y sin <)(x z, y z), where <)(x z, y z) [0, π] denotes the angle between the vectors x z and y z. Then we see from (.8) that ds dt (.) I p () = ϱ p (s, t). S L In order to analyze the local behaviour of the curve we introduce some threedimensional shapes, namely cones, lenses and doughnuts. For x y R 3 and ε (0, π ), C + ε (x; y) := {z R 3 : <)(z x, y x) < ε } is a (one-sided) cone with vertex at x, axis passing through y and opening angle ε. We set (.3) C ε (x; y) := {x + t(z x): t R, z C + ε (x; y)}. Next, for x y R 3 and r > 0, we write (.4) l(x, y; r) := {B r : x, y B r } to denote the lens-shaped region which is formed by the intersection of all open balls B r of radius r that contain both points x, y on their boundary B r. We also write (.5) V (x, y; r) := {B r : x, y B r } to denote the thick (degenerate) doughnut formed by the union of all such balls. Proposition.. (Self-avoidance) Assume that p, L > 0, and I p () <. Then is simple, i.e., its arclength parametrization Γ: S L R 3 is injective. Proof. We argue by contradiction. Assume that Γ(0) = 0 = Γ(s ) for some s S L \ {0}. Let Γ := Γ ( [0, s ] ), Γ := Γ ( [s, L] ). Since Γ is the arc length parametrization of, it has no intervals of constancy. Thus, S L d := min(diam Γ, diam Γ ) > 0. We consider the portion of contained in B d/4 (0). Choose four parameters: σ, σ (0, s ) and t, t (s, L) such that σ < σ, t < t and Γ(σ ), Γ(σ ), Γ(t ), Γ(t ) B d/4 (0). Now, choose a number ε (0, d ) which is smaller than the smallest gap between 0, σ, σ, s, t, t, and L in the natural ordering on S L. Take s (0, ε) S L. If Γ(s) = 0, we set A(s) := (0, s) S L. In this case, since Γ(s) = Γ(0) = 0, we have ϱ(s, σ) Γ(σ) / < s for all σ A(s), and (.6) H (A(s))ϱ(s, σ) p s p for all σ A(s).

6 00 Paweł Strzelecki, Marta Szumańska and Heiko von der Mosel Next, suppose that Γ(s) 0. Then we set (.7) A(s) := {σ (σ, t ) Γ (σ) exists and Γ(σ) B Γ(s) (0)}. For each σ A(s) consider the diameter ball DB(s, σ) defined as follows: DB(s, σ): = B r (a) Two cases are possible now. for r : = Γ(s) Γ(σ), a: = Γ(s) + Γ(σ). Case. Γ intersects DB(s, σ) transversally at Γ(σ). Then we can find a parameter t S L, t close to σ, such that the point Γ(t) int DB(s, σ). Since Γ(σ ), Γ(t ) B d/4 (0) and, by the choice of ε and σ, we have DB(s, σ) B 3ε (0) B d/4 (0), there exists a parameter τ [σ, t ] which is different from σ and satisfies Γ(τ) DB(s, σ). Thus, ϱ(s, σ) R ( Γ(s), Γ(σ), Γ(τ) ) Γ(s) Γ(σ) (.8) = Γ(s) as Γ(σ) B Γ(s) (0). (Notice that by definition of R one has ϱ(s, σ) Γ(s) Γ(σ) / which together with (.8) implies ϱ(s, σ) = Γ(s) Γ(σ) / in this case.) Case. Γ touches DB(s, σ) at Γ(σ), i.e., Γ (σ) (Γ(σ) Γ(s)). In this case, take a ball B = B r with radius r slightly larger than that of DB(s, σ) and such that Γ(s), Γ(σ) B r. Any such B r is intersected transversally by Γ at σ. Mimicking the reasoning for Case, one checks that ϱ(s, σ) r. Taking the infimum over all r > Γ(s) Γ(σ) /, we obtain also in this case. ϱ(s, σ) Γ(s) Now, for each s (0, ε) S L with Γ(s) 0 we have H (A(s)) Γ(s) > Γ(s) since Γ is differentiable a.e. and Γ(s ) = 0. Thus, since Γ(s) s, condition (.6) holds also when Γ(s) 0, i.e., when A(s) is defined as in (.7). Therefore, ε ( ) ε I p () ϱ(s, σ) dσ ds s p ds = + for all p. p 0 A(s) This contradiction completes the proof. 3. The existence of tangents In this section, we prove that for p > a closed curve with finite I p -energy has an arclength parametrization which is everywhere differentiable. In fact with our method of proof we obtain estimates for the Hölder norm of the tangent in terms of the energy. This yields convergence and compactness results for sequences with equibounded energy, and leads to variational applications. All of this is based on the following key result. Theorem 3.. (Uniform cone flatness) Assume that p > and L > 0. There exists a constant c = c(p) such that if ε (0, π ) and η > 0 satisfy (3.) ε p+4 η p c(p)e, diam η, 0

