Li-Yau type inequalities for curves in any codimension
aa r X i v : . [ m a t h . DG ] F e b LI–YAU TYPE INEQUALITIES FOR CURVES IN ANYCODIMENSION
TATSUYA MIURA
Abstract.
For immersed curves in Euclidean space of any codimension weestablish Li–Yau type inequalities that give lower bounds of the (normalized)bending energy in terms of multiplicity. The obtained inequalities are optimalfor any codimension and any multiplicity except for the case of planar closedcurves with odd multiplicity; in this remaining case we discover a hidden alge-braic obstruction and indeed prove an exhaustive non-optimality result. Theproof is mainly variational and involves Langer–Singer’s classification of elas-ticae and Andr´e’s algebraic-independence theorem for certain hypergeometricfunctions. Applications to elastic flows are also given. Introduction
The classical Li–Yau inequality [15] asserts that if a closed surface Σ ⊂ R n , n ≥
3, has a point of multiplicity k ≥
2, then the Willmore energy W [Σ] := R Σ | H | is bounded below by multiplicity in the form of(1.1) W [Σ] ≥ πk, where H denotes the mean curvature vector. (See also a different proof in R [29].)This estimate is sharp due to a nearly k -times covered sphere. In particular, if W [Σ] < π then Σ must be embedded. This result is used as a fundamental toolin many studies; the Willmore flow [12], the Willmore conjecture [20], and others.In this paper we establish a one-dimensional analogue of the Li–Yau inequality,and reveal that a new phenomenon emerges due to low dimensionality. For animmersed curve γ in R n we let κ denote the curvature vector κ := ∂ s γ , anddefine the normalized bending energy ¯ B [ γ ] as the bending energy B [ γ ] := R γ | κ | ds normalized by the length L = L [ γ ] := R γ ds to be scale-invariant:¯ B [ γ ] := L [ γ ] B [ γ ] = L Z γ | κ | ds. In addition, using the complete elliptic integral of the first kind K ( m ) and of thesecond kind E ( m ), we define a unique modulus m ∗ ∈ (0 ,
1) such that K ( m ∗ ) =2 E ( m ∗ ), and then the key universal constant ̟ ∗ > ̟ ∗ := 32(2 m ∗ − E ( m ∗ ) (= 28 . ... )Finally, we say that a curve γ has a point p ∈ R n of multiplicity k if the preimage γ − ( p ) contains at least k distinct points. Mathematics Subject Classification.
Key words and phrases.
Li–Yau inequality, embeddedness, elastica, bending energy, multiplic-ity, elastic flow.
Our first theorem asserts a general Li–Yau type inequality involving multiplicityfor closed curves γ : T → R n , where T := R / Z . Theorem 1.1 (Multiplicity inequality for closed curves) . Let n ≥ and k ≥ .Let γ : T → R n be an immersed closed curve with a point of multiplicity k . Then (1.3) ¯ B [ γ ] ≥ ̟ ∗ k . In particular, if an immersed closed curve γ has the property that ¯ B [ γ ] < ̟ ∗ ,then γ must be embedded. This threshold is optimal because a figure-eight elasticagives an explicit example of a non-embedded planar closed analytic curve withenergy ¯ B = 4 ̟ ∗ (see Definition 2.2 and Lemma 2.3).We also discuss more on optimality and rigidity in inequality (1.3), where wespecify the natural H -Sobolev regularity for curves. On one hand, our inequalityis optimal for many pairs of ( n, k ), namely either if n ≥ k is even. Wealso prove the rigidity that any optimal curve is a k -leafed elastica , i.e., the curveconsists of k half-fold figure-eight elasticae of same length (see Definition 3.1). Theorem 1.2 (Optimality and rigidity) . Let n ≥ and k ≥ . Suppose eitherthat n ≥ or that k is even. Then there exists an immersed closed H -curve γ : T → R n with a point of multiplicity k such that (1.4) ¯ B [ γ ] = ̟ ∗ k . In addition, equality (1.4) is attained if and only if γ is a closed k -leafed elastica. On the other hand, somewhat interestingly, in the remaining case of n = 2 andodd k ≥ Theorem 1.3 (Non-optimality) . For any odd integer k ≥ there exists a positivenumber ε k > such that for any immersed (planar) closed H -curve γ : T → R with a point of multiplicity k , ¯ B [ γ ] ≥ ̟ ∗ k + ε k . Our results are new for all n ≥ k ≥
3. In their very recent study [23],M¨uller–Rupp obtain Theorems 1.1 and 1.2 for the special pair ( n, k ) = (2 , ̟ ∗ (=: c ∗ = 112 . ... in their notation). Theycrucially use the assumption that ( n, k ) = (2 ,
2) since their proof relies on the factthat any planar closed curve with rotation number = ± n ≥ k the non-optimalestimate that ¯ B ≥ k is previously obtained by Wheeler for planar closed curves[31, Theorem 1.6]; see also a relevant statement [32, Lemma 2.1]. Theorem 1.3highlights a new phenomenon compared to the original Li–Yau inequality since(1.1) is sharp regardless of codimension and multiplicity.The study of the bending energy B is initiated by D. Bernoulli and L. Euler inthe 18th century for modelling elastic rods, but still ongoing; see e.g. [17, 30, 26, 22]and references therein. Corresponding variational solutions are called elastic curvesor elasticae, and the known classification of them plays a key role in our study.Our results have direct applications to elastic flows, which are L -gradient flowsinvolving the bending energy. Indeed, in Section 4, we apply our inequality to I–YAU TYPE INEQUALITIES FOR CURVES IN ANY CODIMENSION 3 obtaining optimal energy thresholds below which all-time embeddedness holds alongsuch flows, as in [23].The normalized bending energy ¯ B is a natural one-dimensional counterpart ofthe Willmore energy in the sense that both are scale-invariant functionals involvingcurvature and minimized only by a round shape. The total curvature T := R | κ | ds isalso similar but not effective for detecting embeddedness of closed curves since boththe infima among embedded and non-embedded closed curves coincide with 2 π ; see[23]. The energy ¯ B is certainly effective since a circle attains ¯ B = 4 π < ̟ ∗ .Recall that 4 π is the minimum of ¯ B among closed curves since ¯ B ≥ T ≥ π holds by the Cauchy-Schwarz inequality and Fenchel–Borsuk’s theorem.We finally discuss the idea of our proof. To perform a variational method weencounter the multiplicity-constraint making the admissible set non-open. M¨uller–Rupp’s proof [23] is mainly devoted to careful analysis of possible self-intersectionsby using rotation number, which has no direct extension to higher codimensionsor multiplicities. Instead, our proof proceeds in such a way that we divide theobjective curve at the point of multiplicity and then apply a variational argumentto each component “independently”. Each variational problem is formulated to bewell posed, and moreover its boundary condition is relaxed to being of zeroth order(although the most natural choice would be of first order since H ֒ → C ). Thisrelaxation allows us to obtain a strong rigidity of optimal configurations, whichbenefits from the celebrated classification of elasticae by Langer–Singer [14]. It issomewhat by chance that such independent and relaxed problems can be translatedback to the original problem (before division) while keeping certain optimality.Indeed, nontriviality of this point is explicitly reflected in our non-optimality result,Theorem 1.3. The non-optimality is caused by an obstruction for constructingoptimal planar closed curves, which is related with the irrationality of a certaingeometric quantity. Although such an issue is quite delicate in general, surprisinglyat least to the author, we can exhaustively verify non-optimality by reducing theproblem to a classical deep result of Andr´e [2] on the algebraic independence ofvalues of certain hypergeometric functions over the field of algebraic numbers. Thecase of higher codimensions stands in stark contrast to the planar case as it allowsunified optimality, Theorem 1.2. The main ingredient here is a simple but keyexample which we introduce in Example 3.12 and call elastic propeller.The above dividing idea motivates us to consider not only closed curves but alsoopen curves. In fact, we mainly deal with open curves in our proof, and obtain veryparallel results to Theorems 1.1 and 1.2, see Theorems 2.5 and 3.16. As for opencurves, our results are fully optimal for all codimensions and multiplicities.This paper is organized as follows: In Section 2 we recall and discuss classicalelasticae and prove Theorem 1.1 via the open-curve counterpart, Theorem 2.5. InSection 3 we introduce leafed elasticae and mainly discuss their rigidity, whichis then applied to the proof of Theorems 1.2 and 1.3 again via the open-curvecounterpart, Theorem 3.16. Section 4 is about applications to elastic flows. Acknowledgements.
