A Li-Yau inequality for the 1-dimensional Willmore energy
AA Li–Yau inequality for the 1-dimensionalWillmore energy
Marius M¨uller ∗ and Fabian Rupp † March 2, 2021
Abstract:
By the classical Li–Yau inequality, an immersion of a closedsurface in R n with Willmore energy below 8 π has to be embedded. Wediscuss analogous results for curves in R , involving Euler’s elastic energy andother possible curvature functionals. Additionally, we provide applicationsto associated gradient flows. Keywords:
Li–Yau inequality, Willmore functional, elastic energy, embeddedness.
MSC(2020) : 53A04 (primary), 49Q10, 53E40 (secondary).
1. Introduction and main results
For an immersion f : Σ → R n of a surface Σ, its Willmore energy is defined by W ( f ) := 14 (cid:90) Σ | H | d µ. (1)Here H denotes the mean curvature vector and µ is the Riemannian measure inducedby pulling back the Euclidean metric to Σ. In their fundamental work [LY82], Li andYau proved an inequality which yields that an immersion with not too large Willmoreenergy must in fact be an embedding. More specifically, if Σ is compact, then W ( f ) < π implies that f is an embedding. (2)Moreover, as a doubly covered round sphere shows, the constant 8 π in (2) is optimal. ∗ Mathematisches Institut, Albert–Ludwigs–Universit¨at Freiburg, 79104 Freiburg im Breisgau, Ger-many. [email protected] † Institute of Applied Analysis, Ulm University, Helmholtzstraße 18, 89081 Ulm, Germany. [email protected] a r X i v : . [ m a t h . DG ] M a r n this article, we study the question whether an analogous result as in (2) is true forplanar curves. For a closed smooth curve γ : S → R , which is immersed, i.e. | γ (cid:48) | > κ , its elastic energy is defined by E ( γ ) := (cid:90) S | κ | d s. This formally resembles the Willmore energy. However, in contrast to (1), E is not scalinginvariant, whereas the property of being embedded is. A natural scaling invariant one-dimensional version of the Willmore energy is the total curvature , defined by K ( γ ) := (cid:90) S | κ | d s. It has a wide range of geometric applications and has been studied extensively, forinstance in [Fen29, F´ar49, Mil50, Mil53].We will show that the total curvature does not allow for a non-trivial version of (2).This will be a consequence of the following observation.
Theorem 1.1.
We have π = inf {K ( γ ) | γ ∈ C ( S ; R ) non-embedded immersion } = inf {K ( γ ) | γ ∈ C ( S ; R ) immersion } . Moreover, the infimum among non-embedded immersions is not attained.
In order to obtain a non-trivial version of (2) we need to identify a different quantity.Our main result shows that the elastic energy provides a positive answer, when restrictedto curves of fixed length, or — equivalently — multiplied with the length functional L . Theorem 1.2 (Main theorem) . If γ ∈ W , ( S , R ) is an immersed curve with E ( γ ) L ( γ ) < c ∗ := E ( γ ∗ ) L ( γ ∗ ) , then γ is an embedding. Here γ ∗ is the figure eight elastica (see Definition 5.1). Remark 1.3.
The value of c ∗ is sharp since γ ∗ itself is not an embedding, cf. Lemma 5.6for the details. A numerical computation yields c ∗ (cid:39) . . We have thus identified a geometric quantity of curves whose smallness ensures that thecurve is embedded.Note that it was already observed in [Woj20, Lemma 2.1], that if γ ∈ W , ( S ; R n )has a point with multiplicity k ∈ N , then E ( γ ) L ( γ ) > π k . Such a relation betweenmultiplicity and energy is also part of the statement of the original Li–Yau inequalityfor the Willmore energy [LY82]. However, we remark that the bound in [Woj20] cannotbe used to ensure embeddedness, since any curve γ with E ( γ ) L ( γ ) ≤ π has to bea one-fold covered circle, cf. Lemma 5.4. Moreover, with the methods of geometric2easure theory, in [Poz20, (16)] the bound k ≤ K ( γ ) was established. As K ( γ ) ≥ π ,this estimate cannot answer the question of embeddedness either.The idea for proving Theorem 1.2 is to look at the variational probleminf {E ( γ ) L ( γ ) | γ ∈ W , ( S ; R ) non-embedded immersion } . (3)Minimizing among non-embedded immersions is a non-standard condition, because theadmissible set is not open. This causes difficulties in applying Euler–Lagrange methods.However, we will be able to deduce that the minimizer is an interior point of the ad-missible set and thus satisfies an Euler–Lagrange equation. This can be achieved by adetailed analysis of the self-intersections of minimizers. The main ingredient here is theclassification of planar elastic curves (see for instance [LS84c, DHMV08, DP17]).As a future extension of Theorem 1.2 one could also try to find such embeddedness-ensuring quantities in other ambient manifolds than R .In the hyperbolic halfplane H an embeddedness-ensuring quantity can indeed be iden-tified. By [LS84a] one has for all immersed curves γ ∈ C ∞ ( S ; H ) (cid:90) S | κ H [ γ ] | d s = 2 π W ( S ( γ )) , where κ H [ γ ] denotes the hyperbolic curvature of γ and S ( γ ) denotes the immersion thatarises from revolution of γ around the x -axis. This and (2) yield that (cid:90) S | κ H [ γ ] | d s <
16 implies that γ is an embedding . The threshold of 16 is also sharp for this implication, cf. [MS20, Corollary 6.4]. Noticethat this does not immediately follow from the sharpness of the inequality in [LY82].In S the elastic energy of curves γ ∈ C ∞ ( S ; S ) given by γ (cid:55)→ (cid:90) S | κ S [ γ ] | d s is not an embeddedness-ensuring quantity since any two-fold cover of a closed geodesicin S is non-embedded and has vanishing energy.It would be interesting to investigate whether Theorem 1.2 generalizes to curves in R ,see Remark 5.14 for some ideas in this context.
2. Notational preliminaries
In the following, we will view the 1-sphere as S = [0 , / ∼ , where ∼ denotes the equiv-alence relation that identifies 0 ∼ S ∼ = R / Z . Consequently, an interval [ a, b ] ⊂ S with a < b has to be understoodwith respect to this equivalence relation, i.e. [ a, b ] = { [ x ] ∼ : x ∈ [ a, b ] } . For the sake ofsimplicity of notation, we define the interval [ b, a ] := [ b, ∪ [0 , a ] for a, b ∈ [0 ,
1] with a < b . In the same fashion, the open and half-open intervals are defined.3 efinition 2.1.
We define for k, (cid:96) ∈ N , p ∈ [1 , ∞ ] the Sobolev space W k,p ( S ; R (cid:96) ) as W k,p ( S ; R (cid:96) ) := { u ∈ W k,p ((0 , R (cid:96) ) | u ( m ) (0) = u ( m ) (1) ∀ m = 1 , ..., k − } , where u ( m ) denotes the continuous representative of the m -th weak derivative. Moreover,we denote by W k,pImm ( S ; R (cid:96) ) the set of W k,p -immersions. Remark 2.2.
It can be seen that this definition coincides with the general definition ofSobolev spaces on manifolds, cf. [Heb96, Definition 2.1]. This is why we can also usegeneral results about these spaces and also talk about Sobolev spaces on open subsets of S . We will refer to curves in C k ( S ; R ) as C k -closed, which is also due to the fact that C k ( S ; R ) = { u ∈ C k ([0 , R ) | u ( m ) (0) = u ( m ) (1) ∀ m = 0 , ..., k } . In particular, each curve W k,p ( S ; R ) is C k − -closed. Observe also that each curve in C k ( S ; R ) possesses an extension to a -periodic curve in C k ( R , R ) . Remark 2.3.
Another noticable property of W k,p ( S ) is glueing, i.e. if u ∈ W k,p (( a, b )) and v ∈ W k,p (( b, a )) are such that u ( m ) ( a ) = v ( m ) ( a ) and u ( m ) ( b ) = v ( m ) ( b ) for all m = 0 , ..., k − then w ( x ) := (cid:40) u ( x ) x ∈ ( a, b ) ,v ( x ) x ∈ ( b, a ) lies in W k,p ( S ) . We now review some basic geometric definitions of planar curves. For an immersion γ : S → R we write γ ( x ) = ( γ ( x ) , γ ( x )) , x ∈ S for the components and γ (cid:48) = ∂ x γ forthe derivative. Moreover, we write κ := | γ (cid:48) | − det ( γ (cid:48) , γ (cid:48)(cid:48) ) for its (signed) curvature . An-other important geometric object is the arc-length derivative , denoted by ∂ s = | γ (cid:48) | − ∂ x and the arc-length element d s := | γ (cid:48) | d x . The curvature vector field is (cid:126)κ = ∂ s γ = κ(cid:126)n ,where (cid:126)n denotes the unit normal , obtained by rotating ∂ s γ counterclockwise by π .
3. A non-existence result for the total curvature
In this section, we will prove Theorem 1.1 and show why it implies that there is nonon-trivial generalization of (2) involving the total curvature.
Proof of Theorem 1.1.
By Fenchel’s theorem, cf. Theorem A.1, we have K ( γ ) ≥ π forall γ ∈ C ( S ; R ). Consequentlyinf (cid:8) K ( γ ) | γ ∈ C ( S ; R ) non-embedded immersion (cid:9) ≥ π. (4)To prove equality, we take some angle β ∈ (0 , π ). First, take a segment S β of a circle ofradius 1 of length 2 π − β and place it symmetrically with respect to the x -axis. Extendthe segment by the tangent lines at its endpoints. For β ∈ (0 , π ), they will intersect ina point on the x -axis. Reflecting everything with respect to that point gives a closedcurve γ which is not embedded, cf. Figure 1a.4 a) The construction with β = π (b) The curve becomes longer as β (cid:37) π Figure 1.: A non-embedded W , -curve with approximate total curvature 2 π as β (cid:37) π .By the glueing property in Remark 2.3, we have γ ∈ W , ( S ; R ). In order to compute K ( γ ) we first note that | κ | ≡ κ ≡ K ( γ ) = 2 (cid:90) S β d s = 2(2 π − β ) = 4 π − β (cid:38) π, as β (cid:37) π . Using the characterization of the equality in Fenchel’s theorem (Theorem A.1), we mayconclude that the infimum in (4) is not attained by a non-embedded immersion.Theorem 1.2 shows that K cannot distinguish between embedded and non-embeddedimmersions, since the least possible energy among all closed curves can be approximatedby non-embedded ones. This justifies that (2) has no non-trival generalization to thetotal curvature.