7 A geometric curvature double integral of Menger type for space curves 0 where E = I p (), then for every s, t S L such that Γ(s) Γ(t) = η we have (3.) Γ(S L ) B η ( Γ(s) ) Cε (Γ(s); Γ(t)). (See (.3) for the definition of the cone C ε.) Notice that Theorem 3. easily implies differentiability of Γ at all points of S L and gives control of the modulus of continuity of Γ ; see Corollary 3. below. It also allows us to prove that limits of convergent sequences of curves with uniformly bounded I p - energy have no double points (Corollary 3.3). Thus I p can be used as a cost function or as a side condition in topologically constrained variational problems for curves and rods. As a model example of such a variational application we establish in Theorem 3.4 the existence of I p -minimizing curves in prescribed (tame) knot classes. Let us discuss these consequences of the uniform cone flatness first before giving a detailed proof of Theorem 3.. Corollary 3.. Assume that p >. If I p () <, then Γ is defined everywhere on S L and moreover (3.3) Γ (s) Γ (t) M s t α, where α = (p )/(p + 4) (0, ) and M c(p)i p () /(p+4). Proof. To begin with, pick s < t S L such that Γ (s) and Γ (t) exists. Set η := Γ(s) Γ(t) and ε := [ c(p)i p ()η p ] /(p+4), where c(p) denotes the constant from Theorem 3.. If c(p)i p () <, assume ( π ) (p+4)/(p ) (3.4) s t < ; otherwise assume that (3.5) s t < ( c(p)i p () ) ( /( p) π ) (p+4)/(p ). In both cases we then have ε < π/ and we may apply Theorem 3. twice to conclude that [ ( ) ( ) ] Γ(S L ) B η Γ(s) Bη Γ(t) C ε (Γ(s); Γ(t)) C ε (Γ(t); Γ(s)). Since Γ is injective 3, this easily gives Γ (s) Γ (t) ε c(p)i p () /(p+4) s t (p )/(p+4). Such an estimate is obviously valid also when (3.4) or (3.5), respectively, is violated. Thus, Γ has a unique Hölder continuous extension g to the whole parameter circle S L. As Γ is Lipschitz, we have t Γ(t ) Γ(t ) = Γ (τ) dτ = g(τ) dτ t t for all t, t S L ; it is now a routine matter to check that Γ exists and is equal to g at each point of S L. The estimate (3.3) is also satisfied. 3 In principle, Γ (s) and Γ (t) could both point into the diamond-shaped subset of the intersection of the cones with tips at Γ(s) and Γ(t). But then Γ would have to leave this diamond-shaped region at either tip at some parameter τ s, t producing a double point. This would contradict Proposition.. t

8 0 Paweł Strzelecki, Marta Szumańska and Heiko von der Mosel Corollary 3.3. Assume that K, L > 0 and p >. Let Q be a fixed point in R 3. If a family of rectifiable closed curves j : S R 3 satisfies (3.6) Q j (S ) and H ( j ) = L for all j, and sup I p ( j ) K, j=,,... then there exists ε 0 = ε 0 (p, K) > 0 such that the arclength parametrizations Γ j of j satisfy ( s t ) (3.7) Γ j (s) Γ j (t) min ε 0, for all j and all s, t S L. Moreover, the family of functions Γ j : S L S R 3 is equicontinuous and {Γ j } contains a subsequence {Γ jk } which for j k converges in the C -topology to a simple arclength parametrized closed curve Γ C,(p )/(p+4) (S L, R 3 ) with Q Γ(S L ). Proof. The existence of a convergent subsequence follows easily from (3.6) in combination with Corollary 3. and the Arzela Ascoli compactness theorem. Once (3.7) is established, injectivity of the limit curve Γ follows from (3.7) upon passing to the limit j k. Thus it is enough to prove (3.7). Consider g j C (S L S L ) given by g j (s, t) := Γ j (s) Γ j (t). Since Γ j are uniformly bounded in C,α, it is easy to show that there is a constant ε = ε (p, K) > 0 such that s t (3.8) g j (s, t) for all j and all s, t such that s t ε (p, K). 4 (See, e.g., the next section, Lemma 4., for details.) Since Σ = S L S L \ {(s, t): s t < ε (p, K)} is compact, we find for each j a pair (s j, t j ) Σ such that g j (s j, t j ) g j (s, t) for all (s, t) Σ. Now, we either have s j t j = ε (p, K) in which case (3.8) implies (3.9) g j (s, t) ε (p, K) for all s, t Σ, 4 or we have g j (s j, t j ) = 0, which is equivalent to (3.0) Γ j(s j ) ( Γ j (s j ) Γ j (t j ) ) and Γ j(t j ) ( Γ j (s j ) Γ j (t j ) ). For each j, this implies that Γ j (S L ) [B ηj (Γ j (s j )) B ηj (Γ j (t j ))] is not contained in the intersection C π/4 (Γ j (s j ); Γ j (t j )) C π/4 (Γ j (t j ); Γ j (s j )), where η j := Γ j (s j ) Γ j (t j ) ; see figure below. Left: The plane which passes through two points Γ j (s j ), Γ j (t j ), and contains Γ j (s j). The planar cross-section of ( C π/4 Γj (s j ); Γ j (t j ) ) ( C π/4 Γj (t j ); Γ j (s j ) ) is shaded. The tangent vectors to j at these two points are perpendicular to the common axis of the cones.