The author would like to thank Marius M¨uller, Fabian Rupp,and Ryotaro Sakamoto for their helpful comments and discussions. This work is inpart supported by JSPS KAKENHI Grant Numbers 18H03670 and 20K14341, andby Grant for Basic Science Research Projects from The Sumitomo Foundation.
T. MIURA Elastica and Li–Yau type multiplicity inequality
The goal of this section is to prove Theorem 1.1. To this end we review andprove some results concerning classical elasticae; the most essential step is Lemma2.4. In particular, the so-called figure-eight elastica plays a key role throughout inthis paper. To define this we need to use some properties of elliptic integrals andfunctions, which we first address below for the sake of logical order.2.1.
Elliptic integrals and functions.
The incomplete elliptic integral of the firstkind F ( x, m ) and of the second kind E ( x, m ) with modulus m ∈ (0 ,
1) are definedby F ( x, m ) := Z x dθ p − m sin θ , E ( x, m ) := Z x p − m sin θdθ, respectively. The complete elliptic integral of first kind K ( m ) and of second kind E ( m ) are then defined by K ( m ) := F ( π/ , m ) , E ( m ) := E ( π/ , m ) , respectively. The (Jacobi) amplitude function is defined byam( · , m ) := F − ( · , m ) on R . The (Jacobi) elliptic functions are then given bycn( x, m ) := cos(am( x, m )) , sn( x, m ) := sin(am( x, m )) . Note in particular that cn( · , m ) and sn( · , m ) are 4 K ( m )-periodic, have zeroes (2 Z +1) K ( m ) and 2 Z K ( m ), respectively, and changes their sign at the zeroes (like cosineand sine).To define a figure-eight elastica we need to define a unique modulus m ∗ ∈ (0 , K ( m ∗ ) = 2 E ( m ∗ ); numerically, m ∗ ≈ . K ( m ) − E ( m )is increasing from − π/ ∞ . In addition, one can also easily check that(2.1) m ∗ > . K ( ) − E ( ). This is enough sharp forour argument in this section, but later we need to improve this estimate, cf. Lemma3.10 below.2.2. Classical elastica.
Here and hereafter n ≥ γ : [ a, b ] → R n minimizes the bending energy B in a suitable class of fixed-length curves, thenthere is some λ ∈ R such that the curve γ is a critical point of the energy(2.2) E λ [ γ ] := B [ γ ] + λL [ γ ] = Z γ ( | κ | + λ ) ds. By calculating the first variation of E λ (cf. [8, 6]) we obtain a fourth-order ODE,(2.3) 2 ∇ s κ + | κ | κ − λκ = 0 , where ∇ s denotes the normal derivative along γ ; more precisely, ∇ s ψ := ∂ s ψ −h ∂ s ψ, T i T , where T := ∂ s γ denotes the unit tangent, and here and hereafter h· , ·i denotes the Euclidean inner product. A smooth curve γ that solves (2.3) for some λ ∈ R is called an elastica . The classification of planar elasticae is essentially I–YAU TYPE INEQUALITIES FOR CURVES IN ANY CODIMENSION 5 obtained by Euler in 1744 and thus very classically known (cf. [16, 26]). Concern-ing general elasticae, Langer–Singer’s landmark study [14] provides an exhaustiveclassification result. Here we just collect the facts what we use later for our maintheorems:
Theorem 2.1 (Langer–Singer [14]) . The following statements hold. (i)
Any elastica is contained in an at-most-three-dimensional affine subspaceof R n . (Therefore, we may only consider planar elasticae n = 2 or spatialelasticae n = 3 .) (ii) If an elastica is non-planar, then it has everywhere non-zero curvature(and torsion). (iii)
Any planar elastica with a point of vanishing curvature is either a straightline or a wavelike elastica (see below for definition). (iv)
Suppose that a wavelike elastica γ : [ a, b ] → R satisfies the Navier bound-ary condition that γ ( a ) = γ ( b ) and κ ( a ) = κ ( b ) = 0 . Then there is apositive integer N such that γ is an N -fold elastica (see Definition 2.2). All the statements except for (iv) directly follow by Langer–Singer’s classificationresult [14] (see also an excellent lecture note [28]). Recall that a planar curve γ ⊂ R is called a wavelike elastica if there exist m ∈ (0 ,
1) and s ∈ [0 , K ( m )] such that,up to dilation, the curve γ parameterized by the arclength s ∈ [0 , L ] has signedcurvature k of the form k( s ) = 2 √ m cn( s − s , m ). Statement (iv) also follows ifone checks the fact that a figure-eight elastica is the only wavelike elastica that hasa self-intersection at their inflection points (where curvature k changes the sign).This seems to be a folklore fact but for example one may refer to the explicitrepresentation of wavelike elasticae in [23] to deduce that 2 E ( m ) − K ( m ) = 0.A figure-eight elastica is a particular example of a planar elastica. Here we recallthe definition in terms of curvature represented by an elliptic function. Definition 2.2 (Figure-eight elastica) . Given a positive integer N , we call animmersed curve γ : [ a, b ] → R n N -fold figure-eight elastica if γ is contained ina plane and its signed curvature k( s ) parameterized by the arclength s ∈ [0 , L ]satisfies (up to the choice of the sign) that(2.4) k( s ) = 2Λ √ m ∗ cn (Λ s − K ( m ) , m ∗ ) , Λ := 2 K ( m ∗ ) NL > . Similarly, we call a closed curve γ : T → R n N -fold closed figure-eight elastica ifthere is t ∈ T such that the curve ˜ γ : [0 , → R n defined by ˜ γ ( t ) := γ ( t + t ) isan N -fold figure-eight elastica.We also call a -fold (resp. 1-fold closed) figure-eight elastica half-fold figure-eight elastica (resp. one-fold figure-eight elastica , or simply figure-eight elastica ).Note that in Definition 2.2 the constant Λ just plays the role of a scaling factor;namely, if we let the curve γ represent the case of Λ = 1, then the general curve γ Λ is represented by γ Λ ( s ) = γ (Λ s ).The basic properties of figure-eight elasticae what we use are summarized asfollows, cf. Figure 1. Lemma 2.3 (Basic properties of figure-eight elasticae) . Let γ be an N -fold figure-eight elastica in R n . Then, up to an isometric transformation, γ is contained inthe plane R ≃ R × { } ⊂ R n and its arclength parameterization γ : [0 , L ] → R Figure 1.
