On the Convergence of the Elastic Flow in the Hyperbolic Plane

Abstract We examine the L2-gradient flow of Euler’s elastic energy for closed curves in hyperbolic space and prove convergence to the global minimizer for initial curves with elastic energy bounded by 16. We show the sharpness of this bound by constructing a class of curves whose lengths blow up in infinite time. The convergence results follow from a constrained sharp Reilly-type inequality.


Introduction . History and Context
Our object of study -Euler's elastic energy -measures the bending of a curve in some Riemannian manifold. Since Euler's characterization of its critical points in Euclidean space in 1744 it was widely studied and led to a variety of mathematical methods, see for instance [37]. For a smooth curve γ : I → M in a Riemannian manifold M it is de ned by where κ denotes the curvature and ds denotes the integration with respect to the arclength parameter of γ in M. Its L -gradient ow (1.3) is called elastic ow. Critical points of E are called free elastica.
In [11,15,24,27,34] long time existence of the elastic ow in Euclidean (i.e. M = R d , d ≥ ) and hyperbolic (i.e. M = H ) space was shown, but convergence results are only established under length penalization, i.e. for the L -gradient ow of E λ (γ) := E(γ)+λL(γ) for some λ > , where L(γ) denotes the length of γ. E orts were made to understand the behavior of the penalized ow as λ → . In [26] it is shown that for energies satisfying a Palais-Smale condition, the convergence behavior is preserved provided there are no 'migrating critical points', which is suggested by [36] for the elastic ow in the hyperbolic half plane M = H . However, as [26] points out, the Palais-Smale condition fails to hold true in H . This article answers the open problem posed by Linnér in [26,Section 1.13], namely whether a positive penalization λ > is necessary for the convergence of the gradient ow in the hyperbolic plane. The answer we give is the following: For full convergence of the ow length penalization is necessary, whereas small initial energies still lead to convergent evolutions without length penalization.
Our particular interest of closed curves in H is due to the close connection to the Willmore energy of surfaces of revolution, namely E(γ) = π S(γ) H dA, (1.1) where S(γ) denotes the toroidal surface that arises from revolving γ about the x-axis, see [28] or [16,Theorem 4.1]. Moreover, free elastica in hyperbolic space de ne Willmore surfaces of revolution which were extensively investigated for instance in [3,10,17,30,31]. In [28] the relation between the Willmore energy and the elastic energy was used to show that the global minimum of the Willmore energy of all surfaces of revolution is attained at the stereographic projection of the Cli ord torus. This torus can be obtained by the revolution of (any rescaled and translated version of) around the x-axis. This is the reason we call the curve in (1.2) Cli ord elastica in the following. Note that it is the global minimum of the elastic energy of closed curves in the hyperbolic plane. Consequences of the results of this paper for the Willmore ow of tori in Euclidean three-space will be the content of future research. The aforementioned articles together with [19,29] lay the methodological groundwork for our approach. We however work directly with the unpenalized ow, examining evolution and asymptotic behavior of the length. A large part of this examination will be an explicit parametrization and a close examination of constrained elastic curves in the hyperbolic plane. Previously found parametrizations, for instance in [23,29,31], either apply only for free elastica or are too general for our purposes.

. Overview and Main Results
In the following we present the two main results of this paper: The rst one is the optimal Reilly-type inequality for curves with small energy in Theorem 1.1 and its consequence, the convergence of the elastic ow for initial values with small energy in Theorem 1.2. The second one is the existence of non-converging evolutions of initial values with higher energy in Theorem 1. 3.
First of all note that smooth long-time existence of the elastic ow ∂ t f = −∇ L E λ (f ), on S × ( , ∞), f (·, ) = f , on S . (1. 3) has already been proved for λ ≥ , see [15,Theorem 1.1 (i)] for each f ∈ C ∞ (S , H ). Another striking insight that [15] reveals is that each solution f ∈ C ∞ (S × [ , ∞); H ) with uniformly-in-time bounded hyperbolic length already subconverges up to isometries of H and reparametrization. For details see Theorem 7.1. For the penalized ow with λ > the length is naturally bounded, since it is obviously controlled by the energy E λ . For λ = however one has to answer the main question: Do evolutions by unpenalized elastic ow in H have uniformly bounded hyperbolic length?
To get a avor for the question let us remind the reader of the necessity of length penalization in Euclidean space. If we start the ow with an initial curve that is a circle of radius r , then the solution of the elastic ow with λ = is given by a circle of radius r(t) = (r + t) , whose length is unbounded as t → ∞. In H one does not expect the same behavior as in the Euclidean case. This is mainly due to the fact that scaling, i.e. z → θz for some θ > , is an isometry in the hyperbolic plane. In particular streching up the curve does not decrease the energy as it would do in the Euclidean case.
In [16] the evolution of circles under the ow (1.3) was studied: It was shown that the unpenalized ow converges for each initial datum f that parametrizes a circle. The limit is always an isometric image of (1.2).
In the sphere it is unknown whether length penalization is necessary, but for λ > convergence is shown in [12,Theorem 1.1 (ii)].
Since the gradient ow structure yields a natural bound on the elastic energy E, an idea would be to bound the quotient of elastic energy and length from below, to ensure that the length remains bounded along the ow as well. Such a bound is called Reilly inequality as it was rst obtained in [35] for Euclidean hypersurfaces. A stronger version of the inequality also holds true for hypersurfaces in hyperbolic space [20,Theoreme 1], but in general not for curves (see [29,Fig. 8]). Our rst result shows that such an inequality holds below a certain energy level, which we have also shown to be sharp. A similar result for open curves was obtained in [17,Section 5]. (1.5) The proof of Theorem 1.1 is given in Section 7. For the rst part we apply the direct method in Section 4 to show that the in mum of the elastic energy over curves with xed length is attained at a constrained elastica, i.e. a critical point of E + λL for some λ ∈ R. A detailed analysis of all possible constrained elastica in Section 2 and Section 3 shows (1.4). To show the second part we construct curves of large length with energy arbitrarily close to in Sections 5 and 6. Note that we can replace C ∞ in Theorem 1.1 by W , , as we show this in Theorem 4.3 for (1.4), and this replacement is trivial for assertion (1.5). As an application of Theorem 1.1 we can show the convergence of the elastic ow for initial values below the energy level of 16. From (1.1) it follows that elastic curves with elastic energy or less correspond to surfaces of revolution with Willmore energy π or less. Hence the convergence result can be matched to the energy bound given in [25], where the authors show long time existence and convergence of the L -Gradient ow of the Willmore energy provided that the initial surface is an immersion of S (i.e. has genus 1) and has Willmore energy below π. The content of this paper di ers from these results in two ways: Firstly, the Willmore ow of a surface of revolution and the H -elastic ow of the pro le curve di er by a factor, see [16,Theorem 4.1]. Secondly, since we consider the rotation of closed curves, the obtained surface of revolution is of di erent topology than S , namely it has genus .
As already mentioned, the energy threshold in Theorem 1.2 is optimal, which is discussed in the following theorem. Theorem 1.3 (Nonconvergence of the Unpenalized Elastic Flow). For all ε > there exist a smooth initial curve f with < E(f ) < + ε such that for its evolution f t by the unpenalized elastic ow we nd that L(f t ) is unbounded. In particular the solution does not converge as t → ∞.
The proofs are given in Section 7. As we mentioned before all circular evolutions converge, but their initial energy can be arbitrarily large, in particular larger than 16 (see [16,Lemma 3.1]), so evolutions of high initial energy do not necessarily have to be divergent. In Sections 5 and 6 we however identify a class of initial curves whose ow never converges, namely curves of vanishing total curvature. The reason that these evolutions cannot converge is that the total curvature is a ow invariant and there is no free elastica of vanishing total curvature, hence there is no critical point available to converge to. We shall discuss this ow invariant in Section 5.