9 A geometric curvature double integral of Menger type for space curves 03 By virtue of Theorem 3. for ε := π/4 this means, however, that [ (3.) (π/4) p+4 p g j (s, t) Γ j (s j ) Γ j (t j ) = ηj > (c(p)i p ( j )) p [ ] (π/4) p+4 p =: ε (p, K) > 0 for all j N, (s, t) Σ. c(p)k Summarizing (3.8), (3.9), and (3.), we obtain (3.7) with ε 0 := min { ε (p, K)/, ε (p, K) }. As one of various possible variational applications we consider the variational problem of minimizing the I p -energy for an arbitrary fixed p > in a prescribed knot class. To be more precise, we look for a minimizer of I p in the class C L,k := { C 0 (S, R 3 ) : length () = L, isotopic to k}, where k is a given representative of a particular tame knot or isotopy class. Recall that two continuous closed curves, R 3 are isotopic, if there are open neighborhoods N of and N of, and a continuous map Φ: N [0, ] R 3 such that Φ(, t): N Φ(N, t) is a homeomorphism for all t [0, ], Φ(x, 0) = x for all x N, Φ(N, ) = N, and Φ(, ) =. Theorem 3.4. Let p > and L > 0. In any given isotopy class represented by a closed curve k there is an arclength parametrized curve Γ C,(p )/(p+4) (S L, R 3 ) C L,k such that I p (Γ) = inf CL,k I p ( ). Proof. Scaling a smooth representative of the given knot class to have length L we observe that the class C L,k is not empty and contains smooth curves of finite energy. Thus we find a minimal sequence { i } C L,k with arclength parametrizations Γ i C 0, (S L, R 3 ) satisfying I p ( i ) inf C L,k I p ( ) < as i. Translating the curves we may assume that 0 i (S ) for all i N. By virtue of Corollary 3.3 we find a subsequence again denoted by Γ i converging in the C - topology to a simple arclength parametrized closed curve Γ C,(p )/(p+4) (S L, R 3 ). This implies that also the limit Γ is in the prescribed knot class, since isotopy is stable with respect to C -topology; see [R05]. Consequently, Γ C L,k and we have inf I p ( ) I p (Γ) lim inf I p(γ i ) = lim I p ( i ) = inf I p ( ), C L,k i i C L,k where we used the fact that I p is lower-semicontinuous as stated in Lemma 3.5 below. Lemma 3.5. Suppose that the sequence {Γ j } of arclength parametrized curves Γ j : S L R 3 converges uniformly to Γ, i.e., Γ j Γ in C 0 (S L, R 3 ) as j and assume that Γ is simple. Then I p (Γ) lim inf j I p(γ j ) for all p. Proof. If ρ[γ j ](s, t) δ for some δ > 0, s t, for infinitely many j, then ρ[γ](s, t) δ, since otherwise we would find a parameter τ distinct from s and t, such that R(Γ(s), Γ(t), Γ(τ)) < δ. This would imply that R(Γ j (s), Γ j (t), Γ j (τ)) < δ ]

10 04 Paweł Strzelecki, Marta Szumańska and Heiko von der Mosel for j, since Γ is simple and the function R(,, ) is continuous near mutually different non-collinear points in space. Hence if lim sup j ρ[γ j ] p (s, t) δ for some s t, then ρ[γ] p (s, t) δ ε for all ε > 0. If, on the other hand, lim sup j ρ[γ j ] p (s, t) = 0 then obviously ρ[γ] p (s, t) lim sup j ρ[γ j ] p (s, t). In any case, which implies and I p (Γ) = S L S L ρ[γ] p (s, t) lim sup ρ[γ j ] p (s, t) for all s t, j ρ[γ] p (s, t) S L lim inf j S L \{s} lim inf j ρ[γ j ] p (s, t) dt ds ρ[γ] p (s, t) = dt ds S L S L \{s} ρ[γ] p (s, t) lim inf dt ds = j ρ[γ j ] p (s, t) S L S L ρ[γ j ] p (s, t) S L for all s t lim inf S j L dt ds = lim inf j I p(γ j ), where we used the Lemma of Fatou for the last inequality. dt ds ρ[γ j ] p (s, t) Proof of Theorem 3.. Without loss of generality we assume that s = 0 and Γ(0) = 0. Fix ε > 0 and η > 0 satisfying (3.). We need to introduce some notation first. Choose ε > 0 