Figure-eight elastica. is given by (2.5) γ ( s ) = 1Λ (cid:18) − E (am(Λ s − K ( m ∗ ) , m ∗ ) , m ∗ ) + (Λ s − K ( m ∗ ))2 √ m ∗ cn(Λ s − K ( m ∗ ) , m ∗ ) (cid:19) , where Λ := K ( m ∗ ) NL as in (2.4) . In addition, the above representation satisfies thefollowing properties. (i) The curve γ takes the origin if and only if s = 0 , N L, N L, . . . , L . Inaddition, these points also characterize those where the signed curvature k vanishes. (ii) The curve γ possesses the periodicity that γ ( N L − s ) = P γ ( s ) for s ∈ [0 , N L ] and γ ( s + N L ) = P P γ ( s ) for any s ∈ [0 , N − N L ] , where P i denotes the reflection with respect to the i -th component. (iii) ¯ B [ γ ] = ̟ ∗ N holds. In particular, ̟ ∗ and ̟ ∗ are the normalized bendingenergy of a half-fold and one-fold figure-eight elastica, respectively. (iv) h γ ′ (0) , γ ′ ( N L ) i = cos 2 φ ∗ holds, where φ ∗ ∈ (0 , π/ is defined by a uniqueangle such that cos φ ∗ = 2 m ∗ − > .Proof. Curve representation (2.5) is explicitly obtained in [23] from curvature rep-resentation (2.4) up to π -rotation, rescaling, and parameter-shifting. Our repre-sentation is chosen so that γ (0) is the origin and γ ′ (0) is contained in the firstquadrant, i.e., ( γ ) ′ (0) > γ ) ′ (0) >
0, where γ i denotes the i -th component.Indeed, thanks to the particular property that 2 E ( m ∗ ) = K ( m ∗ ), the curve γ takesthe origin at s = 0 (by periodicity, also at s = N L, . . . , L ). The positivity that( γ ) ′ (0) = 2 m ∗ − > m ∗ > .
5, cf. (2.1), while the other one that( γ ) ′ (0) = 2 √ m ∗ √ − m ∗ > m ∗ ∈ (0 , m ∗ and by the periodicproperties of elliptic functions and integrals. More precisely, from the origin thecurve goes convexly by the positivity of curvature, and then reach the e -axis at s = L N , and then turn back to the origin in a symmetric way. If N ≥
2, then itjust proceeds periodically afterwards.
I–YAU TYPE INEQUALITIES FOR CURVES IN ANY CODIMENSION 7
Property (iii) is essentially shown in [23] but we also give a proof since it is akey quantity and easily computed by (2.4). Up to rescaling we may only computethe case of Λ = 1, that is L = 2 K ( m ∗ ) N ; also, we only need to compute the case of N = 1, since other cases are directly computed by periodicity. By representation(2.4) with Λ = 1 and N = 1, and by periodicity, we compute¯ B [ γ ] = 2 K ( m ∗ ) Z K ( m ∗ )0 (cid:0) √ m ∗ cn( s − K ( m ∗ ) , m ∗ ) (cid:1) ds = 16 m ∗ K ( m ∗ ) Z K ( m ∗ )0 cn ( s, m ∗ ) ds = 16 m ∗ K ( m ∗ ) Z π/ cos ( x, m ∗ ) p − m ∗ sin x dx ( s = F ( x, m ∗ ))= 16 K ( m ∗ ) (cid:0) E ( m ∗ ) + ( m ∗ − K ( m ∗ ) (cid:1) . Since 2 E ( m ∗ ) = K ( m ∗ ), we see that ¯ B [ γ ] = ̟ ∗ , cf. (1.2).Property (iv) follows in the following way: By definition of φ ∗ and by the abovecomputation of ( γ ) ′ (0) we have cos φ ∗ = 2 m ∗ − γ ) ′ (0). Since φ ∗ ∈ (0 , π/ γ ′ (0) and the vector e = (1 , ⊤ , thereflection symmetry in (ii) implies that 2 φ ∗ is the desired angle made by γ ′ (0) and γ ′ ( N L ). (See also [7] for a different derivation.) (cid:3) We now state a key lemma for our main theorems, which is formulated in termsof a minimizing problem of the bending energy subject to a zeroth-order boundarycondition. Hereafter we mainly deal with open curves with H (= W , ) Sobolevregularity, where this regularity is natural in view of the bending energy. Note thatby Sobolev embedding H ֒ → C such curves still possess pointwise meaning up tofirst order; in particular, both immersedness and multiplicity are well defined. Lemma 2.4 (Minimality of -fold figure-eight elasticae) . Let γ : [ a, b ] → R n be animmersed H -curve such that γ ( a ) = γ ( b ) . Then ¯ B [ γ ] ≥ ̟ ∗ , where equality is attained if and only if γ is a half-fold figure-eight elastica. As is already emphasized in the introduction, the choice of this zeroth-orderboundary condition is a key idea in our strategy. Since Lemma 2.4 is later usedfor each part of the original curve after division at a multiplicity point, it seems tobe natural to choose an up-to-first-order (clamped) boundary condition in view of H ֒ → C . However for such a condition the minimum value sensitively dependson a first-order quantity so that there remains an additional issue of complicatedenergy competition. Instead, here we first relax the boundary condition to de-duce a geometrically unique minimizer, and then (in Section 3) consider whether acollection of such minimizers can be applied to the original problem.For convenience we introduce a class of unit-speed curves. Let I := (0 ,
1) and¯ I := [0 , X be the class of all unit-speed curves γ ∈ H ( I ; R n ) ֒ → C ( ¯ I ; R n ),that is, X := { γ ∈ H ( I ; R n ) | | γ ′ | ≡ } . Note that for any γ ∈ X we have L [ γ ] = 1 and thus ¯ B [ γ ] = B [ γ ]. This setting ofarclength parameterization does not lose generality thanks to the invariance of our T. MIURA problem up to rescaling and reparameterization. Finally, let X := { γ ∈ X | γ (0) = γ (1) = 0 } . Proof of Lemma 2.4.