Explicit Parametrization of Elastica in the Hyperbolic Plane
In the following, we shall give an explicit parametrization of elastica in the hyperbolic plane, which we will then use throughout the rest of the paper. We will adapt many concepts from [29] most of which we will state for the reader's convenience. Here, our manifold of interest is the hyperbolic plane H = R × ( , ∞) equipped with the usual metric tensor g(x, y) = y id (c.f. [15, Subsection 2.1]).

. The Elastica Equation
Let M be a smooth manifold. We denote by V(M) the set of all smooth vector elds on M. If (M, g) is a Riemannian manifold with Levi-Civita connection ∇ and c : I → M be an immersed curve with velocity vector c : I → M then we denote by T : I → M the unit tangential eld de ned by We de ne the curvature vector eld can also be seen as a vector eld in R along ι • γ = (γ , γ ). The formula for ∂xγ, see [15, (12)].
As H is di eomorphic to the upper half plane with an orthogonality-preserving di eomorphism, each smoothly immersed curve c : I → H has a smooth normal vector eld N along c such that g c(t) (T(t), N(t)) = and g c(t) (N(t), N(t)) = for each t ∈ I.
The vector eld N is unique up to a sign. Note in particular that {T(t), N(t)} forms an orthonormal basis of T c(t) H . We will now x a choice of N in the following Remark 2.2. The normal eld N becomes unique when we prescribe that e i π ι • T = ι • N, where ι : H → C is the canonical inclusion. We will do this in the following and write N = iT as shorthand notation, which makes actually sense when we look at the curve 'with Euclidean eyes'.
For a smooth immersed curve c : I → H one can not only de ne the curvature vector eld − → κ but also the scalar curvature κ : I → R to be the unique function such that − → κ = κN, see [38,Section 4.4]. Note that we sometimes write κ[c] to emphasize the dependency of the curve c, see e.g. Proposition 2.16.

De nition 2.3 (Elastic Energy
). Let (M, g) be a Riemannian manifold, λ ∈ R, and c : I → M be a smooth immersion. We de ne

De nition 2.4. We de ne
Remark 2.5. Note that E λ is well-de ned also on W , (S , H ) by using (2.1) to make sense of κ . However, note that the way we de ne it, there is no obvious metric on these sets without using a Nash embedding in the sense of [33,Theorem 3]. This de nition of the Sobolev space might appear strange at rst sight, but has the advantage that we can use the complex structure of R ∼ = C.
The critical points of E λ are called elastica and satisfy the following Euler-Lagrange equation (see [29, (1.3

)]).
De nition 2.6 (Elastica in H ). A curve γ ∈ C ∞ (( , L), H ) is called elastica or elastic curve in H if it is parametrized with hyperbolic arclength and satis es for some λ ∈ R. If λ = the curve is called free elastica, otherwise it is called λ-constrained elastica or just elastica.
and u := κ is a nonnegative solution of The elastica equation is solved explicitly in the following proposition, which is a major part of [36] and has been obtained before in [29]. A detailed proof is included in Appendix A.1 for the reader's convenience. [29,36]). Let λ ∈ R be given. Then, every nonnegative solution u = κ of (2.4) is global and attains a global maximum κ := sup x∈R u(x). Therefore, all nonnegative solutions of (2.4) are translations of solutions with the following initial conditions u( ) = κ and u ( ) = .
(2) (Orbitlike Elastica) κ ∈ (λ + , λ + ), C < and u(s) = κ dn (rs, p), where r = λ+ −p and p ∈ ( , ) is such that κ = λ+ −p , more explicitly p = (3) (Asymptotically Geodesic Elastica) κ = λ + , C = and u(s) = κ sech (rs), where r = We want to derive an explicit parametrization for elastic curves. For this, we have to prescribe initial data. In Proposition 2.8 we xed the curvature and its derivative at s = . Initial data for γ( ) and γ ( ) can be chosen in a computationally convenient way. This choice has to be made in a way that elastic curves with any initial data can be retrieved. One would hope that the retrieving process only involves isometries, since then the curvature changes only up to a sign. In R , Euclidean motions de ne isometries and for each p ∈ R and v ∈ TpR there exists a Euclidean motion Φ such that Φ(p) = ( , ) and Φ(v) = (|v|, ). This means that each elastic curve with initial value p and initial tangent vector v is(Euclidean) isometric to an elastic curve starting at the origin with a horizontal tangent line. We shall prove a similar result for H , inspired by [18]. For this note that Φ : H → H is an isometry of H if and only if there exist a, b, c, d ∈ R such that ad − bc = and where ι : H → C denotes the canonical inclusion. Lemma 2.9 (Reduction of the Initial Value Problem). Let z ∈ H and v ∈ TzH such that gz(v, v) = . Then for each y > there exists an isometry Φ of H such that ι(Φ(z)) = iy and d(ι Proof. We tacitly identify ι • Φ ≡ Φ and ι(z) ≡ z. We can without loss of generality assume that z = ir for some r > since we can compose with a translational Möbius transformation that translates z to the imaginary axis and leaves the di erential invariant. Note that Φ(w) = aw+b cw+d for some a, b, c, d ∈ R and we identify dΦw with Φ (w) via complex multiplication. We obtain that Φ is the desired Möbius transformation if and only if Note that condition (2) makes (4) redundant since gz(v, v) = implies that |v| C = r . Indeed, if ( ) is satis ed then whence ( ) holds true. Plugging ( ) into ( ) gives (cir+d) = y v . Note that (2) can easily be solved for c and d. Indeed, if √ · denotes some branch of the complex root we obtain that icr + d = y v and therefore c = r Im y v and d = Re y v . Using that Re(z) = equations ( ) and ( ) yield the following linear system which has a unique solution once c, d are known since Finally we have found a, b, c, d such that ( ), ( ), ( ), ( ) are satis ed. The claim follows.