such that (3.) ε for our purposes, ε = ε/0 will do. Set N ε; (3.3) η N := η N and r N := ε N η N, N =,,.... Next, pick a point p N = Γ(t N ) B ηn (0), which is possible since η diam. For N = we take p := Γ(t); hence p = η. Introduce the lenses and doughnuts (compare with (.4) and (.5) for the definition) (3.4) l N := l(0, p N ; r N ), V N := V (0, p N ; r N ), and finally let (3.5) K N := B ηn (0) \ V N. Our general aim will be to show first that for each N =,,... the part of which is in B ηn (0) is either in K N or in l N or very close to one of the points 0, p N. To be more precise, let α N denote the opening angle of the smallest cone with vertex at 0 containing l N ; then (3.6) sin α N = η N r N.

11 A geometric curvature double integral of Menger type for space curves 05 Let (3.7) (3.8) (3.9) ( ) ( pn + Γ(0) h N := dist, l N = r N cos α ) N, ϕ N := arc tan h N, η N h N := h N sin ϕ N. Figure. (See left.) A plane passing through two points 0 = Γ(0) and p N = Γ(t N ), located at the centers of two tiny shaded balls in the middle part of the picture. Big arcs represent the boundaries of two of the balls of radius r N whose union is equal to the doughnut V N. (Note: for ε small and N large, the ratio r N /η N is in fact much larger than the figure shows.) Later on, cf. (3.) and (3.3), we prove that B ηn (0) must be contained in l N B hn (p N ) B hn (0) K N that is, in that portion of the ball B ηn (0) which is formed by rotation of the shaded region in Fig. around the axis passing through 0 and p N. Figures and 3. Enlarged view of parts of Fig., showing the location of 0 and p N, the distances η N, h N, h N, and the angles α N, ϕ N that are defined by (3.6) (3.9). Finally, let δ N be the smallest angle such that the cone C δn (0; p N ) contains the union l N K N (see Fig. above). An elementary geometric argument shows that δ N = α N + α N, where α N = arcsin η N r N = arcsin ε. Thus, since arcsin x πx/ on N [0, ], we have (3.0) δ N = α N + α N 3π ε N and δ N 3π N= N= if ε/ε is chosen appropriately. The main tool in the proof of Theorem 3. is the following ε N < ε,

12 06 Paweł Strzelecki, Marta Szumańska and Heiko von der Mosel Claim. For each N =,,... the following is true: (3.) Γ(S L ) B ηn (0) C δn (0; p N ) B hn (0). (Applying (3.) iteratively as will be done later, and invoking injectivity of Γ, one easily concludes the whole proof of Theorem 3..) To prove Claim, we need the following elementary relations between h N, h N, η N and r N. Lemma 3.6. (Relations between distances on Figures 3) For all N =,,... we have (3.) ηn h N η N, 4πr N 3r N (3.3) h N η N h N h N h N η N, (3.4) ε η N 4πN. Proof. We use the elementary inequalities (3.5) π x sin x x tan x, 0 < x < π, x (3.6) π (3.7) π By definition of h N, we have ( h N = r N cos α ) (3.6) N cos x x, 0 < x < π, x arc tan x x, 0 < x <. 4 r N α N 8 Moreover, (3.6) h N r ( N αn π This proves (3.). Next, we have (3.5) ) (3.5) π 8 r N sin α N r N π sin α N = η N 4πr N. ( ) (3.6) = π 8 r ηn N < η N. r N 3r N h N = h N sin ϕ N h N tan ϕ N = h N η N and (3.5) ϕ N (3.8) h N h N = h N π π arc tan h (3.7) N h N. η N η N This yields (3.3). Finally, (3.4) follows from (3.) and the definition of r N. Lemma 3.7. (Estimate of R(,, ) when Γ leaves the grey zone) Fix N =,,... and assume that for some τ 0 S L the point Γ(τ 0 ) B ηn (0) but Γ(τ 0 ) l N B hn (p N ) B hn (0) K N. Then for all parameters s A and σ A, where A = {s S L : Γ(s) B hn /0 (p N)}, A = {σ S L : Γ(σ) B hn /0 (0)},