Up to similarity and reparameterization, we may only arguewithin the class X . Thus, it is sufficient to consider the (unnormalized) bendingenergy B and prove the following properties: There exists ¯ γ ∈ X such that B [¯ γ ] =inf X B , such a minimizer ¯ γ must be a half-fold figure-eight elastica, and B [¯ γ ] = ̟ ∗ .The existence of a minimizer follows from the standard direct method, which wedemonstrate here for the reader’s convenience (and for using a similar argumentlater). Let { γ j } ⊂ X be a minimizing sequence B [ γ j ] → inf X B . Then, combiningthis limit with the fact that γ j (0) = 0 (= γ j (1)) and | γ ′ j | ≡
1, we find that { γ j } is bounded in H ( I ; R n ) so that there is a subsequence (without relabeling) thatconverges in the senses of H -weak and C . The limit curve ¯ γ is thus a unit-speedcurve in H ( I ; R n ) such that ¯ γ (0) = ¯ γ (1) = 0, i.e., ¯ γ ∈ X , and the weak lowersemicontinuity ensures thatinf X B = lim inf j →∞ B [ γ j ] ≥ B [¯ γ ] . This means that ¯ γ is a minimizer, completing the proof of existence.Now we prove that any minimizer must be a half-fold figure-eight elastica. Fixany minimizer γ ∈ X (note that | γ ′ | ≡ λ ∈ R such that γ is a critical point of the functional E λ , cf. (2.2); that is, for any η ∈ H ( I ; R n ) ∩ H ( I ; R n ) we have the H -continuous Fr´echet derivatives given by DL [ γ ]( η ) = Z I h γ ′ , η ′ i dt, DB [ γ ]( η ) = Z I h γ ′′ , η ′′ i dt − Z I | γ ′′ | h γ ′ , η ′ i dt, and the minimality of γ implies that DE λ [ γ ] = DB [ γ ]+ λDL [ γ ] = 0 for some λ ∈ R .A bootstrap argument then shows that γ ∈ C ∞ ( ¯ I ; R n ). For all η ∈ C ∞ c ( I ; R n ) wededuce from integration by parts that DE λ [ γ ]( η ) = R h ∇ s κ + | κ | κ − λκ, η i = 0,and hence γ is an elastica, cf. (2.3); in addition, by considering all η ∈ C ∞ ( ¯ I ; R n )such that η (0) = η (1) = 0 we also find that DE λ [ γ ]( η ) = [ h κ, ∇ s η i ] = 0, and hence κ (0) = κ (1) = 0 (see also [6, Lemma A.1]). Combining these facts with the originalboundary condition γ (0) = γ (1) = 0, and using Theorem 2.1 (i)–(iv), we find thatthe minimizer γ must be an N -fold figure-eight elastica for some positive integer N . By Lemma 2.3 (iii) and the energy-minimality of γ , we have N = 1. We thusconclude that any minimizer is a half-fold figure-eight elastica.Finally, by Lemma 2.3 (iii), we deduce that the minimum is ̟ ∗ . (cid:3) Li–Yau type multiplicity inequality.
We now turn to the proof of Theorem1.1. For later use it is convenient to first prove an open-curve counterpart ofTheorem 1.1.
Theorem 2.5 (Multiplicity inequality for open curves) . Let γ : [0 , → R n be animmersed curve with a point of multiplicity k ≥ . Then (2.6) ¯ B [ γ ] ≥ ̟ ∗ ( k − . In particular, if an immersed curve γ has the property that ¯ B [ γ ] < ̟ ∗ , then γ isembedded; the threshold ̟ ∗ is optimal due to a half-fold figure-eight elastica. I–YAU TYPE INEQUALITIES FOR CURVES IN ANY CODIMENSION 9
Proof of Theorem 2.5.
By the assumption on multiplicity, there are 0 ≤ a < · · ·
0, then we may cut off the part γ | [0 ,a ] and create a new curve whose normalized bending energy is strictly lessthan the original one, and hence without loss of generality we may assume that a = 0. Similarly, we may assume that a k = 1. Up to translation we may assumethat the multiplicity point is the origin, and hence can apply Lemma 2.4 to each γ i := γ [ a i ,a i +1 ] , where i = 1 , . . . , k −
1, to deduce that(2.7) L [ γ i ] B [ γ i ] = ¯ B [ γ i ] ≥ ̟ ∗ . Noting that B [ γ ] = P k − i =1 B [ γ i ] and L [ γ ] = P k − i =1 L [ γ i ], we have(2.8) ¯ B [ γ ] = k − X i =1 L [ γ i ] k − X i =1 B [ γ i ] ≥ k − X i =1 L [ γ i ] k − X i =1 L [ γ i ] ̟ ∗ ≥ ( k − ̟ ∗ , where the last estimate follows by the elementary HM-AM inequality. (cid:3) Theorem 1.1 can be regarded as a special consequence of Theorem 2.5.
Proof of Theorem 1.1.
Given any closed curve γ with a point of multiplicity k ,we can create an open curve with a point of multiplicity k + 1 after cutting γ atthe original point of multiplicity and opening the domain T to [0 , (cid:3) Leafed elastica and optimality
The goal of this section is to prove Theorems 1.2 and 1.3. To this end weintroduce the notion of leafed elastica , which is compatible with our problem. Wefirst discuss some basic properties of leafed elasticae, from which we observe howthe difference depending on the pair ( n, k ) occurs in optimality.3.1.
Leafed elastica.
Leafed elasticae are defined by connecting leaves of halffigure-eight elasticae.
Figure 2.
Leaf: A half-fold figure-eight elastica.