4. The variational problem and existence of a minimizer
In order to prove Theorem 1.2, we wish to minimize the functional E ( γ ) L ( γ ) among allcurves γ ∈ W , Imm ( S ; R ) which are not embeddings.As a first step, we want to characterize embeddedness in a way that is useful for varia-tional discussions. Lemma 4.1.
A curve γ ∈ W , Imm ( S , R ) is an embedding, if and only if γ is injective.Proof. See for instance [Lee03, Proposition 5.4].This implies the following very useful characterization of embeddedness.
Proposition 4.2 (Characterization of Embeddedness) . Let γ ∈ W , ( S ; R ) be animmersion. Then γ is embedded if and only if A [ γ ] := inf x (cid:54) = y | γ ( x ) − γ ( y ) || x − y | > .Proof. Suppose A [ γ ] >
0. Then, we have | γ ( x ) − γ ( y ) | ≥ A [ γ ] | x − y | > x (cid:54) = y ∈ S .Thus γ is injective, hence an embedding by Lemma 4.1.Conversely, suppose γ is an embedding and A [ γ ] = 0. Then, there exist x n (cid:54) = y n suchthat | γ ( x n ) − γ ( y n ) || x n − y n | →
0. Passing to a subsequence, we have x n → x, y n → y for x, y ∈ S by compactness. If x (cid:54) = y , we have0 = lim n →∞ | γ ( x n ) − γ ( y n ) || x n − y n | = | γ ( x ) − γ ( y ) || x − y | , γ ( x ) = γ ( y ). This is a contradiction to the embeddedness of γ , cf. Lemma 4.1.Hence x = y in S and for i = 1 , γ i ( x n ) − γ i ( y n ) = γ (cid:48) i ( ξ i,n )( x n − y n )for some ξ i,n ∈ S between x n and y n . Dividing by x n − y n and using the assumption,we find γ (cid:48) i ( x ) = lim n →∞ γ (cid:48) i ( ξ i , n ) = lim n →∞ γ i ( x n ) − γ i ( y n ) x n − y n = 0 , for i = 1 ,
2, a contradiction to γ being an immersion.An important consequence is the following lemma. Lemma 4.3.
The set of C -embeddings is an open subset of C ( S ; R ) .Proof. Suppose γ ∈ C ( S ; R ) is an embedding. We claim that for ε > γ ∈ C ( S ; R ) with (cid:107) ˜ γ − γ (cid:107) C < ε is an embedding. By Proposition 4.2, it sufficesto show A [˜ γ ] >
0, since ˜ γ is clearly an immersion for ε > x (cid:54) = y ∈ S | ˜ γ ( x ) − ˜ γ ( y ) || x − y | ≥ | γ ( x ) − γ ( y ) | − | ˜ γ ( x ) − γ ( x ) − (˜ γ ( y )) − γ ( y )) || x − y |≥ A [ γ ] − sup x (cid:54) = y | ˜ γ ( x ) − γ ( x ) − (˜ γ ( y )) − γ ( y )) || x − y |≥ A [ γ ] − (cid:107) ˜ γ − γ (cid:107) C ≥ A [ γ ] − ε > ε > Remark 4.4.
In the proofs of Proposition 4.2 and Lemma 4.3, we did not really use thespecific structure of S . In particular, the statements of Proposition 4.2 and Lemma 4.3remain true if one replaces S by any compact interval [ a, b ] ⊂ R . We now consider a minimization problem, cf. (3). Define the set A := { γ ∈ W , ( S ; R ) | γ is a non-injective immersion } ⊂ W , ( S ; R ) . Theorem 4.5 (Existence of a Minimizer) . There exists ¯ γ ∈ A with E (¯ γ ) L (¯ γ ) = inf γ ∈A E ( γ ) L ( γ ) > . (5) Proof.
We consider the set ˜ A := { γ ∈ A | L ( γ ) = 1 } (cid:54) = ∅ . As a first step, we show thatthere exist ¯ γ ∈ ˜ A with E (¯ γ ) = inf γ ∈ ˜ A E ( γ ) . (6)6et (cid:0) γ ( n ) (cid:1) n ∈ N be a minimizing sequence for (6). Without loss of generality, we mayassume γ ( n ) to be unit speed parametrized for all n ∈ N , cf. Lemma A.6. By reflexivityand the compactness of the embedding W , ( S ; R ) (cid:44) → C ( S ; R ), we have γ ( n ) (cid:42) ¯ γ in W , ( S ; R ) and γ ( n ) → ¯ γ in C ( S ; R ) for some ¯ γ ∈ W , ( S ; R ), passing to asubsequence. We have | γ (cid:48) n ( x ) | = 1 for all x ∈ S , n ∈ N , thus ¯ γ is parametrized with unitspeed and L (¯ γ ) = 1. Moreover, by Lemma 4.3, ¯ γ cannot be an embedding as γ ( n ) → ¯ γ in C ( S ; R ). Consequently we have ¯ γ ∈ ˜ A by Lemma 4.1.For unit speed curves γ ∈ W , ( S ; R ), the elastic energy is given by E ( γ ) = (cid:90) S (cid:12)(cid:12) γ (cid:48)(cid:48) ( x ) (cid:12)(cid:12) d x. Since the L ( S ; R )-norm is weakly lower semicontinuous, we conclude E (¯ γ ) ≤ lim inf n →∞ E ( γ n ) = inf γ ∈ ˜ A E ( γ ) , so ¯ γ is a minimizer. Since R does not allow for closed geodesics one infers that E (¯ γ ) > γ ∈ W , ( S ; R ) is an arbitrary immersion, we may consider the rescaled curve˜ γ := L ( γ ) γ , so ˜ γ ∈ ˜ A . Note that γ is an embedding if and only if ˜ γ is an embedding.Consequently, we have E ( γ ) L ( γ ) = E (˜ γ ) ≥ E (¯ γ ) = E (¯ γ ) L (¯ γ ) , and hence inf γ ∈A E ( γ ) L ( γ ) = E (¯ γ ) L (¯ γ ).
5. The Euler-Lagrange equation
In this section, we will study the properties of a minimizer ¯ γ from Theorem 4.5 in orderto prove our main theorem. The most important property we will derive is that γ is a constrained elastica , i.e. ¯ γ ∈ C ∞ ( S ; R ) and ∂ s κ + 12 κ − λκ = 0 for some λ ∈ R . (7)Solutions of the constrained elastica equation have been classified in previous works, eg.by [LS84c, DHMV08]. What one needs to show for this is that ¯ γ satisfies the Euler–Lagrange equation , i.e. for all φ ∈ C ∞ ( S ; R ) one has L (¯ γ ) D E (¯ γ )( φ ) + E (¯ γ ) D L (¯ γ )( φ ) = 0 . (8)Indeed, as one can see following the lines of [EG19, Section 5], (8) implies that γ issmooth and (7) holds. If ¯ γ is an interior point of A then (8) follows from the fact thatfor all φ ∈ C ∞ ( S ; R ) one has0 = dd ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε =0 E (¯ γ + εφ ) L (¯ γ + εφ ) , (9)7igure 2.: A tangential self intersection vanishing under small perturbations.since ( − ε , ε ) (cid:51) ε (cid:55)→ E (¯ γ + εφ ) L (¯ γ + εφ ) attains a minimum at ε = 0. The problem isthat we do not know to begin with whether the infimum is attained at an interior pointof A . The goal of this section is to prove exactly this.We remark that A ⊂ W , ( S ; R ) is not open. Indeed, like in Figure 2 it is possibleto leave the set A by a variation that eliminates self-intersections. To show that eachminimizer is an interior point we need to examine these self-intersections, denoted by S [ γ ] := { p ∈ R : ∃ x (cid:54) = x s.t. γ ( x ) = γ ( x ) = p } . We also define the multiplicity of p ∈ S [ γ ] to be mult[ γ ]( p ) := H ( γ − ( { p } )) where H denotes the counting measure. Moreover we have to pay special attention to thetangential self-intersections given by S tan [ γ ] := { p ∈ R : ∃ x (cid:54) = x s.t. γ ( x ) = γ ( x ) = p, det( γ (cid:48) ( x ) , γ (cid:48) ( x )) = 0 } . Before we can proceed with the proof we need some preparations that exclude certainconfigurations.
In this section, we will discuss properties of closed elasticae. We remark that at thispoint we do not know whether the minimizer ¯ γ is a constrained elastica. However, theanalysis of the energy and the self-intersection properties of closed elasticae will playa crucial role in proving Theorem 1.2 later. More precisely, we will be able to excludeself-intersection properties of minimizers ¯ γ once we can exclude them for non-embeddedconstrained elasticae. Define B := { γ ∈ C ∞ ( S ; R ) : γ is a non-embedded constrained elastica } ⊂ A . (10)Throughout this section we will use the elliptic functions defined in Appendix B. Wenext define one special elastica, which will be important for our considerations. Definition 5.1 (The Elastic Figure Eight) . Let m ∗ ∈ (0 , be the unique root of themap (0 , (cid:51) m (cid:55)→ E ( m ) − K ( m ) , see Lemma B.4. We define the one-fold cover of the8lastic figure eight by γ ∗ : [0 , K ( m ∗ )] → R , γ ∗ ( s ) := (cid:18) E (am( s, m ∗ ) , m ∗ ) − s − √ m ∗ cn( s, m ∗ ) (cid:19) . (11) Remark 5.2.
The numerical value of m ∗ is m ∗ (cid:39) . . This is not needed in thesequel, since all the computations we provide are analytical. However, it enables us toapproximate the value of c ∗ . Remark 5.3.
We will also look at another parametrization of the figure eight that iseasier for computations e.g. to compute self-intersections. For this observe that by (11) and Appendix B γ ∗ ( F ( x, m ∗ )) = (cid:18) E ( x, m ∗ ) − F ( x, m ∗ ) − √ m ∗ cos( x ) (cid:19) . and note that F ( · , m ∗ ) is strictly monotone with F (0 , m ∗ ) = 0 and F (2 π, m ∗ ) = 4 K ( m ∗ ) .Hence ˜ γ ∗ : [0 , π ] → R , ˜ γ ∗ ( x ) := (cid:18) E ( x, m ∗ ) − F ( x, m ∗ ) − √ m ∗ cos( x ) (cid:19) is also a parametrization of the figure eight. We show next that the elastic figure eight is smoothly closed and minimizes our functionalin B . It will turn out later that it is actually also a minimizer in A . Lemma 5.4 (Characterization of Closed Elasticae) . The only closed constrained elasti-cae are (possibly rescaled, rotated, translated and reparametrized versions of ) multi-foldcoverings of circles and multi-fold coverings of the figure eight. All of these elasticae arenot embedded except for the one-fold covering of the circle. Moreover, inf γ ∈B E ( γ ) L ( γ ) = E ( γ ∗ ) L ( γ ∗ ) , with γ ∗ as in Definition 5.1. Equality holds if and only if γ is a rescaled, translated androtated reparametrization of γ ∗ .Proof. First we show the assertion that the only closed elasticae are given by the figureeight and the circle. By Proposition B.8 there are — up to scaling, isometries in R , andreparametrization — only five different types of elasticae which we all examine seperatelyfor closedness. Type 1: Linear elasticae.