. Killing Fields
Let (M, g) be a Riemannian manifold and X ∈ V(M). We de ne the ow map ϕ X of X the map that associates to a pair (t, p) ∈ R × M the value cp(t) ∈ M where cp is the unique maximal solution of Since it is unclear whether cp(t) exists for given (t, p) ∈ R × M, the domain of de nition need not be R × M. A vector eld J ∈ V(M) is called Killing eld for M if for each p ∈ M the cp is de ned on the whole of R and ϕ t := ϕ J (t, ·) : M → M is an isometry for each t ∈ R.
The reason that we introduce Killing elds is that they one can associate a Killing eld Jγ to each given elastica γ. Since however Killing elds in H can also be characterized explicitly one obtains a representation of Jγ with three parameters. This can be used to perform an order reduction of the (fourth order) elastica equation. Details will be given in the following Then Jγ has a unique extension to a Killing eld in V(H ), which we will denote by Jγ.
Remark 2.11. Since where C is the constant in (2.3), we see that Jγ ≡ implies that κ ≡ const. and this case is already covered by Proposition 2.15. We infer that in the cases of elastica with nonconstant curvature the Killing eld Jγ is not identically zero. For a given elastica γ, our goal is now to nd the parameters a, b, c that are associated to Jγ in the sense of Proposition 2.12. The rest of this section will be dedicated to the following order reduction result, which is a slight re nement of [19,Remark 4.6].
Remark 2.14. Recall that the prescribed initial datum in Proposition 2.13 does not restrict the generality of the classi cation, see Lemma 2.9. Using Möbius transformations might however change the parameters of the Killing eld, hence the order reduction is exclusively applicable for elastica with the given initial data.
For the proof of Proposition 2.13 it is crucial to examine the so-called characteristic integral curves of an elastica. These are de ned to be the solutions cz of (2.5), where X = Jγ and z ∈ H is a point of maximum curvature of γ. First observe by [ In particular we know that each characteristic integral curve cz must be of one of the three kinds.
To detect which of the three cases in the previous section applies to solutions cp one can use a useful criterion, going back to [9, p.81, Exercise 5b] and [18,Theorem 7.3]. It says that for each Killing eld J on a connected and geodesically complete Riemannian Manifold (M, g) that has a zero q one has that ϕ J preserves the geodesic distance to q, i.e. dist(q, p) = dist(q, ϕ J (t, p)) ∀t ∈ R ∀p ∈ M. (2.7) With all the information provided, we can compute the curvature of characteristic integral curves explicitly. One can view these characteristic integral curves as 'external circle' for the given elastica. If z is a point of maximum curvature of γ and cz is the characteristic integral curve of γ at z then for each t ∈ R such that γ(t ) = z, cz is tangential to γ at t = and if we choose the normal eld of cz such that the normal at z coincides with the normal of γ.
Proof of Proposition 2.13. Let γ be a non-circular elastic curve, i.e. κ is nonconstant. It follows from Proposition 2.12 and Lemma 2.10 that for some constants a, b, c ∈ R it holds The second line yields b = and the rst line yields (κ − λ)y = −ay + c. Recall from Remark 2.11 that a and c cannot be both zero at the same time. Consider the characteristic integral curve of γ at z = ( , y), which we will call c in the sequel. The curvature of c is by Proposition 2.16 given by κ[c ] ≡ κ κ −λ , in particular it is constant. If c is a line, then it must be parallel to the x-axis since c is tangential at the vertex γ( ) by Proposition 2.16 and therefore c ( ) || γ ( ) = (y, ). Hence c , (t) = for each t and this implies, using c (t) = Jγ(c (t)) and Proposition 2.12, that a = . On the other hand, if a = one can deduce from the equation and Proposition 2.12 that c (t) = Jγ(c (t)) = c( , ) T and therefore c is a line parallel to the x-axis. We infer that a = if and only if c is a line parallel to the x-axis. In this case however, each integral curve to Jγ is a line parallel to the x-axis, as the Killing equation implies. Since lines parallel to the x-axis have curvature of ± we obtain Therefore λ + C = . In particular we obtain that = ac = − (λ + C). In the remaining case c is not a line and we nd that a ≠ . According to Proposition 2.15 and since c cannot be a line, c must be part of a Euclidean circle through ( , y), which we can reparametrize the Euclidean way: where m ∈ R and r > are such that m + r = y. A short computation shows Additionally, there exists a di eomorphism ϕ ∈ C (R; R) such that c (t) = c (ϕ(t)). From [19, Theorem 2.11, Remark 2.8] we infer g( Jγ(c (t)), Jγ(c (t))) = const. Plugging in t = ϕ − ( ) we obtain from Lemma 2.10 We can compute this quantity in a di erent way, namely using Proposition 2.12 Plugging in x = r cos(t) and y = m + r sin(t) and using cos (t) − sin (t) = cos( t) and sin (t) = −cos( t) we nd together with (2.12) Multiplying with the denominator and using once again that sin (t) = −cos( t) we obtain Because of the identity theorem for holomorphic functions, the above identity holds true for any t ∈ C. Using linear independence of trigonometric polynomials to compare coe cients we nd (2. 13) In case that m ≠ , dividing the second equation by m and summing with the rst we nd (2.14) Plugging this into the third identity in (2.13) we nd Completing the square and factoring out the brackets we obtain (ac − κ ) = (κ −λ) , and eventually We will continue showing that the case '−' always applies. Suppose that '+' is true for some elastica γ. Then ac > which implies that Jγ has a zero on the y-axis since Thus, (2.7) implies that the characteristic integral curve cannot reach the x-axis. In particular we obtain that m > . Note also that Computing the absolute value of the Killing eld at γ( ) = c ( π ) = ( , m + r) T we obtain that Using (2.14) we nd The last case to consider is m = , but this would imply together with (2.11) that κ = . Therefore, there exists some t such that κ(t ) = κ (t ) = . Using that κ = − κ − λ+ κ we nd that also κ (t ) = and bootstrapping we nd that every derivative of κ attains the value at t . Since all the solutions for u = κ extend to a holomorphic function on an open neighborhood of the real line, we infer that κ ≡ , contradicting the non-circularity.
Having eliminated the parameter b and expressed a, c with quantities in Proposition 2.8 we can now use the quantities to classify elastica by their Killing elds.
De nition 2.17 (Classi cation of Elastica by their Killing Field). Let γ : I → H be an elastic curve parametrized with hyperbolic arclength and the same initial data as in Proposition 2.13. If the extended Killing eld of γ is given by we say that γ has a rotational Killing eld (or simply γ is rotational) if ac > , a translational Killing eld if ac < and a horocyclical Killing eld if ac = .
Remark 2.18. By now we have introduced some parameters that describe elastic curves, which we will use in the following. For the sake of the reader's convenience we include in Table 1 the references that will be missing. We always consider λ ∈ R, κ ∈ R, y > to be the 'original' parameters from which we compute the following new parameters. We also include those which will be de ned later. We conclude this section with some useful facts about elastica with rotational Killing eld which will turn out to be the most relevant for the proof of (1.4).  Proof. If γ has a rotational Killing eld the integral curve c starting at γ( ) satis es Therefore λ > κ − |κ |, and adding to both sides we get