Definition 3.1 (Leafed elastica) . Let n ≥ k ≥
1. We call an immersed H -curve γ : [ a, b ] → R n k -leafed elastica if there are a = a < a < · · · < a k = b such that for each i = 1 , . . . , k the curve γ i := γ | [ a i − ,a i ] is a half-fold figure-eightelastica, and also L [ γ ] = · · · = L [ γ k ]. Similarly, we call a closed curve γ : T → R n closed k -leafed elastica if there is t ∈ T such that the curve ˜ γ : [0 , → R n definedby ˜ γ ( t ) := γ ( t + t ) is a k -leafed elastica. In addition, to specify the dimension n of the target space, sometime we also use the term (closed) ( n, k ) -leafed elastica . Remark . By definition, a k -leafed elastica γ has a point of multiplicity k +1, namely γ ( a ) = · · · = γ ( a k ). Also, a closed k -leafed elastica has a point ofmultiplicity k . We call such a point joint of a leafed elastica.An easy consequence from the definition is the following Proposition 3.3 (Regularity and energy) . Let k ≥ . Then any k -leafed (resp.closed k -leafed) elastica is of class C , and piecewise analytic, has a point of mul-tiplicity k + 1 (resp. k ), and has the energy ¯ B [ γ ] = ̟ ∗ k .Proof. The piecewise analyticity follows by the representation in (2.5). The whole C , -regularity follows in this way; first, any k -leafed elastica γ is automatically ofclass C since H ֒ → C ; also, the curvature of γ vanishes at each joint of leavesby Lemma 2.3 (i) so that γ is also of class C ; finally, the third derivative of γ isbounded in L ∞ since each (planar) leaf of γ has the signed curvature given in (2.4)(in an affine subspace ≃ R ) so that, by the derivative formula(3.1) ∂∂x cn( x, m ) = − sn( x, m ) p − m sn ( x, m ) , its derivative is bounded; consequently, γ is of class W , ∞ = C , . The multiplicityat the joint is already discussed. Finally, the normalized bending energy can beexplicitly computed, cf. Lemma 2.3 (iii). (cid:3) We mention two obvious examples of classical figure-eight elasticae.
Example . Let k ≥
1. Then a k -fold figure-eight elastica is an analytic planarexample of an ( n, k )-leafed elastica. Example . Let k ≥ k -fold figure-eight elastica is ananalytic planar example of a closed ( n, k )-leafed elastica.Leafed elasticae have flexibility due to possible discontinuity of their third deriva-tives at the joints, and in particular these examples do not exhaust all possibleconfigurations. For example, it is easy to imagine that for any k ≥ k -leafed elasticae by reflecting or twisting leaves at their joints arbitrarily.On the other hand, to obtain closed k -leafed elasticae, we are required to close itup in the first-order sense. This requirement causes non-negligible rigidity. Indeed,it is easy to observe Proposition 3.6.
No closed -leafed elastica exists. A closed -leafed elastica is(up to similarity and reparameterization) uniquely given by a figure-eight elastica.Proof. This follows since the angle made by the tangent vectors at the endpointsof one leaf is given by 2 φ ∗ ∈ (0 , π ), cf. Lemma 2.3 and Figure 2. (cid:3) In general, whether there exists a closed ( n, k )-leafed elastica can be characterizedby whether the endpoints of leaves can be joined up to first order. This fact issummarized as in the following lemma, the proof of which is straightforward andsafely omitted.
Lemma 3.7 (Characterization of closed ( n, k )-leafed elasticae) . Let n ≥ and k ≥ . Let Ω ∗ ( n, k ) be the set of all k -tuples ( ω , . . . , ω k ) of n -dimensional unit-vectors ω , . . . , ω k ∈ S n − ⊂ R n such that h ω i , ω i − i = cos 2 φ ∗ holds for any i = 1 , . . . , k ,where we interpret ω := ω k . I–YAU TYPE INEQUALITIES FOR CURVES IN ANY CODIMENSION 11 If γ : T → R n is a unit-speed closed ( n, k ) -leafed elastica, then the k -tuple ofvectors ω i := γ ′ ( ik ) , i = 1 , . . . , k , is an element of Ω ∗ ( n, k ) .Conversely, for any element ( ω , . . . , ω k ) ∈ Ω ∗ ( n, k ) there exists a unique unit-speed closed ( n, k ) -leafed elastica γ : T → R n such that γ (0) = 0 and the k -tupleof ω i := γ ′ ( ik ) , i = 1 , . . . , k , is an element of Ω ∗ ( n, k ) . This characterization is useful both for ensuring nonexistence and for construct-ing concrete examples.We first verify nonexistence of planar closed leafed elasticae with odd leavescaused by an algebraic obstruction for closing the leaves (it always exists if k is even,cf. Example 3.5). This obstruction is related to an irrationality or transcendenceproblem involving special values of hypergeometric functions. Such a problem isin general difficult in spite of its classicality, see e.g. [10] and references therein.However, fortunately, our problem can be reduced to the following theorem ofAndr´e [2]. Let Q ⊂ C denote the field of algebraic numbers. Theorem 3.8 (Andr´e [2]) . The values of the Gaussian hypergeometric functions F [ , ; 1; z ] and F [ − , ; 1; z ] are algebraically independent over Q for any z ∈ Q with < | z | < . With the aid of this result we prove the following
Proposition 3.9.
For any odd k ≥ there exists no closed (2 , k ) -leafed elastica.Proof. We first note that for any given ω ∈ S there are only two possibilitiesof ω ′ ∈ S such that h ω, ω ′ i = cos 2 φ ∗ , namely ω ′ = R ± ω , where R ± ∈ SO(2)denotes the (counterclockwise) rotation matrix through angle ± φ ∗ . Combiningthis fact with Lemma 3.7, we deduce that the assertion is equivalent to the followingstatement: For any odd integer k ≥ k -tuple of rotation matrices R , · · · , R k ∈ SO(2) through angle either 2 φ ∗ or − φ ∗ such that R k · · · R = I ,where I denotes the identity matrix; in other words, for any odd k ≥ σ , . . . , σ k ∈ {− , } such that P ki =1 σ i φ ∗ ∈ π Z .Now we prove that the above equivalent statement holds true. It is suffi-cient to prove the irrationality that φ ∗ π Q . By the well-known relation (e.g.found in [1, 17.3.9, 17.3.10]) that F [ , ; 1; m ] = π K ( m ) and F [ − , ; 1; m ] = π E ( m ) for m ∈ (0 , m ∗ , we deduce that F [ , ; 1; m ∗ ] − F [ − , ; 1; m ∗ ] = 0. By Andr´e’s theorem, the number m ∗ ∈ (0 ,
1) is tran-scendental, and hence so is cos φ ∗ (= 2 m ∗ − φ = ( p/q ) π with p/q ∈ Q we have T q (cos φ ) = cos( qφ ) ∈ {± } , where T q denotes the q -th Chebyshev polynomial ofthe first kind, and hence cos φ ∈ Q . (cid:3) In contrast, if a codimension is positive, n ≥
3, then we can still construct a closed( n, k )-leafed elastica for any multiplicity k ≥
2. We may again focus on the case ofodd k ≥ φ ∗ ∈ (0 , π/
2) ( ≈ . ◦ ) in the form of 45 ◦ < φ ∗ < ◦ . Theupper bound plays a crucial role in Example 3.12 but the lower bound is just usedin Remark 3.18. To this end we first prove that 0 . < m ∗ < .