Since lines are not closed they cannot generate closedelasticae.
Type 2: Wavelike elasticae.
For closedness it is necessary that both componentsof γ are periodic with the same period L , cf. Remark 2.2. The period itself does notmatter since we can always reparametrize the curve. In the wavelike case we have byProposition B.8 γ ( s ) = 2 E (am( s, m ) , m ) − s γ ( s ) = − √ m cn( s, m ) .
9y Proposition B.3 all periods L of γ are given by L ∈ { lK ( m ) : l ∈ N } . We investigate with the aid of Proposition B.3 for which values of m one of these periodsis also a period of γ . For l ∈ N we compute γ ( s + 4 lK ( m )) = 2 E (am( s, m ) + 2 lπ, m ) − s − lK ( m ) (12)= 2 E (am( s, m ) , m ) + 8 lE ( m ) − s − lK ( m )= γ ( s ) + 4 l (2 E ( m ) − K ( m )) . Hence γ and γ share a period if and only if 2 E ( m ) − K ( m ) = 0. In this case 4 K ( m ) isalready a joint period of γ and γ , hence a period of γ . By Lemma B.4, 2 E ( m ) − K ( m )has only one zero m ∗ ∈ (0 , Type 3: Borderline elastica.
Can not be periodic since the second component hasno real period.
Type 4: Orbitlike elasticae.
We proceed similar to the wavelike case. Recall that byProposition B.8 γ ( s ) = 2 m E (am( s, m ) , m ) + (cid:18) − m (cid:19) s, γ ( s ) = − m dn( s, m ) . By Proposition B.3 all periods L of γ are given by L ∈ { lK ( m ) : l ∈ N } Next we look at the behavior of γ , which we can characterize by Proposition B.3 to be γ ( s + 2 lK ( m )) = γ ( s ) + l E ( m ) − K ( m ) + 2 mK ( m ) m . Hence γ and γ share a period if and only if 4 E ( m ) − K ( m ) + 2 mK ( m ) = 0, which isimpossible because of Lemma B.5. Hence there do not exist closed orbitlike elasticae. Type 5: Circular elasticae.
Circles are trivially closed. Their period is given by 2 π when using the standard arclength parametrization x (cid:55)→ (cos( x ) , sin( x )).Summarizing our findings for all the types we obtain that the only closed elasticae arecircles and the figure eight as well as multiple coverings of these. We have also foundtheir periods. Next we show that the one-fold cover of the figure eight is not injective,so that it is admissible for the minimization problem in the statement. For this let γ ∗ = ( γ ∗ , γ ∗ ) be the figure eight. A crucial oberservation is that the first component γ ∗ of γ ∗ is already 2 K ( m ∗ )-periodic as one could compute with the same techniques as in(12). Note that γ ∗ ( K ( m ∗ )) = (cid:18) E ( m ∗ ) − K ( m ∗ ) − √ m ∗ cos( π ) (cid:19) = (cid:18) (cid:19) K ( m ∗ ) periodicity of γ ∗ we obtain γ ∗ (3 K ( m ∗ )) = (cid:18) E ( m ∗ ) − K ( m ∗ ) − √ m ∗ cos(3 π ) (cid:19) = (cid:18) (cid:19) . Hence γ ∗ ( K ( m ∗ )) = γ (3 K ( m ∗ )). Next we show that the figure eight is minimizing. Forthis we compare the energy of the one-fold cover of the figure eight to the energy of thedoubly-covered circle. The energy of the doubly covered circle γ dcc is given by E ( γ dcc ) L ( γ dcc ) = 4 π · π = 16 π . (13)For the energy of the figure eight note that L ( γ ∗ ) = 4 K ( m ∗ ) as γ ∗ is arc-lengthparametrized by the construction in Appendix B. Now E ( γ ∗ ) = (cid:90) K ( m ∗ )0 m ∗ cn ( s, m ∗ ) d s = 4 m ∗ (cid:90) π cos ( θ ) (cid:112) − m ∗ sin ( θ ) d θ = 16[( m ∗ − K ( m ∗ ) + E ( m ∗ )] . Hence E ( γ ∗ ) L ( γ ∗ ) = 64( E ( m ∗ ) K ( m ∗ ) + ( m ∗ − K ( m ∗ ) ) . Using that by Definition 5.1 2 E ( m ∗ ) = K ( m ∗ ) we have E ( γ ∗ ) L ( γ ∗ ) = 64(4 m ∗ − E ( m ∗ ) . (14)We show that this quantity is smaller than 16 π . For this we use that by Lemma B.7one has 64(4 m ∗ − E ( m ∗ ) ≤ m ∗ − − m ∗ ) π . (15)With standard arguments it can be shown that g ( z ) := (2 z − − z ) is strictly monotoneon (0 ,
1] and g (1) = 1 which makes (2 m ∗ − − m ∗ ) = g ( m ∗ ) <
1. Therefore, by (14)and (15) E ( γ ∗ ) L ( γ ∗ ) = 64(4 m ∗ − E ( m ∗ ) < π , (16)which implies by (13) that E ( γ ∗ ) L ( γ ∗ ) < E ( γ dcc ) L ( γ dcc ) . Remark 5.5.
From (16) and Remark 5.2 one may compute c ∗ (cid:39) . . Lemma 5.6.
Let γ ∗ be as in Definition 5.1. Then S [ γ ∗ ] = { } with mult[ γ ∗ ](0) = 1 .Moreover, S tan [ γ ∗ ] = ∅ .Proof. Note that the assertion is not affected by reparametrization. We will work withreparametrizations in the sequel. More exactly, we work with the parametrization ˜ γ ∗ from Remark 5.3, given by˜ γ ∗ : [0 , π ] → R , (cid:18) ˜ γ ∗ ( x )˜ γ ∗ ( x ) (cid:19) = (cid:18) E ( x, m ∗ ) − F ( x, m ∗ ) − √ m ∗ cos( x ) (cid:19) . x , x ∈ [0 , π ) be such that ˜ γ ∗ ( x ) = ˜ γ ∗ ( x ) and, without loss of generality, x < x . Note that ˜ γ ∗ ( x ) = ˜ γ ∗ ( x ) yields that cos( x ) = cos( x ) which implies — since x , x ∈ [0 , π ) that x = 2 π − x . We infer that˜ γ ∗ ( x ) = ˜ γ ∗ ( x ) = ˜ γ ∗ (2 π − x ) = 2 E (2 π − x , m ∗ ) − F (2 π − x , m ∗ )= 2 E ( − x , m ∗ ) − F ( − x , m ∗ ) + (2 E ( m ∗ ) − K ( m ∗ ))= 2 E ( − x , m ∗ ) − F ( − x , m ∗ ) = − (2 E ( x , m ) − F ( x , m )) = − ˜ γ ∗ ( x ) . Hence ˜ γ ∗ ( x ) = 0 and therefore also ˜ γ ∗ ( x ) = 0. This means that x and x are solutionsof 2 E ( x, m ∗ ) − F ( x, m ∗ ) = 0By Lemma B.6 this implies that x , x ∈ { , π , π, π } . Since also x = 2 π − x this leaves the only possibility of x = π , x = π . We obtainthat the only self intersection point occurs at ˜ γ ∗ ( π ) = ˜ γ ∗ ( π ) = (0 , T . Consequently, S [˜ γ ∗ ] = { (0 , T } with mult[˜ γ ∗ ]((0 , T ) = 2. It remains to show that S tan [˜ γ ∗ ] = ∅ , i.e.the self-intersection is not tangential. To do so we compute˜ γ ∗(cid:48) ( x ) = 1 − m ∗ sin ( x ) (cid:112) − m ∗ sin ( x ) ˜ γ ∗(cid:48) ( x ) = 2 √ m ∗ sin( x )and therefore we have ˜ γ ∗(cid:48) ( π ) = ( − m ∗ √ − m ∗ , √ m ∗ ) T , ˜ γ ∗(cid:48) ( π ) = ( − m ∗ √ − m ∗ , − √ m ∗ ) T and thusdet(˜ γ ∗(cid:48) ( π ) , ˜ γ ∗(cid:48) ( π )) = − − m ∗ ) √ m ∗ √ − m ∗ (cid:54) = 0 , as m ∗ (cid:54) = by Lemma B.4. In this section, we will prove that every minimizer in (5) is a constrained elastica andthus has to be the figure eight by Lemma 5.4, after rescaling, rotation, translation andreparametrization. This is a crucial and non-standard step in proving our main result.First, we examine the number of intersection points and their multiplicities. Then weshow that minimizers have no tangential self-intersections. This allows us to concludethat all minimzers are interior points of A , and hence elasticae. The key idea here iscomparing the minimizer to the figure eight elastica defined in Definition 5.1.Since self-intersection points are delicate to examine we will always localize. We say that¯ γ solves the Euler-Lagrange equation weakly on an open set U ⊂ S if (8) holds for all φ ∈ C ∞ ( U ; R ). Lemma 5.7.
Let γ ∈ A be a minimizer of (5) . Suppose that x ∈ S is such that ¯ γ ( x ) (cid:54)∈ S [ γ ] . Then there exists an open neighborhood U of x in S such that for all φ ∈ C ∞ ( U ; R ) one has L (¯ γ ) D E (¯ γ )( φ ) + E (¯ γ ) D L (¯ γ )( φ ) = 0 . (17)12 roof. Fix x as in the statement. Since ¯ γ ∈ A we have that S [¯ γ ] (cid:54) = ∅ . Hence there existssome p ∈ S [ γ ] and two distinct values x (cid:48) , x (cid:48)(cid:48) ∈ S such that γ ( x (cid:48) ) = γ ( x (cid:48)(cid:48) ) = p . Fix onechoice of such p, x (cid:48) , x (cid:48)(cid:48) . Now set U := S \ { x (cid:48) , x (cid:48)(cid:48) } which is open in S and contains x as γ ( x ) (cid:54) = p . Now fix φ ∈ C ∞ ( U ; R ). We prove that φ satisfies (17). To do so we showthat ¯ γ + εφ ∈ A for all ε ∈ R . As x (cid:48) , x (cid:48)(cid:48) (cid:54)∈ U we obtain that φ ( x (cid:48) ) = φ ( x (cid:48)(cid:48) ) = 0 and thusfor all ε ∈ R (¯ γ + εφ )( x (cid:48) ) = ¯ γ ( x (cid:48) ) = p = ¯ γ ( x (cid:48)(cid:48) ) = (¯ γ + εφ )( x (cid:48)(cid:48) ) . In particular ¯ γ + εφ ∈ A as it has a self-intersection. Equation (17) follows then fromthe consideration in (9).Once we have this tool at hand, we can start to study the self-intersections. Lemma 5.8.