. Explicit Parametrization
So far, we have reduced the elastica equation to a rst order system, see Proposition 2.13. To get an explicit parametrization we exploit the structure of this system further: Remarkably, the system becomes separable if we rewrite it as an equation in C. The reason is, that the Killing elds come from isometries in H , all of which are also isometries in the Riemann sphere CP . This being a Riemann surface, the Killing elds should have some holomorphic structure. The elastica equation has been examined in a more general setting using this structure in [23], where the author provides explicit parametrizations of elastica in arbitrary space forms. Unfortunately these parametrizations are not very useful examining the limiting cases, as we will do in the later sections. A slight disadvantage of our approach is that we can only parametrize globally de ned elastic curves. Since our main focus lies on closed elastic curves, this is not restrictive for our application. From now on, we will assume unless not explicitly stated otherwise, that γ ∈ C ∞ (R, H ) is a globally de ned smooth immersed elastic curve. The proof is a laborious extension of the same conceptual avor of the next proof. For readability we postpone the proof to Appendix A.2.

Theorem 2.22 (An Explicit Parametrization of Elastica in H )
. Let γ : R → H be an elastic curve parametrized by hyperbolic arclength with curvature κ so that for y > . Then either κ ≡ const. or there exist a, c ∈ R with |a| + |c| ≠ determined uniquely by ac = − (λ + C), Moreover, there exists z ∈ iR ≠ such that γ is parametrized by 18) and the meromorphic function f is given by (2.19) Proof. Rewrite (2.6) as equation of complex numbers to obtain where a, c are as in the statement. With the relations in the statement it is an easy computation that y determines a, c uniquely. Now parametrization by arclength implies that T = γ . This immediately implies (2.17) using Proposition 2.21. Note that (2.17) is a rst order ODE with locally Lipschitz right hand side, so its maximal solution is unique when we specify γ( ) = iy. Let now y > be given. We verify now that (2.18) indeed yields a solution of (2.17) with γ( ) = iy. For this we need to distinguish several cases depending on the value of a, c, which depend on y according to the paramter identities in the statement. Case 1 a, c > . We take f as given in (2.19) and di erentiate the expression (2.18) yields for γ. De ning z := √ acz we compute Hence, γ indeed solves the equation. It remains to show that z [= z √ ac ] ∈ C can be chosen such that To nd such z we need to invert the tanh in the expression. For this we rst observe that y < c a , which can easily be obtained by the parameter identities in the statement and the fact that by Proposition 2.8 (2.21). Case 2, a, c < , works similarly. In this case one can observe that y > c a . Case 3, ac < . De ning z := Note that − c a tanh(z ) = iy can be solved using that tanh(z ) = i tan(−iz ). In the end we obtain z = i arctan √ ay √ −c . The other cases can be solved analogously. Remark 2.23. Note that not every curve given by (2.18) for some κ from Proposition 2.8 is an elastica. The reason for that is that any such curves are not necessarily parametrized by hyperbolic arclength. We shall see counterexamples in Appendix A.2. This makes the analysis more complicated.