85. Our proof isbased on the series expansion and provides very simple machinery to estimate thevalue of m ∗ , which is easy to calculate and as accurate as one needs in principle. Lemma 3.10. . < m ∗ < . . Proof.
Recall the known series expansions of E and K (e.g. found in [1, 17.3.11,17.3.12]) that are absolutely convergent for any m ∈ (0 , K ( m ) = Z π/ p − m sin θ dθ = π ∞ X n =0 (cid:18) (2 n − n )!! (cid:19) m n ,E ( m ) = Z π/ p − m sin θdθ = π ∞ X n =0 (cid:18) (2 n − n )!! (cid:19) − n m n . Letting f ( m ) := π ( K ( m ) − E ( m )), we deduce from these expansions that f ( m ) = − ∞ X n =1 A n m n , where A n := (cid:18) (2 n − n )!! (cid:19) n + 12 n − . Since m ∗ is a unique root of the increasing function f , it is now sufficient to provethat f ( ) < f ( ) >
0. To this end we first state general criteria forclarity. Let S N ( m ) := N X n =1 A n m n , T N ( m ) := S N ( m ) + 11 − m m N +1 . Then the following general estimate holds for any m ∈ (0 ,
1) and any integer N ≥ S N ( m ) ≤ f ( m ) + 1 ≤ T N ( m ) . Indeed, the lower bound is trivial since A n m n ≥ S N ( m ) is a partial sumof P A n m n , while the upper bound also follows since A n ≤ P A n m n ≤ S N ( m ) + P ∞ n = N +1 m n = T N ( m ). We thus obtain generalcriteria to compare f ( m ) and 1; namely, it is now sufficient to find some examplesof N, N ′ ≥ S N ( ) > T N ′ ( ) <
1. An explicit computation(of finite operations multiplying integers) shows that S (cid:18) (cid:19) = 17398658471271717986918400 > , T (cid:18) (cid:19) = 7174004775396983172057594037927936 < , thus completing the proof. (cid:3) This immediately implies
Lemma 3.11. π/ < φ ∗ < π/ .Proof. By definition cos φ ∗ = 2 m ∗ − φ ∗ , it is sufficient to prove that cos( π/
3) =1 / < m ∗ − < / √ π/ (cid:3) The key example is the following
Example . Let ω , ω , ω ∈ S ⊂ R be taken so that thetriple of them is an element of Ω ∗ (3 , φ ∗ ∈ (0 , π/ ω , ω , ω need to make a triangularpyramid whose one face is an equilateral triangle and the others are congruent to asame isosceles triangle of angle 2 φ ∗ , cf. Figure 3 (left). By using such a triple we canconstruct as in Lemma 3.7 an example of a closed (3 , n ≥ elastic propeller . I–YAU TYPE INEQUALITIES FOR CURVES IN ANY CODIMENSION 13
Figure 3.
Elastic propeller: A unique 3-leafed elastica ( n ≥ Proposition 3.13.
Let n ≥ . Then a closed ( n, -leafed elastica is (up to simi-larity and reparameterization) uniquely given by an elastic propeller.Proof. This follows by the uniqueness of a closed (3 , n, (cid:3) An elastic propeller can be also used for constructing (non-symmetric) closedleafed elasticae with any higher odd number of leaves in a unified manner.
Example n, k )-leafed elastica for n ≥ k ≥ . For any n ≥ k ≥ n, k )-leafed elastica by connecting an elastic propeller and a k − -fold closedfigure-eight elastica.In summary, we obtain Proposition 3.15.
Let n ≥ and k ≥ . Either if n ≥ or if k is even, thenthere exists a closed ( n, k ) -leafed elastica.Proof. It follows from Example 3.5, Example 3.12, and Example 3.14. (cid:3)
Optimality and rigidity in multiplicity inequality.
From now on weprove Theorems 1.2 and 1.3 by using the above results on leafed elasticae. Wefirst state and prove an open-curve counterpart of Theorem 1.2, which holds in fullgenerality.
Theorem 3.16 (Optimality and rigidity for open curves) . Let n ≥ and k ≥ .Then there exists an immersed H -curve γ : [0 , → R n with a point of multiplicity k such that (3.2) ¯ B [ γ ] = ̟ ∗ ( k − . In addition, equality (3.2) is attained if and only if γ is a ( k − -leafed elastica.Proof. The existence of an optimal curve follows since a k − -fold figure-eight elas-tica attains equality. In addition, any ( k − γ attains (3.2). Then, as in the proof ofTheorem 2.5, the curve γ can be divided into k curves γ , . . . , γ k − . In addition, equality holds for all the inequalities in the proof of Theorem 2.5. In view of theHM-AM inequality (2.8) we have L [ γ ] = · · · = L [ γ k − ]. In addition, in view of(2.7) we also have L [ γ i ] B [ γ i ] = ̟ ∗ for all i , and hence by Lemma 2.4 each curve γ i needs to be a half-fold figure-eight elastica. This means that γ is a ( k − (cid:3) Theorem 1.2 is now a direct consequence of Theorem 3.16 and the contents inSection 3.1.
Proof of Theorem 1.2.
The existence of a closed curve in R n with multiplicity k satisfying (1.4) is ensured by Proposition 3.15 combined with Proposition 3.3. Also,we deduce that any of such closed curves must be a k -leafed elastica by opening thegiven closed curve at a multiplicity point and applying Theorem 3.16 to the openedcurve. Finally, any closed k -leafed elastica attains (1.4) by Proposition 3.3. (cid:3) We finally prove Theorem 1.3. Proposition 3.9 combined with Theorem 3.16 isalready sufficient for asserting the (weaker) statement that there exists no closed H -curve in R that has a point of multiplicity k and attains (1.4). However, thisdoes not rule out existence of a minimizing sequence such that ¯ B [ γ j ] → ̟ ∗ k . Inorder to rule out this phenomenon we ensure general existence of planar optimalcurves (which are not necessarily leafed elasticae).Let C k denote the class of all closed H -curves in the plane R with a point ofmultiplicity k . Let β k := inf γ ∈ C k ¯ B [ γ ] . We have β = 4 π obviously. Theorems 1.1 and 1.2 imply that β k = ̟ ∗ k for anyeven k . For any odd k ≥ ̟ ∗ k ≤ β k ≤ ̟ ∗ ( k + 1) . Indeed, the former directly follows by Theorem 1.1 and the last trivial upper boundfollows due to the k +12 -fold figure-eight elastica having multiplicity k (in fact k + 1).In fact, we have a slightly improved (non-optimal) upper bound for odd k = 2 ℓ + 1: β k = β ℓ +1 ≤ (2 √ ̟ ∗ ℓ + 2 π ) = (cid:0) √ ̟ ∗ ( k −
1) + 2 π (cid:1) . This can be obtained by wrapping a unit-length figure-eight elastica ℓ times andattaching a circle of radius R to its center; the resulting curve γ is a closed H -curvewith multiplicity k , whose energy can be explicitly computed, ¯ B [ γ ] = ( ℓ · πR )( ℓ · ̟ ∗ + 2 π/R ), so that optimizing in R yields the desired upper bound. Althoughthe exact value of β ℓ +1 is remained open, here we prove that at least there existsa minimizer attaining the infimum in β k , which can be decomposed into k elasticae(this decomposition is however not used in this paper). Proposition 3.17.