13 we have A geometric curvature double integral of Menger type for space curves 07 (3.8) R(Γ(s), Γ(σ), Γ(τ 0 )) 4r N. In particular, ϱ(s, σ) 4r N for (s, σ) A A. Proof. If τ 0 satisfies the assumptions of Lemma 3.7, then there exists a unique point q R 3 determined by the following three conditions (i) q p N = q = r N, (ii) Γ(τ 0 ) B rn (q) \ l N, (iii) the four points Γ(τ 0 ), q, p N and 0 are co-planar. By elementary geometry, α N (3.9) β 0 := <) ( p N Γ(τ 0 ), Γ(0) Γ(τ 0 ) ) π α N. (This is easy to see: draw two circles c, c of radius r N, containing p N and 0 = Γ(0) and lying in the plane determined by p N, 0 and Γ(τ 0 ). Then, β 0 = π α N / when Γ(τ 0 ) lies on the short arc of c connecting p N to 0, and β 0 = α N / when Γ(τ 0 ) lies on the long arc of c connecting p N to 0. When Γ(τ 0 ) is between these two arcs, β 0 takes some intermediate value.) For s A and σ A, let β(s, σ) denote the angle at Γ(τ 0 ) in the triangle with vertices Γ(s), Γ(σ) and Γ(τ 0 ). We then have (3.30) β(s, σ) β 0 β + β, where β : = <) ( p N Γ(τ 0 ), Γ(s) Γ(τ 0 ) ), β : = <) ( Γ(0) Γ(τ 0 ), Γ(σ) Γ(τ 0 ) ). Since the distances of Γ(τ 0 ) to p N and to 0 = Γ(0) exceed h N, and because s A and σ A, we have β i β max, i =,, where sin β max = h N /0h N. Hence, (3.5) β max π sin β max = π h N 0h N (3.3) πh N 0η N (3.) πη N (3.6) = π 30r N 5 sin α N Therefore, by (3.9) and (3.30), we obtain α N 4 β(s, σ) π α N 4 and (3.5) < α N 8. (3.3) sin β(s, σ) sin α N 4. Thus, R(Γ(s), Γ(σ), Γ(τ 0 )) = = Γ(s) Γ(σ) sin β(s, σ) η N sin(α N /4) η N sin(α N /) cos α N η N 4 sin(α N /) (3.6) = 4r N. We are now ready for estimates of the energy which prove Claim by contradiction. Assume that Claim were false. Fix N. Since (3.3) C δn (0, p N ) B hn (0) l N B hn (p N ) B hn (0) K N,

14 08 Paweł Strzelecki, Marta Szumańska and Heiko von der Mosel we would then find a parameter τ 0 satisfying the assumptions of Lemma 3.7. Recall that E = I p (). Shrinking the domain of integration to A A, we obtain ds dσ E ϱ(s, σ) p if we choose Lemma 3.7 (3.3) (3.3) A A A A 4 p r p N 0 4 p h 4 Nη N r p N as A i h N /0 = 0 4 p h 4 Nη p (3.4) N N p ε p 0 4 p (4π) 4 η p 4+p N N 8 p ε = 0 4 p (4π) p η p ε }{{ 4+p (N )(p ) } N 8+p as ε = ε/0 E c(p) > E (3.33) c(p) = 0 4 p (4π) p ( This contradiction ends the proof of Claim. min N=,... ) (N )(p ) > 0. N 8+p Proof of Theorem 3. continued. Noting that h N < η N and applying Claim inductively, we obtain (3.34) Γ(S L ) B η (0) C δ +δ /+ +δ N /(0; p ) B hn (0), N =, 3,..., since the axis of each of the successive cones lies in the preceding cone in the iteration. As the series δ N converges and its sum is smaller than ε by (3.0), this yields Γ(S L ) B η (0) C ε (0; p ) B hn (0), N =, 3,.... Since h N 0 for N, the intersection of all the sets C ε (0; p ) B hn (0) is equal to the cone C ε (0, p ). This completes the whole proof of Theorem Improved Hölder continuity of tangents In this section we prove that if p > and a curve has finite energy, i.e., I p () <, then in fact its arclength parametrization Γ is of class C,β for β = p. Theorem 4.. If I p () < for some p >, then Γ is contained in the class C, (/p) (S L, R 3 ), and there exists a constant M = M() such that (4.) Γ (u) Γ (v) M u v /p (E [u,v] ) /p for all u, v S L, where E [u,v] := v v u u dσ dτ ϱ(σ, τ) p. Remark. We have no explicit examples showing that (4.) is optimal for curves with bounded I p -energy. However, let us note that for curves of bounded global curvature it implies a qualitatively optimal result: if ϱ(s, σ) θ > 0 for all s, σ S L, then Γ is Lipschitz with a constant proportional to /θ; see [SvdM03a, Lemma ].