Let ¯ γ ∈ A be a minimizer of (5) . Then S [¯ γ ] = { p } for some p ∈ R .Proof. Note first that S [¯ γ ] (cid:54) = ∅ as ¯ γ is not embedded. We proceed showing that S [¯ γ ]is a singleton. Assume that there exist p , p ∈ S [¯ γ ] such that p (cid:54) = p . In particularthere exists ρ > B ρ ( p ) ∩ B ρ ( p ) = ∅ . We claim that ¯ γ must be a constrainedelastica. For this we show that the Euler-Lagrange equation is globally fulfilled. Thus,we have to discuss the behavior at self-intersection points. Fix any x ∈ S such that¯ γ ( x ) ∈ S [¯ γ ]. We will derive that ¯ γ solves the Euler-Lagrange equation in a neighborhoodof x . We show first that there exists δ = δ ( x ) > γ (( x − δ, x + δ )) hasempty intersection with one of B ρ ( p ) or B ρ ( p ). For this we distinguish two cases,the first one being ¯ γ ( x ) (cid:54)∈ B ρ ( p ). By continuity there exists some δ > | ¯ γ ( x ) − ¯ γ ( x ) | < ρ for all x ∈ ( x − δ, x + δ ) and hence by the triangle inequality¯ γ ( x ) (cid:54)∈ B ρ ( p ) for all x ∈ ( x − δ, x + δ ). The second case is ¯ γ ( x ) ∈ B ρ ( p ), it couldas well be equal to p . By the construction of ρ this implies that ¯ γ ( x ) (cid:54)∈ B ρ ( p ) andthus we can repeat the above continuity argument to find that there exists δ > γ ( x ) (cid:54)∈ B ρ ( p ) for all x ∈ ( x − δ, x + δ ). With this case distinction we havecompleted the construction of δ = δ ( x ). We will without loss of generality assume that¯ γ (( x − δ, x + δ )) has empty intersection with B ρ ( p ), otherwise we switch the roles.We show now that for all φ ∈ C ∞ (( x − δ, x + δ ); R ) one has L (¯ γ ) D E (¯ γ )( φ ) + E (¯ γ ) D L (¯ γ )( φ ) = 0 . (18)To this end we fix φ ∈ C ∞ (( x − δ, x + δ ); R ). We show that for all ε ∈ R one has¯ γ + εφ ∈ A which implies (18) by minimality of ¯ γ . Recall that p ∈ S [¯ γ ] and hence thereexist x (cid:48) , x (cid:48)(cid:48) ∈ S such that ¯ γ ( x (cid:48) ) = ¯ γ ( x (cid:48)(cid:48) ) = p . We claim that x (cid:48) , x (cid:48)(cid:48) (cid:54)∈ ( x − δ, x + δ ).Indeed, if we assume e.g that x (cid:48) ∈ ( x − δ, x + δ ), we obtain by choice of δ that p = ¯ γ ( x (cid:48) ) ∈ ¯ γ (( x − δ, x + δ )). This is a contradiction to the fact that ¯ γ (( x − δ, x + δ ))does not intersect B ρ ( p ). Similarly one obtains that x (cid:48)(cid:48) (cid:54)∈ ( x − δ, x + δ ). Now we cancompute for each ε ∈ R using that supp φ ⊂ ( x − δ, x + δ )(¯ γ + εφ )( x (cid:48) ) = ¯ γ ( x (cid:48) ) = p = ¯ γ ( x (cid:48)(cid:48) ) = (¯ γ + εφ )( x (cid:48)(cid:48) ) . This implies that ¯ γ + εφ ∈ A and — as we discussed — also (18).13ecalling that x was arbitrary we have shown that for each x ∈ S such that x ∈ S [¯ γ ]there exists an open neighborhood U x of x in S such that for all φ ∈ C ∞ ( U x )(18) holds true, under the assumption that there exist p (cid:54) = p ∈ S [¯ γ ]. Since suchneighborhood exists also for non-intersection points by Lemma 5.7 we obtain that foreach x ∈ S there exists an open neighborhood U x of x in S such that the Euler-Lagrangeequation (8) holds for all φ ∈ C ∞ ( U x ; R ). Now we can choose a partition of unity of S subordinate to the cover { U x } x ∈ S which yields finitely many { x , ..., x N } and non-negative functions η x , ..., η x N ∈ C ∞ ( S ; R ) such that the support of η x i is compactlycontained in U x i for all i = 1 , ..., N and N (cid:88) i =1 η x i = 1 . Now fix φ ∈ C ∞ ( S ; R ). We obtain by (18) and the linearity of the Frech´et derivative L (¯ γ ) D E (¯ γ )( φ ) + E (¯ γ ) D L (¯ γ )( φ ) = N (cid:88) i =1 L (¯ γ ) D E (¯ γ )( η x i φ ) + E (¯ γ ) D L (¯ γ )( η x i φ ) = 0 . As discussed after (8), ¯ γ is a constrained elastica. Since ¯ γ minimizes (5), it must thenalso minimize among non-embedded constrained elasticae. However, the one-fold coverof the figure eight γ ∗ is the unique minimizer of EL among all constrained elasticae, upto rescaling, rotation and reparametrization, cf. Lemma 5.4. Therefore, ¯ γ has to be asuitably rescaled, rotated and reparametrized version of γ ∗ . This yields that S [¯ γ ] is asingleton, since S [ γ ∗ ] is a singleton, a contradiction.The ideas of the following proofs will be very similar to the preceding one. We assumethat a certain configuration exists in ¯ γ and then conlude that ¯ γ has to be a constrainedelastica. We then use the classification of those in Section 5.1 to rule out this configura-tion. The next lemma is in the same spirit. Lemma 5.9.
Let ¯ γ ∈ A be a minimizer of (5) . Suppose that p ∈ S [ γ ] . Then mult[¯ γ ]( p ) = 2 . In particular there exist exactly two values x , x ∈ S such that x (cid:54) = x and γ ( x ) = γ ( x ) .Proof. As S [¯ γ ] is a singleton by Lemma 5.8 the last sentence of the claim follows im-mediately from the multiplicity result. As p ∈ S [¯ γ ] we have mult[¯ γ ]( p ) ≥
2. Assumethat mult[¯ γ ]( p ) >
2. We show that then ¯ γ must be a constrained elastica. To thatend, let x ∈ γ − ( { p } ) be arbitrary. We claim that there exists a neighborhood U x on which the Euler Lagrange equation is fulfilled. By the assumption on the multi-plicity there exist two distinct values x (cid:48) , x (cid:48)(cid:48) ∈ γ − ( { p } ) \ { x } . Fix a choice of such x (cid:48) , x (cid:48)(cid:48) . Choose δ = δ ( x ) > x (cid:48) , x (cid:48)(cid:48) (cid:54)∈ ( x − δ, x + δ ). We claim that thenfor all φ ∈ C ∞ (( x − δ, x + δ ); R ) and ε ∈ R one has ¯ γ + εφ ∈ A . This is true since φ ( x (cid:48) ) = φ ( x (cid:48)(cid:48) ) = 0 and thus(¯ γ + εφ )( x (cid:48) ) = ¯ γ ( x (cid:48) ) = p = ¯ γ ( x (cid:48)(cid:48) ) = (¯ γ + εφ )( x (cid:48)(cid:48) ) .
14s a result we may conclude L (¯ γ ) D E (¯ γ )( φ ) + E (¯ γ ) D L (¯ γ )( φ ) = 0 ∀ φ ∈ C ∞ (( x − δ, x + δ ); R ) . We have shown that for all x ∈ γ − ( { p } ) there exists a neighborhood U x such that ¯ γ solves the Euler Lagrange equation weakly on U x . By Lemma 5.8, we have S [¯ γ ] = { p } and hence we infer that for all p = γ ( x ) ∈ S [¯ γ ] there exists a neighborhood U x suchthat ¯ γ solves the Euler Lagrange equation weakly on U x . Together with Lemma 5.7 weconclude that each x ∈ S has an open neighborhood U x such that ¯ γ solves (8) weaklyon U x . One can repeat the partition of unity argument in the proof of Lemma 5.8 tofind that ¯ γ solves (8) globally in S . Hence ¯ γ is by the discussion after (8) a constrainedelastica. By minimality of ¯ γ , it follows that ¯ γ must minimize EL also among non-embedded elasticae and hence is a rescaled, rotated and translated reparametrization ofthe one-fold covered figure eight γ ∗ , cf. Lemma 5.4. As mult[ γ ∗ ]( p ) ≤ p ∈ S [ γ ∗ ]we infer that mult[¯ γ ]( p ) ≤
2, a contradiction.Next we show that ¯ γ has no tangential self-intersections. As a preparation for this, wediscuss an important quantity, the winding number T , which is defined in Definition A.3in Appendix A. Proposition 5.10.
Let γ ∈ A such that T [ γ ] (cid:54) = ± . Then γ is an interior point of A with respect to the W , ( S ; R ) -norm.Proof. Let γ ∈ A . Since T : W , Imm ( S ; R ) → R is continuous and Z -valued, it is locallyconstant near γ . Thus, if T [ γ ] (cid:54) = ±
1, we have T [˜ γ ] = T [ γ ] (cid:54) = ± γ ∈ W , ( S ; R )with (cid:107) ˜ γ − γ (cid:107) W , < δ for δ > γ is not an embedding, hence ˜ γ ∈ A .The winding number can now be used to detect interior points of A , which we will usenext to exclude tangential self-intersections. Lemma 5.11.