Closing and Simplicity Conditions
In this section we want to investigate whether the elastic curves parametrized in Theorem 2.22 are (smoothly) closed, i.e. whether there is some L > such that γ( ) = γ(L) and all derivatives of γ coincide at and L. Since we consider elastica parametrized by hyperbolic arclength the smallest such L is given by the hyperbolic arclength of γ. The other property of our interest will be simplicity, i.e. whether the curve has no self-intersections in ( , L). The following propositions will reveal why we are interested in these properties: They are related to the energy and to the number of periods the curvature completes in one period of the curve and will be useful.
Since the curvature of γ also periodic, we may denote with n the number of periods the curvature completes within [ , L], i.e. n is given by The following closing conditions are closely related to the conditions in [36]. We present a self-contained proof for the reader's convenience. First note that there are no closed asymptotically geodesic elastica as the curvature of such is not periodic (cf. Proposition 2.8). For wavelike and orbitlike elastica we can derive closing conditions also employing their Killing elds.
for some m ∈ Z and where p, r are given in Proposition 2.8, n ∈ N is given in (3.1) and K denotes the complete elliptic integral of rst kind, see Appendix B. Moreover, if m = then n = . In case that |m| > and n > , |m| and n are relatively prime.
Proof. Having a rotational Killing eld implies that ac > and therefore −ac = (λ + C) < by Theorem 2.22. Thus, C < and therefore γ is orbitlike. Since κ( ) = κ(L) = κ the formula for L follows from Proposition 2.8 and Proposition B.4 (4). An easy computation shows that for two complex numbers z, w ∈ C, tan(z) = tan(w) holds if and only if z = w + mπ for some m ∈ Z. The same periodicity holds true for the complex cotangent function. Therefore, in the relevant rst two cases of (2.19), the function f in Theorem 2.22 is π √ ac -periodic.
Using Theorem 2.22, we obtain another necessary condition, namely for some m ∈ N. Rearranging we obtain Substituting u = κ(s) and using κ( ) = κ(L) we nd that the last integral is zero. This substitution is justi ed when we use that λ + C + κ > because of Proposition 2.21. We obtain that (3.2) and (3.3) are necessary for the closedness of γ, and their su ciency follows from Theorem 2.22. Note that in the case of m = , the and therefore Due to the su ciency of (3.2) and (3.3), γ closes smoothly after K(p) r . Since n has to be minimal, we infer that n = . Now assume that |m| and n are larger than 1. If they had a common prime factor q > then and dividing by q we obtain Since m q and n q are integers, the su ciency of (3.2) and (3.3) again implies that γ already closes up smoothly after n q periods, contradicting the minimality of n. It follows that gcd(|m|, n) = . Additionally, γ is not simple. Furthermore, γ can not be free, i.e. λ = is not allowed.
as a computation similar to (3.5) reveals. Note now that κ = κ cn (rs, p) is K(p) r -periodic. Therefore We conclude that γ( K(p) r ) = γ( ) and therefore γ has a self-intersection. Hence, γ is not simple. To show that γ is not free we look at (3.6) with λ = . In order for this equation to be satis ed, one would need κ ≡ , which is not possible. Additionally, γ is not simple.
Proof. The claim that n = can be shown following the lines of the corresponding part of the proof of Proposition 3.4, more precisely one proceeds similar to (3.7) and (3.8) with the minor di erence of the periodicity of cn instead of dn, see Proposition B.5. If γ were simple, Proposition 3.2 would imply that n ≥ which is a contradiction.
Proposition 3.6 (Closed orbitlike elastica). Let γ be an orbitlike elastica with parameter λ < π − ≈ . . If γ is closed, then γ is rotational and satis es m ≠ . Proof. If γ is not rotational or m = , then either Proposition 3.5 or Proposition 3.3 imply that n = . Therefore γ can not be simple since otherwise Proposition 3.2 would be violated. Notice that E(γ) ≥ according to Proposition 3.1. Then, by Proposition 2.8 and (B.1) we nd where we used Proposition B.5 in the last step. Solving the inequality for λ we obtain that λ ≥ π − .
Remark 3.7. A close examination of the proof of Proposition 3.6 reveals that we have actually shown a stronger result: Orbitlike elastica with n = exist only for λ ≥ π − .

The Ratio of Energy and Length
In this section we show the announced Reilly type inequality (1.4). This inequality will account for the fact that for each evolution (f t ) t≥ by elastic ow with E(f ) < , the hyperbolic length (L(f t )) t≥ is uniformly bounded in time. This already implies that (f t ) t≥ is a convergent evolution as we shall see in Section 7.   Proof. Fix ε > . Note that according to Proposition 3.4 any wavelike elastica is nonsimple and therefore E(γ) ≤ − ε < can only hold for simple orbitlike or for circular elastica, see Proposition 3.1. In the case of a circular elastica, i.e. κ ≡ const we obtain that γ is a circle in H since according to Proposition 2.15 this is the only closed curve with constant curvature. However then |κ[γ]| > and this implies In the case of a simple orbitlike elastica with n ≥ (see Proposition 3.2) we obtain using Proposition 2.8 and (B.1) where we used in the last step that λ ≥ − . This is true since according to Proposition 3.5 γ has to be rotational which allows us to apply the estimate in Proposition 2.20. Now assume that the statement is false. Then there exists a sequence (λ l , C l ) l∈N such that for each l ∈ N there is an orbitlike elastica γ l with parameters λ l , C l satisfying E(γ l ) ≤ − ε and E(γ l ) L(γ l ) ≤ l . According to Lemma 4.1 implies that − ε ≥ λ l + , and hence (λ l ) l∈N de nes a bounded sequence. Since the sequence (C l ) l∈N is bounded as well. Hence, there is a subsequence (k l ) ⊂ N and (λ, C) such that λ k l → λ and C k l → C. Passing to the limit in (4.3) we obtain + λ + ( + λ) + C = ( + λ) + C, which implies that C = . As we discussed above, γ l is rotational and hence λ l + C l < . Consequently, ≥ lim l→∞ (λ k l + C k l ) = λ and thus λ = . But then Lemma 4.1 yields the contradiction The following theorem is a precise formulation of Theorem 1.1 in the space W , (S , H ).