15 A geometric curvature double integral of Menger type for space curves 09 (Note that even when /ϱ is uniformly bounded, Γ does not have to be of class C, as in the example of a stadium curve, where two coplanar semicircles are connected with two parallel straight segments.) Proof of Theorem 4.. If E [u,v] = 0, then Γ([u, v]) is a segment of a straight line and there is nothing to prove. From now on we assume E [u,v] 0. We know from Corollary 3. that Γ C,α (S L, R 3 ) for α := (p )/(p + 4). This implies that Γ is a local homeomorphism in the following sense: Lemma 4.. Assume that (4.) Γ (t) Γ (s) C Γ t s α, t, s S L. Then for every λ (0, ) there exists a positive number δ = δ(λ, C Γ, α) < such that (4.3) Γ(t) Γ(s) λ t s for all t, s S L with t s < δ. Proof. Fix s S L and without loss of generality assume that Γ (s) = (, 0, 0). We then have for all t satisfying Γ (t) Γ (s) Γ (s) Γ (t) C Γ t s α λ ( λ t s < δ := C Γ ) /α (w.l.o.g. we may assume that C Γ so that δ <.) Thus t Γ(t) Γ(s) Γ (t) Γ (s) = Γ (τ) dτ λ t s. This ends the proof of Lemma 4.. From now on we fix λ = λ 0 : = ( 3 )p/ and choose δ = δ(λ 0, C Γ, α). In our situation we have α = (p )/(p + 4) and C Γ = M c(p)i p () /(p+4) ; see Corollary 3.. Therefore δ depends on p and I p () so far. Throughout the rest of the proof, we assume that u, v S L are fixed and 0 < v u < δ <. Take any c > 0 such that (4.4) ( 8c p E [u,v] ) α 4. For example, (4.5) c: = ( 8 +/α E [u,v] ) /p, satisfies this smallness condition and will serve our purposes later on. Remark. Note that one could replace E [u,v] in (4.4) by E [u,v ], where [u, v ] [u, v], and keep the same constant c as in (4.5). Since ( E [u,v] ) /p ( E[u,v ]) /p, the smallness condition (4.4) would still be satisfied. Shrinking δ if necessary and using the absolute continuity of the integral, we may also assume that E [u,v] 8 (+/α) whenever u v < δ. Then c >. s

16 0 Paweł Strzelecki, Marta Szumańska and Heiko von der Mosel (4.6) (4.7) Let T s := {t [u, v] : ϱ(s, t) c Γ(s) Γ(t) p }, S := {s [u, v] : H (T s ) u v }. 8 (An informal word about the structure of the whole proof might be of some use here: These sets in fact do depend on u, v but we do not make this explicit in the notation in order to avoid too many indices. Think of parameters in S as bad. The overall strategy is as follows. First we prove in Lemma 4.4 that the tangent Γ behaves nicely when, for given u, v, one restricts it to good parameters [u, v] \ S. Then we notice that there are good parameters very close to u, v, and this observation gives an improvement of the initial Hölder estimate (4.). It turns out that the whole reasoning can be iterated; in the limit we obtain the desired conclusion (4.).) In order to find an upper bound for the measure of S we estimate v v E [u,v] = ϱ p (s, t) dt ds ϱ p (s, t) dt ds u u S T s c p Γ(s) Γ(t) dt ds S T s > u v ds as Γ(s) Γ(t) u v < 8H (T 8c p s ) for s S Thus S H (S) ( 8c p u v ). (4.8) H (S) < 8c p E [u,v] u v. In the sequel, we use primes to denote complements of various sets in S L, i.e., S : = S L \ S etc. Lemma 4.3. If s, s [u, v] \ S, then T s T s [u, v] is nonempty; in fact, H (T s T s [u, v]) 3 u v. 4 Proof. For i =, we have, by definition, H (T s i [u, v]) 7 u v. 8 Thus (4.9) H (T s T s [u, v]) H (T s [u, v]) + H (T s [u, v]) H ( (T s T s ) [u, v] ) 3 u v. 4 This observation completes the proof of Lemma 4.3. Lemma 4.4. If u v < δ, then for s [u, v] \ S and t [u, v] \ T s we have (4.0) Γ (s) Γ (t) c Γ(s) Γ(t) /p. Proof. Fix s [u, v] \ S and t [u, v] \ T s. W.l.o.g. suppose s < t. Set r = c Γ(s) Γ(t) /p and let l := l ( Γ(s), Γ(t); r ), V := V ( Γ(s), Γ(t); r ), A := V \ l;