Let ¯ γ ∈ A be a minimizer in (5) . Then S tan [¯ γ ] = ∅ .Proof. By Lemma 5.9, we have S [¯ γ ] = { p } with mult[¯ γ ]( p ) = 2. Assume that p is a tangential self-intersection with multiplicity two and ¯ γ − ( { p } ) = { x , x } . Af-ter reparametrization, we may assume that ¯ γ is parametrized with constant speed, i.e. | ¯ γ (cid:48) ( x ) | = L (¯ γ ) for all x ∈ S . Hence, we have ¯ γ (cid:48) ( x ) = ± ¯ γ (cid:48) ( x ).We first consider the case ¯ γ (cid:48) ( x ) = ¯ γ (cid:48) ( x ). Then ¯ γ := ¯ γ | [ x ,x ] and ¯ γ := ¯ γ | [ x ,x ] are C -closed embedded curves and after an appropriate reparametrization, we have¯ γ , ¯ γ ∈ W , ( S ; R ) using Remark 2.3. We obtain T [¯ γ ] = T [¯ γ ] + T [¯ γ ] . Since ¯ γ k is simple closed, we obtain T [¯ γ k ] = ± k = 1 ,
2. Hence T [¯ γ ] (cid:54) = ±
1, so¯ γ is an interior point of A by Proposition 5.10. Consequently ¯ γ satisfies (9) and thus(8). Consequently, ¯ γ is an elastica by the discussion after (8). Therefore, by minimality,15ust be a rescaled translated and rotated reparametrization of γ ∗ by Lemma 5.4. Thiscontradicts Lemma 5.6.For the case ¯ γ (cid:48) ( x ) = − ¯ γ (cid:48) ( x ) we can without loss of generality assume that x = 0 and¯ γ (cid:48) (0) = ( L (¯ γ ) , T . Let θ ∈ W , ((0 , R ) be the angle function from Lemma A.2 with θ (0) = 0. Then L (¯ γ ) κ ( x ) = θ (cid:48) ( x ) by (31). Suppose now that ¯ γ is not an interior point of A . Hence T [¯ γ ] = ± T [¯ γ ] = 1 and hence1 = 12 π (cid:90) γ κ d s = θ (1) − θ (0)2 π . In particular θ (1) = 2 π . As θ (0) = 0 and ¯ γ (cid:48) ( x ) = − ¯ γ (cid:48) (0) = ( −L (¯ γ ) , T we infer that θ ( x ) = kπ for some odd number k ∈ Z + 1. Now we define (cid:101) γ ( x ) := (cid:40) ¯ γ ( x ) x ∈ [0 , x ]¯ γ (1 − ( x − x )) x ∈ [ x , . The curve (cid:101) γ is well-defined since ¯ γ (0) = ¯ γ ( x ) = ¯ γ (1) using Remark 2.3. Note inparticular that x (cid:54) = 0 in S and hence (cid:101) γ is not injective. We claim that (cid:101) γ is anotherminimizer. To this end, we show that (cid:101) γ ∈ W , ( S ; R ) and E ( (cid:101) γ ) L ( (cid:101) γ ) = E (¯ γ ) L (¯ γ ) . ByRemark 2.3 it suffices to show that the zeroth and first derivatives of the two casescoincide at x = 0 = 1 and at x = x . This is easy to check using that ¯ γ (cid:48) (0) = ¯ γ (cid:48) (1) = − ¯ γ (cid:48) ( x ). It is also immediate to check that E ( (cid:101) γ ) = E (¯ γ ) and L ( (cid:101) γ ) = L (¯ γ ). Hence (cid:101) γ isanother minimizer as claimed. Observe also that S tan [ (cid:101) γ ] = S tan [¯ γ ] (cid:54) = ∅ since (cid:101) γ (0) = (cid:101) γ ( x )and (cid:101) γ (cid:48) (0) = − (cid:101) γ (cid:48) ( x ). We now claim that (cid:101) γ (cid:48) ( x ) = L ( (cid:101) γ ) (cid:32) cos( (cid:101) θ ( x ))sin( (cid:101) θ ( x )) (cid:33) ∀ x ∈ (0 , , where ˜ θ is given by (cid:101) θ ( x ) = (cid:40) θ ( x ) x ∈ (0 , x ]( k − π + θ (1 − ( x − x )) x ∈ ( x , , (19)with k ∈ Z + 1 as before. To show that (cid:101) θ is the angle function of (cid:101) γ , we observe thatfor x ∈ ( x , (cid:101) γ (cid:48) ( x ) = − ¯ γ (cid:48) (1 − ( x − x )) = −L (¯ γ ) (cid:18) cos( θ (1 − ( x − x )))sin( θ (1 − ( x − x ))) (cid:19) = L (¯ γ ) (cid:18) cos( π + θ (1 − ( x − x )))sin( π + θ (1 − ( x − x ))) (cid:19) . Hence, by Lemma A.2, (cid:101) θ and π + θ (1 − ( · − x )) differ only by a constant multiple of 2 π .The multiple has to be chosen in such a way that (cid:101) θ ∈ W , (0 , θ ( x ) = kπ =16 k − π + θ (1), the only possible choice for (cid:101) θ is the one we defined in (19). Using thisand the relation κ | γ (cid:48) | = θ (cid:48) (cf. (31)) we compute12 π (cid:90) S κ [˜ γ ] d s ˜ γ = 12 π (cid:18)(cid:90) x θ (cid:48) ( x ) d x − (cid:90) x θ (cid:48) (1 − ( x − x )) d x (cid:19) = 12 π (cid:18)(cid:90) x θ (cid:48) ( y ) d y − (cid:90) x θ (cid:48) ( y ) d y (cid:19) = 12 π (2 θ ( x ) − θ (0) − θ (1)) = k − , since θ (0) = 0 , θ (1) = 2 π . In particular by Proposition A.4 T [ (cid:101) γ ] (cid:54)∈ { +1 , − } since k − k is odd. We infer that (cid:101) γ is an interior point of A . Since itis also a minimizer, it must by the same arguments as in the beginning of the proof be arescaled, translated and reparametrized version of γ ∗ . However, this is a contradiction,since S tan [ γ ∗ ] = ∅ by Lemma 5.6 but S tan [ (cid:101) γ ] = S tan [¯ γ ] (cid:54) = ∅ .Once we can rule out tangential self-intersections (as in Figure 2), we can finally showthat a minimizer is an interior point. The following lemma shows that non-tangentialself-intersections are stable under C -small perturbations. Lemma 5.12.
Let ¯ γ ∈ C ( S ; R ) and assume ¯ γ has a single non-tangential self inter-section with multiplicity two, i.e. S [¯ γ ] = { p } with ¯ γ − ( { p } ) = { ¯ x , ¯ x } for ¯ x (cid:54) = ¯ x and S tan [¯ γ ] = ∅ . Then, there exists δ > , ε > such that (¯ x − δ, ¯ x + δ ) ∩ (¯ x − δ, ¯ x + δ ) = ∅ and every curve γ ∈ B := { η ∈ C ( S ; R ) | (cid:107) γ − ¯ γ (cid:107) C < ε } has a unique self inter-section, i.e. there exist unique x (cid:54) = x ∈ S such that γ ( x ) = γ ( x ) . Moreover, thisself-intersection is non-tangential, satisfies x i ∈ [¯ x i − δ, ¯ x i + δ ] for i = 1 , and thefunction B (cid:51) γ (cid:55)→ ( x , x ) ∈ R is of class C .Proof. Let δ > x − δ, ¯ x + δ ) ∩ (¯ x − δ, ¯ x + δ ) = ∅ . Moreover,taking ε > γ ∈ B is immersed and injective when restricted to S \ (¯ x i − δ, ¯ x i + δ ) for i = 1 ,
2. Now, we define U := (¯ x − δ, ¯ x + δ ) × (¯ x − δ, ¯ x + δ ) × B and the functionΦ : U → R , Φ( x , x , γ ) := γ ( x ) − γ ( x ) . Then Φ(¯ x , ¯ x , ¯ γ ) = 0 by assumption. Moreover, Φ is of class C since the map ϕ : ( a, b ) × B → R , ( x, γ ) (cid:55)→ γ ( x ) is C for any a < b with derivative Dϕ ( x, γ )[ z, η ] = η ( x ) + γ (cid:48) ( x ) z for x ∈ ( a, b ) , z ∈ R , γ ∈ B and η ∈ C ( S ; R ). Now, the partial derivative D ( x ,x ) Φ(¯ x , ¯ x , ¯ γ ) : R → R given by D ( x ,x ) Φ(¯ x , ¯ x , ¯ γ )[ z , z ] = ¯ γ (cid:48) (¯ x ) z − ¯ γ (cid:48) (¯ x ) z for z , z ∈ R . is invertible, since ¯ γ (cid:48) (¯ x ) and ¯ γ (cid:48) (¯ x ) are linearly independent. Since Φ is C , we mayhence assume that D ( x ,x ) Φ( x , x , γ ) is invertible for all ( x , x , γ ) ∈ U . (20)17y the implicit function theorem [Zei96, Theorem 4.B], after possibly reducing ε > δ >
0, for all γ ∈ B there exists unique x i = x i ( γ ) ∈ (¯ x i − δ, ¯ x i + δ ) , i = 1 , , withΦ( x ( γ ) , x ( γ ) , γ ) = 0 for all γ ∈ B . Moreover, the map γ (cid:55)→ ( x ( γ ) , x ( γ )) is of class C . The uniqueness of the self-intersection follows from the local uniqueness and thefact that γ is injective on S \ (¯ x i − δ, ¯ x i + δ ) for both i = 1 and i = 2. Furthermore, by(20) this self-intersection is always non-tangential.Equipped with this result, we can show that our minimizer satisfies the Euler–Lagrangeequation. Lemma 5.13.
Let ¯ γ ∈ A be a minimizer in (5) . Then ¯ γ is a constrained elastica.Proof. By Lemma 5.9 and Lemma 5.11, there exist exactly two values x (cid:54) = x ∈ S with ¯ γ ( x ) = ¯ γ ( x ) = p such that ¯ γ (cid:48) ( x ) and ¯ γ (cid:48) ( x ) are linearly independent. Let φ ∈ C ∞ ( S ; R ). By Lemma 5.12, there exists ε > ε ∈ ( − ε , ε ) the curve¯ γ + εφ has a self-intersection, so ¯ γ + εφ ∈ A . Then, ( − ε , ε ) (cid:51) ε (cid:55)→ E (¯ γ + εφ ) L (¯ γ + εφ )has a local minimum in ε = 0. We conclude L (¯ γ ) D E (¯ γ )( φ ) + E (¯ γ ) D L (¯ γ )( φ ) = 0 ∀ φ ∈ C ∞ ( S ; R ) . Finally, we can prove our main result.
Proof of Theorem 1.2.
By Theorem 4.5 there exists ¯ γ ∈ A such that E (¯ γ ) L (¯ γ ) = inf γ ∈A E ( γ ) L ( γ ) . By Lemma 5.13 we infer that ¯ γ ∈ B , where B is defined as in (10). This implies togetherwith Lemma 5.4 thatinf γ ∈A E ( γ ) L ( γ ) = E (¯ γ ) L (¯ γ ) = inf γ ∈B E ( γ ) L ( γ ) = E ( γ ∗ ) L ( γ ∗ ) . The claim follows.
Remark 5.14.