Theorem 4.3 (A Reilly-Type Inequality). For each ε > there exists
Furthermore, c(ε) can be chosen as in Lemma 4.2.

E(γ)
L(γ) is attained by an elastica. Clearly, since γ ∈ A(γ) ≠ ∅ we have inf A(γ) E(·) ≤ E(γ) ≤ − ε. Note that since the length is kept xed L is constant and it su ces to show that inf A(γ) E(γ) is attained by an elastica. To prove this, we proceed in three steps.
Step 2: Any minimizer in A(γ) is a critical point of E + λL for some λ ∈ R. For this we use [40,Proposition 43.21]  for some λ ∈ R one has to show that G is a Frechét di erentiable submersion (i.e. G (γ) is surjective for all γ ∈ M) and F is Frechét di erentiable on M. We only sketch the proof of the submersion property: It is standard to show that each critical point of G in M satis es κ ≡ . However, in M there exist no closed curves with κ ≡ since geodesics in H are never closed. This implies that G is a submersion on M.
Step 3: It follows for instance from [17, Section 5] (with the same function spaces used in Step 2) that all solutions of (4.10) are smooth and their arclength reparametrizations satisfy the (possibly constained) elastica equation.
To conclude the proof we use Lemma 4.2 to obtain that since γ * is an elastica satisfying E(γ * ) ≤ − ε. Since γ was arbitrary, the claim follows.

A Flow Invariant
In this section we describe the possible limit behavior of the ow by computing the Euclidean total curvature T[γ] := π γ κ R ds for closed elastic curves γ, more precisely for ι • γ, where ι : H → C is the canonical embedding into the upper half plane. Notice once more that κ R denotes the R -curvature of γ and not the hyperbolic curvature of γ and ds denotes (only in this section) the Euclidean arclength parameter. The explicit parametrization given in Theorem 2.22 allows us to look at the curves 'with Euclidean eyes' and therefore to compute T[γ]. The total curvature is such an important quantity since -as it will turn out -for subconvergent evolutions by elastic ow (1.3) in H , the initial curve and the limit curve will have the same total curvature (at least, provided that the initial curve is smooth, see [15,Theorem 1.1]). Therefore, the total curvature allows us to classify subconvergent evolutions by elastic ow and to exclude their existence for certain initial data. Indeed, we will show in this section that there cannot be a subconvergent evolution with initial data of vanishing total curvature.
De nition 5.1 (Regular Homotopy). Let Imm(S , R ) be the set of all immersed curves in C (S , R ). Together with the relative topology of C (S , R ), it becomes a topological space. We say that two curves γ , γ ∈ Imm(S , R ) are regularly homotopic, if they lie in the same path-component of Imm(S , R ). A path in Imm(S , R ) is called a regular homotopy.
Remark 5.2. Let γ ∈ C ∞ (S , H ) be immersed and let (γ t ) t≥ be the evolution of γ under the elastic ow (see Theorem 7.1) in H . Note that for each t ≥ , the canonical Euclidean inclusions of γ and γ t are regularly homotopic in Imm(S , R ), since H is di eomorphic to R . Here we also used that the ow is su ciently smooth, see [16, Theorem 1.1].

Proof. The fact that T[c]
is an integer and that T is continuous with respect to the C (S , R )-topology follows for instance from [21,Theorem 6.11]. The remaining direction is known as the Whitney-Graustein Theorem and proved in [39].
Remark 5.4. The previous proposition actually shows that T de nes a ow invariant for all ows that de ne regular homotopies in Imm(S , R ).
Plugging into the formula for κ R we nd that on [ , L] We obtain The di erential equation in Theorem 2.22 reads θ(s)γ (s) = aγ(s) +c and Proposition 2.21 implies that aγ(s) + c ≠ for all s. Therefore Proof. Using the notation from Proposition 5.5 we rst show that θ z dz = . Recall that z → z has a complex antiderivative on the simply-connected domain C \ R ≤ . We shall show that θ([ , L]) ⊂ C \ R ≤ . Indeed, if Im(θ(s)) = then κ (s) = . Using that κ(s) = κ cn(rs, p) for some r, p this happens only if κ(s) = ±κ . Furthermore, we nd using Proposition 2.8 that Re(θ(s)) = κ(s) − λ = κ − λ ≥ > , which implies that θ(s) ∉ C \ R ≤ . It remains to show that γ az az +c dz = , but this is clear since γ lies entirely in the upper half plane and the roots of the integrand are both on the real axis, remember ac < since C > and ac = − (λ + C), see Proposition 2.13 and Proposition 2.8. The claim follows using Cauchy's Integral Theorem. Proof. We show rst that Recall that a parametrization of θ is given by θ(s) = κ (s) − λ + iκ . We compute using the elastica equation Now κ(s) = ±κ dn(rs, p), where the choice of sign has to be consistent again because of smoothness. We only treat the case '+' here but the other case can be shown similarly. The rst and second derivatives can be simpli ed as follows using κ = r according to the second case in Proposition 2.8, and Proposition B.4: κ (s) = −κ rp sn(rs, p)cn(rs, p) = − r p cos(am(rs, p)) sin(am(rs, p)) = −r p sin( am(rs, p)), κ (s) = −κ r p (cn (rs, p)dn(rs, p) − sn (rs, p)dn(rs, p)) = −κ r p dn(rs, p) cos( am(rs, p)) = −κ(s)r p cos( am(rs, p)).
It becomes obvious that θ is an n-fold cover of ∂B r p ( ). Therefore We write ± since it is not important for our result in which direction the circle is parametrized. Indeed, if we had treated the '−' case in detail, the circle would be parametrized in the opposite direction. Lemma A.3 shows that r p < if and only if κ < + λ. Also, Remark A.9 shows that r p = and κ = + λ can never occur, so the classi cation is indeed complete.
For the rest note that z → az az +c is a logarithmic derivative and therefore all the residues coincide with the orders of the roots of z → az + c. However, since ac > , all poles have order 1. Therefore where ω(γ, ·) denotes the winding number of γ and √ · denotes one branch of the complex square root. Note that exactly one of −c a and − −c a lies in H . Therefore one of these winding number is zero. Let us assume that ω(γ, − −c a ) = . We look to determine ω(γ, − c a ). On the one hand The following Corollary gives a su cient condition for the initial value ensuring the non-convergence of the elastic ow. A natural question is then to nd the minimal energy level on which such phenomena occur. In Corollary 6.4 we present smooth curves γε with energy below + ε satisfying T[γε] = , ε > . Corollary 5.9 (A Class of Bad Initial Data). Let γ be a smoothly closed curve such that T[γ ] = . Let (γ t ) t≥ be the time evolution of the elastic ow with initial value γ . Then (L(γ t )) t≥ is unbounded. In particular (γ t ) t≥ is a nonconvergent evolution.
Proof. Assume on the contrary that L(γ t ) is bounded. Then there is a free elastic curve γ∞ and tn → ∞ such that the constant-hyperbolic-speed reparametrizations of (an(γ tn − (pn , ))) n∈N converge to γ∞ in W m, (S , R ) for each m ∈ N and appropriately chosen an , pn, see Theorem 7.1. Therefore Proposition 5.3 yields that The existence of such γ∞ however would contradict Corollary 5.8.