17 A geometric curvature double integral of Menger type for space curves see (.4) and (.5) for the definition of lenses l( ) and doughnuts V ( ). Since t T s, we have ϱ(s, t) > r by the very definition of T s. Thus Γ ( [s, t] ) R 3 \ A. As is simple, i.e., Γ is, we have either Γ ( (s, t) ) l or Γ ( [s, t] ) R 3 \ V. We now exclude the latter case. Since the shortest curve that joins Γ(s) and Γ(t) in R 3 \ V has length at least πc Γ(s) Γ(t) /p, the inclusion Γ ( [s, t] ) R 3 \ V would imply by our choice of λ = λ 0 = (/3) (p/) in (4.3) that (4.3) /p s t 3c Γ(s) Γ(t) c s t /p. Thus s t /p c > and therefore u v s t, which is absurd, since u v < δ <. Hence, we have Γ ( (s, t) ) l. This yields where the angle ϕ is given by Γ(σ) Γ(s) C + ϕ (0; Γ(t) Γ(s)) ϕ := arcsin Γ(s) Γ(t) c Γ(s) Γ(t) /p. for all σ (s, t), Thus Γ (s) belongs to the closure of C + ϕ (0; Γ(t) Γ(s)). Analogously, we prove that Γ (t) belongs to the closure of C + ϕ (0; Γ(t) Γ(s)). This gives as arcsin τ τ on [0, ]. Γ (s) Γ (t) ϕ c Γ(s) Γ(t) /p, We are now ready to improve the Hölder exponent of Γ. So we continue the Proof of Theorem 4.. Using (4.8), (4.9) and the smallness condition (4.4), we select three points s [u, v] \ S, s [u, v] \ S and t [u, v] \ (T s T s ) such that u < s < t < s < v and (4.) u s < 8c p E [u,v] u v < 8 u v, v s < 8c p E [u,v] u v < u v. 8 Applying (4.0) twice, we obtain (4.) Γ (s ) Γ (s ) Γ (s ) Γ (t) + Γ (s ) Γ (t) 4c s s p. Thus (4.3) Γ (u) Γ (v) Γ (u) Γ (s ) + Γ (s ) Γ (s ) + Γ (s ) Γ (v) C Γ ( 8c p E [u,v] ) α u v α + 4c s s p (4.4) 4 C Γ u v α + 4c u v p. Now, let us note that for c given by (4.5) and s, s selected above we still have ( ) 8c p α E [u,s ] 4 and ( ) 8c p α E [s,v] 4 ; cf. the remark immediately after (4.5). Besides, u s and v s are smaller than δ. Thus, we may repeat the whole reasoning leading from (4.4) to (4.3) (twice, first

18 Paweł Strzelecki, Marta Szumańska and Heiko von der Mosel for the pair u, s instead of u, v, and then for s, v instead of u, v) and obtain two inequalities similar to (4.3): (4.4) (4.5) Γ (u) Γ (s ) 4c u s /p + 4 C Γ u s α, Γ (v) Γ (s ) 4c v s /p + 4 C Γ v s α. Using (4.), (4.4) and (4.5), and keeping in mind that α <, we now obtain p Γ (u) Γ (v) Γ (u) Γ (s ) + Γ (s ) Γ (s ) + Γ (s ) Γ (v) 4c u s /p + 4 C Γ u s α (4.6) (4.) + 4c v s /p + 4 C Γ v s α + 4c u v /p 8c ( 8c p E [u,v] ) α u v /p + 4c u v /p Again, the resulting inequality + C ( ) α u Γ 8c p E [u,v] v α (4.4) 4 C Γ u v α + 4c u v /p ( + 4 ) Γ (u) Γ (v) 4 C Γ u v α + 4c u v /p( + 4 ) is valid for any pair of u, v S L such that u v < δ and any constant c such that the smallness condition (4.4) is satisfied. Thus, as in (4.4) (4.5), we may keep here the c defined by (4.5) but replace