If we seek to generalize Theorem 1.2 in higher codimension, e.g. in R ,the arguments in this section do not immediately carry over as more elasticae would needto be discussed in a generalized version of Lemma 5.4.An exhaustive classification of elasticae in R in [LS84b] shows that all non-planarelasticae γ are embedded and knotted. Therefore, by the Fary–Milnor Theorem (cf.[F´ar49, Mil50]) their energy is bounded from below since E ( γ ) L ( γ ) = (cid:90) S | κ | d s (cid:90) S s ≥ (cid:18)(cid:90) S | κ | d s (cid:19) ≥ π > c ∗ , where we used (16) in the last inequality. Clearly, multi-fold covers of these non-planarelasticae will only have higher energy. Hence, γ ∗ is still the minimizer among all non-embedded elasticae in R , generalizing Lemma 5.4 to higher codimension. Moreover,we can prove as in Lemma 5.8 and Lemma 5.9 that there exists a minimizer ¯ γ of EL mong closed non-embedded curves in R which has only one point of self-intersectionwith multiplicity . However, the methods used in Lemma 5.11 and Lemma 5.13 are onlyavailable in codimension one, making it unclear, whether the minimizer is an elastica.This is the only obstruction to a generalization of Theorem 1.2 into higher codimension.
6. An application: the elastic flow
We consider a family of smooth curves γ : [0 , T ) × S → R evolving with respect to thegradient flow equation ∂ t γ = −∇ s (cid:126)κ − | (cid:126)κ | (cid:126)κ + λ(cid:126)κ, (21)where (cid:126)κ = (cid:126)κ [ γ ] = ∂ s γ is the curvature vector and ∇ s = ∂ ⊥ γ s is the normal part of thearc-length derivative. Here, we either consider the length penalized elastic flow, where λ ≥ length preserving elastic flow, where λ = λ ( γ ( t, · )) ∈ R depends on the solution and is given by λ = (cid:82) S (cid:104)∇ s (cid:126)κ + | (cid:126)κ | (cid:126)κ, (cid:126)κ (cid:105) d s (cid:82) S | (cid:126)κ | d s . (22)It can be easily checked that the length remains constant along solutions of (21) with λ given by (22), sincedd t L ( γ ) = (cid:90) (cid:104)∇L ( γ ) , ∂ t γ (cid:105) d s = − (cid:90) S (cid:104) (cid:126)κ, ∂ t γ (cid:105) d s = 0 , (23)by (22). Both geometric flows have been studied in [DKS02], where long-time existenceand subconvergence as t → ∞ has been established. Using Theorem 1.2, we establish anenergy bound which guarantees embeddedness along the flow. Note that in contrast tosecond order evolutions, this does not follow from a maximum principle, as (21) is of 4-thorder. Moreover as in [DPS16, MP20, RS20] we can apply a suitable (cid:32)Lojasiewicz–Simoninequality to deduce convergence and then Lemma 5.4 to give a precise characterizationof the limit. Theorem 6.1.
Let γ ∈ C ∞ ( S ; R ) be an embedded curve such that E ( γ ) L ( γ ) 0, we have E ( γ ( t )) L ( γ ( t )) < E ( γ ∗ ) L ( γ ∗ ) bythe assumption on the initial datum, so γ ( t ) is embedded for all t ≥ (cid:107) ∂ ms (cid:126)κ (cid:107) L ∞ ≤ C m , for some C m > m ∈ N . Thus, if ˜ γ denotes the reparametrization by arc-length,we get (cid:107) ∂ mx ˜ γ (cid:107) L ∞ ≤ C m for all m ∈ N . (24)19f we define the integral average p ( t ) := (cid:82) S ˜ γ ( t, · ) d x ∈ R , we find (cid:107) ˜ γ ( t, · ) − p ( t ) (cid:107) L ∞ ≤ L (˜ γ ( t )) = L ( γ ) . (25)Now, if t n → ∞ is any sequence, using the Arzel`a–Ascoli theorem and a diagonalsequence argument, after passing to a subsequence, we find ˜ γ ( t n ) − p ( t n ) → γ ∞ in C m ( S ; R ) as n → ∞ for every m ∈ N , where γ ∞ ∈ C ∞ ( S ; R ) is a closed elastica,cf. [DKS02, Theorem 3.2]. Since E ( γ ∞ ) L ( γ ∞ ) < E ( γ ∗ ) L ( γ ∗ ), the curve γ ∞ is embeddedand has to be a (translation and reparametrization) of the one-fold cover of a circle byLemma 5.4. Consequently, its radius has to be L ( γ )2 π since L ( γ ∞ ) = L ( γ ). However,different sequences could still yield circles with different centers.Even so, this cannot happen, since E satisfies a constrained (cid:32)Lojasiewicz–Simon gradientinequality, cf. [Rup20]. This can be proven using [Rup20, Corollary 5.2], since the ener-gies E and L are analytic and the length is of lower order, see also [RS20, Theorem 4.8]for the analogous argument in the case of clamped curves. Hence, there exist constants C LS , σ > θ ∈ (0 , ] such that for all γ ∈ W , ( S ; R ) with (cid:107) γ − γ ∞ (cid:107) W , ≤ σ and L ( γ ) = L ( γ ) we have |E ( γ ) − E ( γ ∞ ) | − θ ≤ C (cid:107)∇E ( γ ) + λ ( γ ) ∇L ( γ ) (cid:107) L , (26)with λ ( γ ) as in (22). To prove the full convergence statement, we furthermore assume (cid:107) ˜ γ ( t n ) − p ( t n ) − γ ∞ (cid:107) W , < σ for all n ∈ N . We define s n := sup { s ≥ t n | (cid:107) ˜ γ ( t ) − p ( t ) − γ ∞ (cid:107) W , < σ for all t ∈ [ t n , s ] } , and observe s n > t n by smoothness. Now, the function G ( t ) := ( E (˜ γ ( t )) − E ( γ ∞ )) θ = ( E ( γ ( t )) − E ( γ ∞ )) θ is decreasing and satisfies lim t →∞ G ( t ) = 0. Moreover, since γ solves (21) we have − dd t G = θ ( E (˜ γ ) − E ( γ ∞ )) θ − (cid:18) − dd t E ( γ ) (cid:19) = − θ ( E (˜ γ ) − E ( γ ∞ )) θ − (cid:104)∇E ( γ ) , ∂ t γ (cid:105) L (d s γ ) = − θ ( E (˜ γ ) − E ( γ ∞ )) θ − (cid:104)∇E ( γ ) + λ ( γ ) ∇L ( γ ) , ∂ t γ (cid:105) L (d s γ ) = θ ( E (˜ γ ) − E ( γ ∞ )) θ − (cid:107)∇E ( γ ) + λ ( γ ) ∇L ( γ ) (cid:107) L (d s γ ) (cid:107) ∂ t γ (cid:107) L (d s γ ) , where we used that (cid:104)∇L ( γ ) , ∂ t γ (cid:105) L (d s γ ) = 0 by (23). Furthermore, using the geometrictransformation of the energy and the L -gradient, we find( E (˜ γ ) − E ( γ ∞ )) θ − (cid:107)∇E ( γ ) + λ ( γ ) ∇L ( γ ) (cid:107) L (d s γ ) = ( E (˜ γ − p ) − E ( γ ∞ )) θ − (cid:107)∇E (˜ γ − p ) + λ (˜ γ − p ) ∇L (˜ γ − p ) (cid:107) L (d s ˜ γ − p ) . Thus, by the definition of s n and the (cid:32)Lojasiewicz–Simon inequality (26) we have − dd t G ( t ) ≥ θC LS (cid:107) ∂ t γ ( t ) (cid:107) L (d s γ ( t ) ) for all t ∈ [ t n , s n ) . − dd t G ( t ) ≥ C (cid:107) ∂ t ˜ γ ( t ) (cid:107) L (d x ) for all t ∈ [ t n , s n ) (27)for some C = C ( L ( γ ) , E ( f ) , θ, C LS ) > 0. Thus, given ε > t ∈ [ t n , s n ) we obtain (cid:107) ˜ γ ( t ) − ˜ γ ( t n ) (cid:107) L (d x ) ≤ C G ( t n ) → , (28)as n → ∞ since lim n →∞ G ( t n ) = 0. Now, we assume that all of the s n are finite. Bycontinuity, (28) also holds for t = s n and we observe (cid:107) p ( t n ) − p ( s n ) (cid:107) L (d x ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ˜ γ ( t n ) d x − (cid:90) S ˜ γ ( s n ) d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) S | ˜ γ ( t n ) − ˜ γ ( s n ) | d x → , using Jensen’s inequality and (28). Moreover, using the bounds in (24) and (25) we may,as at the beginning of the proof, assume that ˜ γ ( s n ) − p ( s n ) → ψ smoothly as n → ∞ for some ψ ∈ C ∞ ( S ; R ). Thus, we find (cid:107) ˜ γ ( s n ) − p ( s n ) − γ ∞ (cid:107) L (d x ) ≤ (cid:107) ˜ γ ( s n ) − ˜ γ ( t n ) (cid:107) L (d x ) + (cid:107) p ( t n ) − p ( s n ) (cid:107) L (d x ) + (cid:107) ˜ γ ( t n ) − p ( t n ) − γ ∞ (cid:107) L (d x ) → , as n → ∞ . Therefore, ψ = f ∞ . However, by the definition of s n and a continuity argument, we have (cid:107) ψ − f ∞ (cid:107) W , = lim n →∞ (cid:107) ˜ γ ( s n ) − p ( s n ) − γ ∞ (cid:107) W , = σ > , a contradiction. Consequently, we have s n = ∞ for some n ∈ N and therefore (cid:107) ˜ γ ( t ) − p ( t ) − γ ∞ (cid:107) W , < σ for all t ≥ t n . But then (27) implies that for any t n ≤ t ≤ t (cid:48) we have (cid:107) ˜ γ ( t ) − ˜ γ ( t (cid:48) ) (cid:107) L (d x ) ≤ (cid:90) t (cid:48) t (cid:107) ∂ t ˜ γ ( τ ) (cid:107) L (d x ) d τ → , as t (cid:48) , t → ∞ by dominated convergence. Therefore, the limit lim t →∞ ˜ γ ( t ) has to exist in L (d x ) and hence equals γ ∞ . A subsequence argument shows that for any m ∈ N wehave (cid:107) ˜ γ ( t ) − γ ∞ (cid:107) C m ( I ; R d ) → t → ∞ , i.e. the convergence is smooth.The bound in Theorem 6.1 is optimal, since the stationary flow γ ( t ) ≡ γ ∗ with γ ∗ as inDefinition 5.1 solves (21) but possesses self-intersections for all times. Theorem 6.2. Let λ > and let γ ∈ C ∞ ( S ; R ) be an embedded curve such that ( E ( γ )+ λ L ( γ )) λ < c ∗ . Then, the length penalized elastic flow (21) remains embedded forall times and converges, as t → ∞ , after reparametrization to a one-fold cover of a circlewith radius √ λ . roof. By [DKS02, Theorem 3.2] the flow exists for all times and also satisfies the bounds(24) and (25). Moreover, we have the bounds E ( γ ( t )) + λ L ( γ ( t )) ≤ E ( γ ) + λ L ( γ )for all t ≥ 0. An easy calculation yields max { xy | x, y ∈ (0 , ∞ ) , x + λy ≤ M } = M λ for any M > 0, so we have E ( γ ( t )) L ( γ ( t )) ≤ ( E ( γ )+ λ L ( γ ) )4 λ < c ∗ by assumption. Thus,using Theorem 1.2 the flow remains embedded.As in the proof of Theorem 6.1, by the uniform estimates (24) and (25), there exists t n → ∞ such that ˜ γ ( t n ) − p ( t n ) → γ ∞ , where ˜ γ is a reparametrization of γ and γ ∞ isa closed elastica. Here p ( t ) = (cid:82) S ˜ γ ( t ) d x ∈ R . The convergence of the flow as t → ∞ has been established in [MP20, Theorem 1.2 and Remark 1.4]. Alternatively, one caneasily modify the arguments in the proof of Theorem 6.1 and apply a (unconstrained)(cid:32)Lojasiewicz–Simon gradient inequality for the penalized elastic energy.Since the limit is necessarily an elastica with E ( γ ∞ ) L ( γ ∞ ) < c ∗ , it can only be a one-foldcover of a circle by Lemma 5.4. Denoting by R > ∂ s κ + 12 κ − λκ = 12 R − λR , which equals zero if and only if R = √ λ . Acknowledgments Marius M¨uller was supported by the LGFG Grant (Grant no. 1705 LGFG-E). FabianRupp is supported by the Deutsche Forschungsgemeinschaft (DFG, German ResearchFoundation) - project no. 404870139. Both authors would like to thank Anna Dall’Acquafor helpful discussions. Appendix A Differential geometry in Sobolev spaces In this section, we will review some standard results from elementary differential geom-etry in the setting of W , -curves. Theorem A.1 (Fenchel’s Theorem) . Let γ ∈ C ( S ; R ) be an immersed curve. Then K ( γ ) ≥ π with equality if and only if γ is embedded and convex.Proof. See [Fen29, Satz 1]. An explicit characterization of the equality case can bededuced from [BH74, Theorem 3], for instance. Lemma A.2 (Angle Function) . For an immersed curve γ ∈ W , ( S ; R ) , there exists θ ∈ W , ((0 , R ) such that γ (cid:48) ( x ) | γ (cid:48) ( x ) | = (cid:18) cos θ ( x )sin θ ( x ) (cid:19) for all x ∈ S . (29) We call θ an angle function for γ . Moreover, any two functions satisfying (29) can onlydiffer by an integer multiple of π . roof. The proof works exactly as in the case of smooth curves, see for instance [B¨ar10,Lemma 2.2.5]. For the regularity of θ , we use local representations of θ . For instance, inthe case γ (cid:48) ( x ) > 0, one has locally θ ( x ) := arctan (cid:18) γ (cid:48) ( x ) γ (cid:48) ( x ) (cid:19) + 2 π(cid:96), for some (cid:96) ∈ Z . (30)Hence θ ∈ W , ( S ; R ) follows from (30) and the chain rule for Sobolev functions. Definition A.3 (Winding Number) . Let γ ∈ W , ( S ; R ) be an immersion with cor-responding angle function θ ∈ W , ((0 , R ) . We define the winding number of γ as T [ γ ] := π ( θ (1) − θ (0)) . Note that T [ γ ] does not depend on the choice of θ and is alwaysan integer. Proposition A.4. Let γ ∈ W , ( S ; R ) be an immersed curve. Then T [ γ ] = 12 π (cid:90) S κ d s. Proof. Let θ ∈ W , ((0 , R ) be an angle function for γ . Differentiating (29) and usingthe chain rule for Sobolev functions and the definition of the unit normal, we obtain κ(cid:126)n = (cid:126)κ = ∂ s γ = θ (cid:48) | γ (cid:48) | (cid:18) − sin θ cos θ (cid:19) = θ (cid:48) | γ (cid:48) | (cid:18) − γ (cid:48) γ (cid:48) (cid:19) = θ (cid:48) | γ (cid:48) | (cid:126)n, consequently κ (cid:12)(cid:12) γ (cid:48) (cid:12)(cid:12) = θ (cid:48) almost everywhere. (31)Moreover, by the fundamental theorem of calculus for W , -functions, we find T [ γ ] = 12 π ( θ (1) − θ (0)) = 12 π (cid:90) θ (cid:48) d x = 12 π (cid:90) κ (cid:12)(cid:12) γ (cid:48) (cid:12)(cid:12) d x = 12 π (cid:90) S κ d s. Proposition A.5 (Hopf’s Umlaufsatz for W , -embeddings) . Let γ ∈ W , ( S ; R ) bean embedding. Then T [ γ ] = ± .Proof. Let γ ( n ) be a sequence of smooth curves with γ ( n ) → γ in W , ( S ; R ). ByProposition A.4, we can easily see that T [ γ ( n ) ] → T [ γ ]. Since the set of embeddings isopen in C ( S ; R ) by Lemma 4.3, we see that γ ( n ) is an embedding for n ≥ N largeenough, hence T [ γ ( n ) ] = ± n ≥ N by Hopf’s Umlaufsatz for smooth curves.However, since the sequence (cid:0) T [ γ ( n ) ] (cid:1) n ∈ N converges, it has to be eventually constant, say T [ γ ( n ) ] = τ ∈ {− , } for all n ≥ N . But then T [ γ ] = lim n →∞ T [ γ ( n ) ] = τ ∈ {− , } . Lemma A.6. Let γ ∈ W , ( S ; R ) be an immersion. Then, there exists a constantspeed reparametrization ˜ γ of γ such that ˜ γ ∈ W , ( S ; R ) .Proof. Follows with the arguments in [B¨ar10, Proposition 2.1.13], using the Sobolevembedding W , ( S ; R ) (cid:44) → C ( S ; R ) and the chain rule for Sobolev functions.23 ppendix B Jacobi elliptic functions and Euler’s elastica B.1 Elliptic functions We provide some elementary properties of Jacobi elliptic functions, which can be foundfor example in [AS64, Chapter 16]. Definition B.1 (Amplitude Function, Complete Elliptic Integrals) . Fix m ∈ [0 , .We define the Jacobi-amplitude function am( · , m ) : R → R with modulus m to be theinverse function of R (cid:51) z (cid:55)→ (cid:90) z (cid:112) − m sin ( θ ) d θ ∈ R We define the complete elliptic integral of first and second kind as K ( m ) := (cid:90) π (cid:112) − m sin ( θ ) d θ, E ( m ) := (cid:90) π (cid:113) − m sin ( θ ) d θ and the incomplete elliptic integral of first and second kind as F ( x, m ) := (cid:90) x (cid:112) − m sin ( θ ) d θ, E ( x, m ) := (cid:90) x (cid:113) − m sin ( θ ) d θ. Note that F ( · , m ) = am( · , m ) − . Definition B.2 (Elliptic Functions) . For m ∈ [0 , the Jacobi elliptic functions aregiven by cn( · , m ) : R → R , cn( x, m ) := cos(am( x, m )) , sn( · , m ) : R → R , sn( x, m ) := sin(am( x, m )) , dn( · , m ) : R → R , dn( x, m ) := (cid:113) − m sin (am( x, m )) . The following proposition summarizes all relevant properties and identities for the ellipticfunctions. They can all be found in [AS64, Chapter 16]. Proposition B.3. 1. (Derivatives and Integrals of Jacobi Elliptic Functions) For each x ∈ R and m ∈ (0 , we have ∂∂x cn( x, m ) = − sn( x, m )dn( x, m ) , ∂∂x sn( x, m ) = cn( x, m )dn( x, m ) ,∂∂x dn( x, m ) = − m cn( x, m )sn( x, m ) , ∂∂x am( x, m ) = dn( x, m ) . 2. (Derivatives of Complete Elliptic Integrals) For m ∈ (0 , E is smooth and dd m E ( m ) = E ( m ) − K ( m )2 m , dd m K ( m ) = ( m − K ( m ) + E ( m )2 m (1 − m ) . . (Trigonometric Identities) For each m ∈ [0 , and x ∈ R the Jacobi elliptic func-tions satisfy cn ( x, m ) + sn ( x, m ) = 1 , dn ( x, m ) + m sn ( x, m ) = 1 . 4. (Periodicity) All periods of the elliptic functions are given as follows, where l ∈ Z and x ∈ R : am( lK ( m ) , m ) = l π , cn( x + 4 lK ( m ) , m ) = cn( x, m ) , sn( x + 4 lK ( m ) , m ) = sn( x, m ) , dn( x + 2 lK ( m ) , m ) = dn( x, m ) ,F ( lπ , m ) = lK ( m ) E ( lπ , m ) = lE ( m )am( x + 2 lK ( m ) , m ) = lπ + am( x, m ) ,F ( x + lπ, m ) = F ( x, m ) + 2 lK ( m ) ,E ( x + lπ, m ) = E ( x, m ) + 2 lE ( m ) . 