Optimality of the Energy Bound
So far, we have shown that the length along the elastic ow remains bounded, provided that the initial datum γ ∈ C ∞ (S , H ) has small elastic energy, more precisely E(γ ) < , see Theorem 4.3. Additionally we have constructed a class of initial data for which the length along the ow is unbounded, namely the class of curves of vanishing Euclidean total curvature. To investigate optimality of the bound of , we look for curves of small energy with vanishing total curvature.
Proof. Fix some λ ∈ ( , π − ). Take an arbitrary curve σ ∈ C ∞ (S , H ) such that T[σ] = and consider the ow for E λ with initial datum σ. Applying [15, Theorem 1.1] we nd that the ow exists and subconverges to an elastic curve γ that satis es (2.2) with λ = λ . This elastic curve has to satisfy T[γ] = (see (5.3)). We now claim that γ cannot be circular or orbitlike, since circular and orbitlike elastic curves with λ < π − have nonvanishing total curvature. Indeed, for circular elastica one can easily compute the total curvature of an kfold cover of a circle, which is exactly k, so nonzero. Now suppose γ is an orbitlike elastica. Since λ < π − , γ is rotational with m ≠ by Proposition 3.6. Then there are two cases to distinguish: if κ < + λ then Corollary 5.7 yields the contradiction = T[γ] = m. If κ > + λ, then according to Corollary 5.7, T[γ] = m ± n, which can be zero only in the case m = n = since m, n are relatively prime otherwise, see Proposition 3.3. However n = is a contradiction to Remark 3.7. Hence, γ must be wavelike which completes the proof.
We now derive a modi ed closing condition for wavelike elastic curves that is more stable to compute for small λ. This has the advantage that the new condition eliminates parameters that can hypothetically become large for small λ and therefore lead to numerical di culties. Proof. If γ is closed, we nd using Proposition 3.4 that Proof. Let (λn) n∈N be a sequence of positive numbers smaller than and converging to zero. Denote by γn a λn-gure eight constructed in Proposition 6.2 and let Cn , pn , rn be its canonical parameters. We show that pn → . Indeed, assume that there is a subsequence, which we will denote again by (pn) which converges to some other p ∈ [ √ , ). We rst show that (κ (n) ) n∈N (which denotes the maximum curvature of γn) is bounded.
Indeed, if there were a subsequence (again denoted by (κ n ) n∈N ) that converges to ∞, then (κ (n) ) ((κ (n) ) −λn) would converge to zero. We can plug all the asymptotics in (6.1) to obtain the contradiction = lim n→∞ π cos (θ) − λn because the denominator can be uniformly bounded and the convergence of all quantities is uniform. Therefore κ (n) remains bounded. In particular, since (κ (n) ) = p n λn+ p n − , it must hold that p ≠ √ . We can also show by a similar contradiction argument, again using (6.1), that given p ≠ , (κ (n) ) ((κ (n) ) −λn) must tend to as n → ∞.

. Behavior at the Critical Energy Level
We have discussed what happens if we start the ow with curves of energy below and we have also identi ed phenomena that occur for curves of energy just slightly above . The only energy level that remains to be understood is the energy level of exactly . Here we distinguish two cases: If the elastic ow (f t ) t≥ does not start at an elastic curve, the energy will instantaneously decrease from the energy level of to an energy level below, as in this case This being so, we can bound (L(f t )) t≥ by restarting the ow at a positive time where we reach an energy level below . If the ow starts at an elastic curve of energy , the ow will not change the curve at all. Hence (L(f t )) remains bounded in any case, which is -as we discussed -su cient for the convergence. In this section we rule out the latter case by showing that there exists no closed free elastica of energy equal to . We show even more: The only closed free elastica of energy less or equal to is -up to reparametrization and isometries -the Cli ord elastica. This leaves it as the only possible limit curve for evolutions with small energy.
According to [29, Proof of Proposition 5.3, p.21] the expression in parentheses is always strictly larger than π.
Since n ≥ this leads to the desired contradiction. Now suppose that E(γ) = . Again because of Proposition 3.4 and Proposition 2.8, γ is either orbitlike or circular. Suppose now that γ is orbitlike. Similar to the Proof of Lemma 4.1 one computes using λ = that = E(γ) = n E(p) However, according to Proposition B.5, the number on the left hand side is stricly between √ π ≈ . and , and hence cannot be natural. We conclude that γ has to be circular, i.e. κ[γ] ≡ const. Corollary 6.6. Let γ be a closed free elastica with E(γ) ≤ . Then γ is the Cli ord elastica (1.2) up to translation, rescaling and reparametrization.