u in (4.4) by s (or v by s ). Iterating computations similar to the proof of (4.6), we check by induction that Γ (u) Γ (v) 4 n+ C Γ u v α + 4c u v /p 4 n+ C Γ u v α + 8c u v /p, n =,,.... Upon letting n, we finally obtain (4.7) Γ (u) Γ (v) 8c u v /p = 8 +(/p)+(/pα)( E [u,v] ) /p u v /p whenever u v < δ. If u v δ, then one simply has to divide the segment [u, v] into k [L/δ] + pieces [u j, u j+ ] of length at most δ, where j = 0,,..., k and u = u 0 < u < < u k = v. Since each of the squares [u j, u j+ ] is contained in [u, v], we have E [uj,u j+ ] E [u,v] for each j, and therefore k Γ (u) Γ (v) Γ (u j ) Γ (u j+ ) (4.7) j=0 C(p, α) ( E [u,v] ) /p k δ /p C(p, α) ( E [u,v] ) /p ( L δ + ) u v /p, n k=0 4 k C(p, α) ( ) /p k E [u,v] u j u j+ /p j=0

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20 4 Paweł Strzelecki, Marta Szumańska and Heiko von der Mosel [Ma98] [Ma04] Mattila, P.: Rectifiability, analytic capacity, and singular integrals. - Proc. ICM, Vol. II (Berlin 998), Doc. Math. 998, Extra Vol. II, (electronic). Mattila, P.: Search for geometric criteria for removable sets of bounded analytic functions. - Cubo 6, 004, 3 3. [Mel95] Melnikov, M.: Analytic capacity: discrete approach and curvature of measure. - Sb. Mat. 86, 995, [Men30] Menger, K.: Untersuchungen über allgemeine Metrik. Vierte Untersuchung. Zur Metrik der Kurven. - Math. Ann. 03, 930, [O H9] O Hara, J.: Family of energy functionals of knots. - Topology Appl. 48, 99, [O H03] O Hara, J.: Energy of knots and conformal geometry. - Ser. Knots Everything 33, World Scientific, River Edge, 003. [P0] Pajot, H.: Analytic capacity, rectifiability, Menger curvature and the Cauchy integral. - Lecture Notes in Math. 799, Springer, Berlin-Heidelberg-New York, 00. [R05] Reiter, P.: All curves in a C -neighbourhood of a given embedded curve are isotopic. - Preprint Nr. 4, Institut f. Mathematik, RWTH Aachen 005; see preprints. [Schu06] Schul, R.: Ahlfors-regular curves in metric spaces. - Ann. Acad. Sci. Fenn. Math. 3, 007, [SvdM03a] Schuricht, F., and H. von der Mosel: Global curvature for rectifiable loops. - Math. Z. 43, 003, [SvdM03b] [SvdM04] [StvdM05] [StvdM06] [StvdM07] [SStvdM07] Schuricht, F., and H. von der Mosel: Euler Lagrange equations for nonlinearly elastic rods with self-contact. - Arch. Rat. Mech. Anal. 68, 003, Schuricht, F., and H. von der Mosel: Characterization of ideal knots. - Calc. Var. Partial Differential Equations 9, 004, Strzelecki, P., and H. von der Mosel: On a mathematical model for thick surfaces. - In: Physical and numerical models in knot theory, edited by Calvo, Millett, Rawdon and Stasiak, Ser. Knots Everything 36, World Scientific, Singapore, 005, Strzelecki, P., and H. von der Mosel: Global curvature for surfaces and area minimization under a thickness constraint. - Calc. Var. Partial Differential Equations 5, 006, Strzelecki, P., and H. von der Mosel: On rectifiable curves with L p -bounds on global curvature: Self-avoidance, regularity, and minimizing knots. - Math. Z. 57, 007, Szumańska, M., P. Strzelecki, and H. von der Mosel: Regularizing and self-avoidance effects of integral Menger curvature. - In preparation. Received 7 March 008