5. (Asymptotics of the Complete Elliptic Integrals) lim m → K ( m ) = ∞ , lim m → K ( m ) = π m → E ( m ) = 1 , lim m → E ( m ) = π . B.2 Some computational lemmas involving elliptic functions We will also need some more advanced identities for elliptic functions, e.g the following Lemma B.4. The map (0 , (cid:51) m (cid:55)→ E ( m ) − K ( m ) has a unique zero m ∗ ∈ (0 , .Moreover m ∗ > .Proof. We define for m ∈ (0 , f ( m ) := E ( m ) K ( m ) − 1. Note that f has the same zeroes as m (cid:55)→ E ( m ) − K ( m ). By Proposition B.3 one haslim m → f ( m ) = 1 , lim m → f ( m ) = − . and hence there has to exists a zero of f . To show that it is unique, we show that f isdecreasing, which follows immediately from the following computationdd m E ( m ) K ( m ) = 1 K ( m ) (cid:18) E ( m ) − K ( m )2 m K ( m ) − E ( m ) ( m − K ( m ) + E ( m )2 m (1 − m ) (cid:19) = 12 m (1 − m ) K ( m ) (cid:0) − m ) E ( m ) K ( m ) − (1 − m ) K ( m ) − E ( m ) (cid:1) 25 12 m (1 − m ) K ( m ) (cid:0) E ( m )(1 − m ) K ( m ) − (1 − m ) K ( m ) − E ( m ) (cid:1) + 12 m (1 − m ) K ( m ) (cid:0) ((1 − m ) − (1 − m )) K ( m ) (cid:1) = 12 m (1 − m ) K ( m ) (cid:0) − ( E ( m ) − (1 − m ) K ( m )) − m (1 − m ) K ( m ) (cid:1) (32) ≤ − . It remains to show that m ∗ > . Indeed2 E ( ) − K ( ) = (cid:90) π (cid:114) − 12 sin ( θ ) − (cid:113) − sin ( θ ) d θ = (cid:90) π cos ( θ ) (cid:113) − sin ( θ ) d θ > , which implies that f ( ) > f we find m ∗ > . Lemma B.5. The expression E ( m ) − K ( m ) + mK ( m ) is strictly positive for all m ∈ (0 , .Proof. Let f ( m ) := E ( m ) K ( m ) − m . Note note that f ( m ) is positive if and only if theexpression in the statement is positive and K ( m ) > 0. Further note thatlim m → f ( m ) = 0 . To show the claim it suffices to prove that f (cid:48) < 0. To do so, it suffices to show that dd m E ( m ) K ( m ) < − for all m ∈ (0 , m E ( m ) K ( m ) = 12 m (1 − m ) K ( m ) (cid:0) − ( E ( m ) − (1 − m ) K ( m )) − m (1 − m ) K ( m ) (cid:1) ≤ − , where the last inequality was obtained by estimating the square with zero. We will showthat this estimate is always with strict inequality, i.e. E ( m ) − (1 − m ) K ( m ) (cid:54) = 0 ∀ m ∈ (0 , . (33)Note again first that lim m → ( E ( m ) − (1 − m ) K ( m )) = 0 . Now an easy computation yieldsdd m ( E ( m ) − (1 − m ) K ( m )) = 12 K ( m ) > . E ( m ) − (1 − m ) K ( m ) > ∀ m ∈ (0 , . Hence (33) is shown and thus ddm E ( m ) K ( m ) < − for all m ∈ (0 , f weobtain f (cid:48) < Lemma B.6. Let m ∗ be the unique zero in Lemma B.4. Then the map [0 , π ) (cid:51) x (cid:55)→ E ( x, m ∗ ) − F ( x, m ∗ ) has exactly four zeroes in [0 , π ) , namely x = 0 , x = π , x = π, x = π .Proof. Let f : R → R be the smooth function defined by f ( x ) := 2 E ( x, m ∗ ) − F ( x, m ∗ ).We show first that f (0) = f ( π ) = f ( π ) = f ( π ) = f (2 π ). Indeed, by Proposition B.3one has f ( l π l (2 E ( m ∗ ) − K ( m ∗ )) = 0 . Next we show that f (cid:48) has four zeroes in [0 , π ]. Indeed, f (cid:48) ( x ) = 1 − m ∗ sin ( x ) (cid:112) − m ∗ sin ( x ) , which is zero if and only if sin ( x ) = m ∗ which happens exactly four times in [0 , π ]since m ∗ > by Lemma B.4. Assume now that there exists some x ∈ (0 , π ) apartfrom 0 , π , π, π , π such that f ( x ) = 0. We can now sort the set { , π , π, π , π, x } = { y , y , y , y , y , y } with 0 = y < ... < y = 2 π . Since f ( y ) = ... = f ( y ) = 0 , by the mean value theorem for all i ∈ { , ..., } there exists some z i ∈ ( y i , y i +1 ) such that f (cid:48) ( z i ) = 0. This however is a contradiction to the fact that f (cid:48) has only 4 zeroes. As aconsequence, there exists no x as in the assumption. The claim follows. Lemma B.7 (cf. [MS20, Proposition B.5]) . For all m ∈ (0 , one has E ( m ) ≤ π √ √ − m. Proof. The proof follows from [MS20, Proposition B.5]. Be aware that the authors thereuse the different notation of m = p , their definition of E ( p ) is actually E ( p ) in ournotation. 27 .3 Explicit parametrization of Euler’s elasticae In the following we shall prove the following classification result. Proposition B.8. Let I ⊂ R be an interval and let γ : I → R be a smooth solution of (7) for some λ ∈ R . Then up to rescaling, reparametrization and isometries of R , γ isgiven by one of the following elastic prototypes .1. (Linear elastica) γ is a line, κ [ γ ] = 0 .2. (Wavelike elastica) There exists m ∈ (0 , such that γ ( s ) = (cid:18) E (am( s, m ) , m ) − s − √ m cn( s, m ) (cid:19) . Moreover κ [ γ ] = 2 √ m cn( s, m ) . 3. (Borderline elastica) γ ( s ) = (cid:18) s ) − s − s ) (cid:19) . Moreover κ [ γ ] = 2sech( s ) . 4. (Orbitlike elastica) There exists m ∈ (0 , such that γ ( s ) = 1 m (cid:18) E (am( s, m ) , m ) + ( m − s − s, m ) (cid:19) Moreover κ [ γ ] = 2dn( s, m ) . 5. (Circular elastica) γ is a circle. We give a proof in the rest of this section. Suppose that γ is parametrized by arc-length.As we know that κ satisfies (7), using [Lin96, Proposition 3.3] we obtain that one out ofthe four following cases occurs.(1) (Constant curvature) κ is constant.(2) (Wavelike elastica) κ ( s ) = ± α √ m cn( α ( s − s ) , m ) for some m ∈ [0 , , α > , s ∈ R . In this case λ = α (2 m − . (3) (Orbitlike elastica) κ ( s ) = ± α dn( α ( s − s ) , m ) for some m ∈ [0 , , α > , s ∈ R .In this case λ = α (2 − m ) . (4) (Borderline elastica) κ ( s ) = ± α sech( α ( s − s )) for some α > , s ∈ R . In thiscase λ = α . 28nce this is known we can find explicit parametrizations of all these elastica. Note thatonce we have parametrized the solutions for s = 0 and ‘+’ instead of ‘ ± ’ we can obtainall other solutions by reparametrization or reflection. Hence we consider only the casesof ‘+’ and s = 0.From [DP17, Proposition 6.1] it is known that each smooth solution γ : I → R of (7)with some parameter λ ∈ R corresponds up to isometries of R and reparametrizationto a solution of γ (cid:48)(cid:48) = σγ γ (cid:48) ,γ (cid:48)(cid:48) = − σγ γ (cid:48) ,γ (cid:48) + γ (cid:48) = 1 , (34)for some σ > 0. One can now compute that if γ is a solution of (34) then κ = γ (cid:48)(cid:48) γ (cid:48) − γ (cid:48)(cid:48) γ (cid:48) = − σγ and γ (cid:48) − σ γ ≡ µ for some constant µ ∈ R . Following the lines of [ ? ,Proposition 6.1] we also obtain that λ = − σµ . We are also free to assume that γ (0) = 0as (34) is not affected by adding a constant to γ . From now on the parameters α and m will be our main parameters. We will express σ, µ in terms of them and use (34) toobtain an explicit parametrization. Case 1: Constant curvature. This yields either lines or circles. Case 2: Wavelike elastica . First we show that σ = α and µ = 1 − m . Note that bypoint (1) of the list of possible curvatures and λ = − σµ we have that µ = α σ (1 − m ).In partcular, since κ = − σγ we have γ ( s ) = − σ α √ m cn( αs, m ) (35)and since γ (cid:48) = σ γ + µ we obtain γ (cid:48) ( s ) = σ γ ( s ) + α σ (1 − m ) = α σ (2 m cn ( αs, m ) + 1 − m ) . (36)Therefore using (34) and Proposition B.3 we obtain1 = γ (cid:48) ( s ) + γ (cid:48) ( s ) = α σ (cid:16)(cid:0) m cn ( αs, m ) + 1 − m (cid:1) + 4 m sn ( αs, m )dn ( αs, m ) (cid:17) = α σ (cid:0) (1 − m sn ( αs, m )) + 4 m sn ( αs, m )(1 − m sn ( αs, m )) (cid:1) = α σ Hence σ = α , which implies by (35) that γ ( s ) = − α √ m cn( αs, m ) . We can moreover improve the formula for µ to µ = 1 − m . Using this and (36) we find γ (cid:48) ( s ) = 2 m cn ( αs, m ) + (1 − m ) = 1 − m sn ( αs, m )29nd integrating using γ (0) = 0 we obtain γ ( s ) = s − m (cid:90) s sn ( αs, m ) d s = s − α (cid:90) am( αs,m )0 m sin θ (cid:112) − m sin θ dθ = s − α (cid:90) am( αs,m )0 (cid:32) (cid:112) − m sin θ − (cid:112) − m sin θ (cid:33) dθ = s − α F (am( αs, m ) , m ) + 2 α E (am( αs, m ) , m ) = 2 α ( E (am( αs, m ) , m ) − αs ) . Hence for fixed α > γ ( s ) = α γ wave ( αs ) where γ wave is given by γ wave ( s ) = (cid:18) E (am( s, m ) , m ) − s − √ m cn( s, m ) (cid:19) . Case 3: Orbitlike elastica. We proceed as in the wavelike case. We first show that σ = α m and µ = m − m . From point (2) in the list of curvatures and λ = − σµ we inferthat µ = α ( m − σ . This leads to γ ( s ) = − σ α dn( αs, m )and by Proposition B.3 γ (cid:48) ( s ) = σ γ ( s ) + α ( m − σ = α mσ (1 − ( αs, m ))Using (34) and Proposition B.3 we obtain1 = γ (cid:48) ( s ) + γ (cid:48) ( s ) = α m σ (cid:0) (1 − ( αs, m )) + 4sn ( αs, m )cn ( αs, m ) (cid:1) = α m σ (cid:0) (1 − ( αs, m )) + 4sn ( αs, m )(1 − sn ( αs, m ) (cid:1) = α m σ Therefore we find that σ = α m and from this follows that µ = m − m . We infer γ ( s ) = − mα dn( αs, m )and γ (cid:48) ( s ) = 1 − ( αs, m ) . Integrating we obtain γ ( s ) = s − (cid:90) s sn ( αs, m ) d s = s (cid:18) − m (cid:19) + 2 αm E (am( αs, m ) , m ) . 30e infer that for fixed α > γ ( s ) = α γ orbit ( αs ), where γ orbit is given by γ orbit = 1 m (cid:18) E (am( s, m ) , m ) + ( m − s − s, m ) (cid:19) . 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