For any k (in particular, any odd k ≥ ) there exists a curve ¯ γ ∈ C k such that ¯ B [¯ γ ] = β k . In addition, the curve γ can be divided into k (open)curves γ , . . . , γ k at a point of multiplicity k and each curve is an elastica.Proof. Fix an arbitrary k . Let { γ j } ⊂ C k be a minimizing sequence of ¯ B . Afterreparameterization, rescaling, and translation, we may assume that for each j thecurve γ j is of unit-speed and thus of unit-length L [ γ j ] = 1, and also there are0 = a j (1) < a j (2) < · · · < a j ( k + 1) = 1 (= a j (1)) in T such that γ ( a j ( i )) = 0 forall i (thanks to multiplicity k ). Then a standard direct method argument, which I–YAU TYPE INEQUALITIES FOR CURVES IN ANY CODIMENSION 15 is parallel to the proof of Lemma 2.4, implies the existence of a unit-speed curve¯ γ ∈ H ( T ; R ) such that β k = ¯ B [¯ γ ].Now we ensure that the limit curve ¯ γ still possesses multiplicity k . Using theabove notation a j ( i ), we prove that all adjacent points a j ( i ) and a j ( i + 1) do notcollide as j → ∞ . Let L j ( i ) := | a j ( i + 1) − a j ( i ) | . By the unit-speed parameteriza-tion, L j ( i ) is nothing but the length of the curve γ j | i := γ j | [ a j ( i ) ,a j ( i +1)] , and henceby the Cauchy-Schwarz inequality ¯ B = LB ≥ T involving the total curvature T [ γ ] = R γ | κ | ds we have B [ γ j ] ≥ B [ γ j | i ] ≥ L j ( i ) T [ γ j | i ] ≥ L j ( i ) π , where the last estimate T [ γ j | i ] ≥ π follows by Fenchel–Borsuk’s theorem (i.e., T ≥ π for closed curves) and by the fact that γ j | i and its suitable reflection at theorigin creates a closed curve whose total curvature is 2 T [ γ j | i ]. By the boundednessof B [ γ j ] we deduce that there is δ > L j ( i ) ≥ δ holds for all i and j .This means that no adjacent points collide, and hence up to a subsequence (withoutrelabeling), all a j (1) , . . . , a j ( k ) converge to distinct k points a (1) , . . . , a ( k ) in T as j → ∞ . By C -convergence we have ¯ γ ( a ( i )) = 0 for all i = 1 , . . . , k . Therefore, ¯ γ still has multiplicity k so that ¯ γ ∈ C k . This ensures the existence of a minimizer.Finally, we prove that any minimizer ¯ γ ∈ C k can be divided into k elati-cae. Let p ∈ R be a point of multiplicity k and choose (ordered) k distinctpoints a , . . . , a k ∈ ¯ γ − ( p ). Divide ¯ γ into k (open) curves ¯ γ , . . . , ¯ γ k by cutting at a , . . . , a k ∈ T . Then we can freely perturb each of ¯ γ i away from the endpointswhile keeping the original curve within the class C k , since such a perturbationpreserves multiplicity of ¯ γ ; here “freely” means that we only need to take the fixed-length constraint into account in terms of the multiplier method. Hence by theminimality of ¯ γ we deduce that each ¯ γ i satisfies (2.3) with some λ i ∈ R , thus beingan elastica. This ensures a decomposition of ¯ γ into k elasticae. (cid:3) We are now in a position to complete the proof of Theorem 1.3.
Proof of Theorem 1.3.
We prove by contradiction. Suppose that for an odd k ≥ { γ j } ⊂ C k such that¯ B [ γ j ] → ̟ ∗ k ; this means that β k = ̟ ∗ k since ̟ ∗ k is a lower bound, cf.Theorem 1.1. By Proposition 3.17 there exists a closed curve ¯ γ ∈ C k attainingequality (1.4). Then by applying Theorem 3.16 (as in the proof of Theorem 1.2) weconclude that ¯ γ must be a closed (2 , k )-leafed elastica. However, this contradictsthe nonexistence result in Proposition 3.9. (cid:3) We close this section by mentioning miscellaneous remarks.
Remark . In contrast that genericuniqueness of closed k -leafed elasticae holds for k ≤
3, cf. Propositions 3.6 and3.13, this is not the case for k ≥
4. Indeed, for ( n, S n − -freedom around the joint-axis of two figure-eight elasticae;similar phenomena occur for any composite number k . In addition, there is anothermechanism of non-uniqueness even for a prime number k ; for any k ≥ k papers of congruent obtuse isosceles triangles(with angle 2 φ ∗ > π/
2) along their equal-length sides even in the three-dimensionalspace (in contrast to k = 3). Remark C -regularity) . By representa-tion (2.4) and derivative formula (3.1) we easily deduce that if a k -leafed elastica γ has C -regularity, then the curve γ is a unique planar curve given by a k − -foldfigure-eight elastica. As a consequence, if k ≥ γ is a closed ( n, k )-leafed elastica of class C , then γ is a unique planar curve (up to similarity andreparameterization) given by a k -fold closed figure-eight elastica; also, if k ≥ n, k )-leafed elastica of class C . Remark . As is indicated in [23], the total (absolute) cur-vature T [ γ ] = R γ | κ | ds is not an effective embeddedness criterion for closed curvessince the total curvatures of both an embedded thin convex curve and a thin figure-eights (closed to a segment) are nearly 2 π . However, we still have the optimal lowerbound T [ γ ] > ( k − π for open curves with multiplicity k via Fenchel–Borsuk’stheorem (as in the proof of Proposition 3.17) and hence T [ γ ] > kπ for closed curveswith multiplicity k . 4. Elastic flows
In this last section we discuss applications to elastic flows. We call the L -gradient flow of the modified (or length-penalized) bending energy E λ := B + λL (as in (2.2)) for a given λ > elastic flow , and the L -gradient flow of the bendingenergy B under the fixed-length constraint L [ γ ] = L for a given L > fixed-lengthelastic flow . One-parameter families of curves γ : T × [0 , ∞ ) → R n describing suchflows solve the PDE in the form of(4.1) ∂ t γ = − ∇ s κ − | κ | κ + λκ, where in the former case λ > E λ , while in the lattercase it depends on the solution and is given in the form of(4.2) λ ( t ) = R γ ( t ) h ∇ s κ + | κ | κ, κ i ds R γ ( t ) | κ | ds . At least from Polden’s 1996 thesis, elastic flows are studied by many authors, see e.g.a recent nice survey [18] and references therein. Concerning these flows, long-timeexistence and smooth convergence to an elastica are valid in general at least fromsmooth closed initial curves. The results follow by combining the fundamental resultby Dziuk–Kuwert–Sch¨atzle [8] with recent developments on the Lojasiewicz-Simoninequality as is demonstrated in [23] (see also [18]); we note that the argument in [23]directly works for higher codimensions as so do the key ingredients [8, 24, 19, 25].Our Li–Yau type inequality can be used for ensuring all-time embeddedness ofsolutions starting from initial curves below certain energy thresholds. Note that insecond-order flows such a property often holds in general (without smallness) bythe maximum principle (see e.g. [11] and also [4]) but for elastic flows or generalhigher-order flows this is not the case [3].