Proof of the Main Results
In this section we show the proofs of the main results. We start with the fundamental result of [15] that settles question of long time existence and identi es the uniform-in-time boundedness of the hyperbolic length as su cient for the convergence  shows that any bound on the length is su cient for the subconvergence.
With a Lojasiewicz-Simon gradient inequality we can actually improve the subconvergence to convergence: If the elastic ow f subconverges to an elastic curve f∞ in the sense of Remark 7.2 (1), then it converges smoothly to f∞.
Since a proof of this result is beyond the scope of this article we only give a sketch here and refer the reader to [14] for details Let δ > and consider a smooth curve γ such that < E(γ ) ≤ + δ and T[γ ] = , whose existence is provided by Corollary 6.4. From Theorem 7.1 we obtain the evolution (γ t ) t≥ of γ by the elastic ow with λ = . Then E(γ t ) ≤ + δ but according to Corollary 5.9 we have L(γ t ) → ∞, at least up to a subsequence. This subsequence produces arbitrarily small values of E L . Proof of Theorem 1.2. Let f be a smooth immersion with E(f ) ≤ . First, we assume that δ : all t by (1.4). Hence, by Theorem 7.1 and Remark 7.3, the ow converges in the sense of Remark 7.2 (1) to some free elastica with energy below . In Corollary 6.6 we show that the only free elastica with energy below 16 is the Cli ord Elastica, which nishes the proof in this case. If E(f ) = , then f is not elastic by Corollary 6.6, thus E(f t ) < for all t > by (7.2), from which we can deduce the claim as above.
Similarly to the proof of (1.5) we show Theorem 1.3.
Proof of Theorem 1.3. Theorem 1.3 is immediate from Corollary 5.9 and Corollary 6.4.

A. Proof of Proposition 2.8
Proof of Proposition 2.8. Remember that u = κ ≥ . Therefore we aim to classify nonnegative solutions of u + u − ( λ + )u − Cu = , see (2.4). This equation is of the form u = P(u) for the polynomial P given by Note that α, β, γ have to be real-valued since otherwise P(u) can only have one real root, which is zero. However then P | ( ,∞) is negative, which contradicts the existence of positive real-valued solutions of u = P(u). Note also that one root of P has to be strictly positive for the very same reason. From now on we adhere to the convention α ≤ β ≤ γ as in [8]. Note that α ≠ γ because otherwise α = β = γ = and the equation reads (u ) = −u . This however has no nonnegative solution except for the trivial one. Observe also that β ≤ u(s) ≤ γ for all s, since otherwise nonnegativity is violated again (since α ≤ ). In particular we nd β ≠ γ. We distinguish between two cases: α ≠ β and α = β. Note that Conversely, note that if there exists a solution u with C = then λ + ≥ since otherwise all roots are nonpositive and u = P(u) cannot be true. Therefore α = λ + − (λ + ) + C = = β. As a conclusion, α = β holds if and only if C = .

A. Proof of Proposition 2.21
Let γ be a globally de ned elastic curve parametrized by hyperbolic arclength. We will need several lemmas to prove the claim. Recall from the proof of Theorem 2.22 (see (2.20) and use T = γ ) that we have a di erential equation for γ in C, namely θ(s)γ (s) = aγ(s) + c for s ∈ R, where θ(s) := κ (s) − λ + iκ . We can not divide by θ a priori and hence the Picard-Lindelöf Theorem is not applicable. Recall also that by (2.15) J γ (z) = az + c for all z ∈ C such that Im(z) > . (A.2) If the Killing eld has a zero in H , then one can infer from (A.2) that ac > . Therefore γ is rotational and hence orbitlike, see De nition 2.17 and Proposition 3.3.
Since γ is an immersion, the following lemma is immediate. Proof. As we discussed in the introduction of this subsection, θ can vanish only provided that the Killing eld has a zero in H , which implies that γ is rotational and orbitlike, see the arguments in the aforementioned introduction. Observe that for each orbitlike elastica γ it holds that κ ≥ κ ( − p ), see Proposition 2. We infer that all inequalities in the above chain have to be equalities. From this follows that κ(t ) = κ ( − p ) which is the minimum possible curvature (see De nition B.2) and parameter identity no. ( ) using that equality holds in the last step. For parameter identity no. (1) observe using 2.8 that Proof. Observe that parameter identity ( ) in Lemma A.3 implies λ + C = − λ. The rest is a short computation using (2.3): Lemma A.6 (Explicit Parametrization near s = ). Let γ be a globally de ned elastic curve with θ vanishing somewhere. Then there exists z ∈ C \ R such that where x(t) = tan √ λ t + z and y(t) = tanh log κ+ Proof. First note that γ( ) ≠ i c a since θ( ) ≠ , see Lemma A.1. Therefore we can use similar arguments as in the proof of Theorem . to obtain that in a neighborhood of t = for some z ∈ C such that γ( ) = c a tan(z ). Such a z exists since γ( ) ∈ iR \ i c a and tan is surjective on C\{i, − }. Observe also that z ∉ R since otherwise γ( ) ∈ R, a contradiction. Since Proof of Proposition 2.21. Assume that there exists a globally de ned curve γ such that a zero of Jγ lies in γ(R) and γ is parametrized with hyperbolic arclength. Therefore, if we look at γ as a curve in C it satis es where x = tan(z + √ λpK(p)) and z is chosen as in Lemma A.6. We infer from (A.4) and (A.5) that | − i x| = | + i x|. Squaring both sides and using |z + w| = |z| + |w| + Re(zw) we infer that Re(i x) = and therefore x ∈ R. We proceed showing that this cannot be true. We distinguish between 3 cases. Case 1: √ λK(p)p = π + lπ for some l ∈ Z. In this case x = tan(z + π / ) = − cot(z ). Assume that cot(z ) = β ∈ R. An easy computation shows that Taking absolute values on both sides we nd e Re(iz ) = |e iz | = iβ− iβ+ = which implies z ∈ R and contradicts the statement of Lemma A.6. Notice that we used here that tan( √ λK(p)p) x ≠ . Observe that the right hand side of (A.6) is real-valued by assumption. With similar arguments as in case 1, it can be shown that tan(z ) ∉ R if z ∉ R. Again, this leads to a contradiction to Lemma A.6.