Theorem 4.1.
Let λ > and γ : T → R n be a closed curve such that λ E λ [ γ ] < ̟ ∗ . Then the elastic flow starting from γ is embedded for all t ≥ . In addition,it smoothly converges as t → ∞ to a one-fold round circle of radius √ λ up toreparameterization. Theorem 4.2.
Let γ : T → R n be a closed curve such that ¯ B [ γ ] < ̟ ∗ . Thenthe fixed-length elastic flow starting from γ is embedded for all t ≥ . In addition, I–YAU TYPE INEQUALITIES FOR CURVES IN ANY CODIMENSION 17 it smoothly converges as t → ∞ to a one-fold round circle of radius L [ γ ]2 π up toreparameterization. These results extend M¨uller–Rupp’s corresponding results in [23] from n = 2 to n ≥
2. The threshold 4 ̟ ∗ is optimal simultaneously for all-time embeddedness andfor convergence to a circle; indeed, to each flow, a figure-eight elastica of suitablesize is a non-embedded stationary solution and attains the threshold 4 ̟ ∗ .We may safely omit the proof of the above theorems since M¨uller-Rupp [23]already provide detailed proofs that completely work for n ≥ once our result(Theorem 1.1) is established. A key point is that ¯ B ≤ λ E λ always holds thanksto the elementary inequality that 4 λab ≤ ( a + λb ) . Hence, along each of the twoflows, the gradient-flow structure implies the preservation of the property that ¯ B < ̟ ∗ for all time. This property with Theorem 1.1 ensures all-time embeddedness.Uniqueness of the limit profile is ensured by the fact that a circle is a uniqueclosed elastica such that ¯ B < ̟ ∗ ; this can be verified by investigating an energyquantization induced by the classification of closed elasticae as follows. (Relativearguments are previously carried out for the Willmore flow [12] through the Li–Yauinequality [15] and Bryant’s classification of Willmore spheres [5].)We argue more on the energy quantization of closed elasticae. Closed elasticaeare now completely classified. Indeed, it is known (essentially from Euler’s 1744study) that any planar closed elasticae is either a round circle or a figure-eightelastica, or their multiple covering. An N -fold circle (resp. figure-eight elastica) γ has the energy ¯ B [ γ ] = 4 π N (resp. 4 ̟ ∗ N , cf. Lemma 2.3 (iii)). In addition,by Langer–Singer’s theorem on spatial closed elasticae [13], any closed non-planarelastica is an embedded nontrivial torus knot or its multiple covering. Hence, theclassical F´ary-Milnor theorem [9, 21] shows that the total curvature R | κ | ds of anynontrivial knot is greater than 4 π , and hence by the Cauchy-Schwarz inequalitywe deduce that any other (spatial) elastica γ has the property that ¯ B [ γ ] > π .Therefore, in order to show that γ is a closed elastica such that ¯ B [ γ ] < ̟ ∗ = ⇒ γ is a circle,it is sufficient to check that 4 ̟ ∗ < π . This is already proved in [27, 23] byestimating elliptic integrals directly. This also follows variationally, since our keyLemma 3.7 combined with the fact that a circle belongs to X and has energy¯ B = 4 π implies that ̟ ∗ < π . Here we give an alternative variational proofwhich only relies on the classification of planar closed elasticae: Remark ̟ ∗ < π ) . Let Z be the class of zero-rotation-number planar closed H -curves of unit-speed and of unit-length. Since Z is closed in H -weak (or C ) and open in H , a direct method for the bending energyensures the existence of a smooth minimizer being an elastica; by classification,this must be a unique figure-eight elastica, and hence min Z B = 4 ̟ ∗ . On theother hand, since the bending energy of a 2-fold circle γ of unit-length is 16 π , byreflecting one of the two circles we can create a competitor curve ˜ γ ∈ Z of energy B [˜ γ ] = 16 π (such that ˜ γ parameterizes osculating circles as a figure-eight). Theuniqueness of a minimizer implies that 4 ̟ ∗ = min Z B < B [˜ γ ] = 16 π .The energy quantization observed here is summarized as follows: Proposition 4.4.
Let C be the set of all closed elasticae in R n , n ≥ . There is astrictly increasing sequence { b k } ∞ k =1 ⊂ [4 π , ∞ ) such that ¯ B ( C ) = { b , b , . . . } and that b = 4 π , b = 4 ̟ ∗ , and b = 16 π . The preimage ¯ B − ( b ) consists of circles, ¯ B − ( b ) figure-eight elasticae, and ¯ B − ( b ) two-fold circles. We also mention some differences between n = 2 and n ≥ n = 2 such a convergence follows just from all-timeembeddedness (but not necessarily below the threshold) since a circle is the onlyembedded planar elastica. This is not the case for n ≥ n = 2 the fact is that convergence to a circle holdseven if we merely assume that an initial curve has rotation number ±
1, since alongthe flow the rotation number is preserved while a circle is the only planar closedelastica with rotation number ±
1. This assumption is certainly weaker than all-timeembeddedness (or our smallness assumptions) since embeddedness implies that therotation number must be ±
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Department of Mathematics, Tokyo Institute of Technology, Meguro,Tokyo 152-8511, Japan
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