The gradient flow of O’Hara’s knot energies

Jun O’Hara invented a family of knot energies Ej,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E^{j,p}$$\end{document}, j,p∈(0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j,p \in (0, \infty )$$\end{document}, O’Hara in Topology Hawaii (Honolulu, HI, 1990). World Science Publication, River Edge 1992. We study the negative gradient flow of the sum of one of the energies Eα=Eα,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E^\alpha = E^{\alpha ,1}$$\end{document}, α∈(2,3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (2,3)$$\end{document}, and a positive multiple of the length. Showing that the gradients of these knot energies can be written as the normal part of a quasilinear operator, we derive short time existence results for these flows. We then prove long time existence and convergence to critical points.


Introduction
Is there an optimal way to tie a knot in Euclidean space? And if so, how nice are these optimal shapes? Is there a natural way to transform a given knot into this optimal shape?
To give a precise meaning to such questions a variety of energies for immersions have been invented and studied during the last twenty five years which are subsumed under the term knot energies.
In this article we deal with the third of the questions above. But first of all, let us gather some known answers to the first two questions.
The first family of geometric knot energies goes back to O'Hara. In [23], O'Hara suggested for j, p ∈ (0, ∞) the energy |c (x)| · |c (y)|dxdy of a regular closed curve c ∈ C 0,1 (R/Z, R n ). Here, d c (x, y) denotes the distance of the points x and y along the curve c, i.e., the length of the shorter arc connecting these two points.
O'Hara observed that these energies are knot energies if and only if j p > 2 [23,Theorem 1.9] in the sense that then both pull-tight of a knot and selfintersections are punished. Furthermore, he showed that minimizers of the energies exist within every knot class if j p > 2. So: Yes, there is an optimal way to tie a knot-actually even several ways to do so. Abrams et al. proved in [1] that for p ≥ 1 and j p − 1 < 2 p these energies are minimized by circles and that these energies are infinite for every closed regular curve if j p − 1 ≥ 2 p. 1 There is a reason why for the rest of our questions we will only consider the case p = 1: For p = 1, we expect that the first variation of E j, p leads to a degenerate elliptic operator of fractional order-even after breaking the symmetry of the equation coming from the invariance under re-parameterizations. We will only consider the nondegenerate case p = 1 and look at the one-parameter family E α (c) := E α,1 (c) = (R/Z) 2 1 |c(x) − c(y)| α − 1 d c (x, y) α |c (x)| |c (y)|dxdy. (1.1) We leave the case p = 1 for a later study.
The most prominent member of this family is E 2 which is also known as Möbius energy due to the fact that it is invariant under Möbius transformations [14, Theorem 2.1]. While for α ∈ (2, 3) the Euler-Lagrange equation is a non-degenerate elliptic sub-critical equation it is a critical equation for the case of the Möbius energy E 2 .
In [14], Freedman, He, and Wang showed that even E 2 can be minimized within every prime knot class. Whether or not the same is true for composite knot classes is an open problem, though there are some numerical experiments that indicate that this might not be the case in every such knot class [19]. Furthermore, they derived a formula for the L 2 -gradient of the Möbius energy [14,Equation 6.12] which was extended by Reiter [24,Theorem 1.45] to the energies E α for α ∈ [2,3). They showed that the first variation of these functionals can be given by Using the Möbius invariance of E 2 , Freedman, He, and Wang showed that local minimizers of the Möbius energy are of class C 1,1 [14]-and thus gave a first answer to the question about the niceness of the optimal shapes. Zheng-Xu He combined this with a sophisticated bootstrapping argument to find that minimizers of the Möbius energy are of class C ∞ [17]. Reiter could prove that critical points c of E α , α ∈ (2, 3) with κ ∈ L α are smooth embedded curves [24], a result we extended to critical points of finite energy in [8] and to the Möbius energy in [9].
Let us now turn to the central theme of this article, the last of the three questions we started this article with: Is there a natural way to transform a given knot into its optimal shape? Since E α is not scaling invariant for α ∈ (2, 3), the L 2 -gradient flow of E α alone cannot have a nice asymptotic behavior. We want to avoid that the curve would get larger and larger in order to decrease the energy. Here, the length L(c) of the curve c will help us.
To transform a given knotted curve into a nice representative, we will look at the L 2 -gradient flow of E = E α + λL for α ∈ (2, 3) and λ > 0 a fixed constant. This leads to the evolution equation (1.4) Another Ansatz for a flow that might transform a given curve into a critical point is to let λ depend on time in such a way that the length of the curve stays fixed during the flow. This leads to the evolution equation where λ(t) is given by Unfortunately, the extra term is much harder to control than in the case for constant λ due to its supercritical nature. We will not discuss this evolution equation further and leave it for a later study.
We will see that the right hand side of Eq. (1.4) can be written as the normal part of a quasilinear elliptic but non-local operator of order α + 1 ∈ [3,4).
The main result of this article is the following theorem. Roughly speaking, it tells us that, given an initial regular embedded curve of class C ∞ , there exists a unique solution to the above evolution equations. This solution is immortal and converges to a critical point. More precisely we have: to (1.4) with initial data c(0) = c 0 that after suitable re-parameterizations converges smoothly to a critical point of E α + λL.
Note that we do not expect the above theorem to hold for the gradient flow of the Möbius energy E 2 . Due to the criticality of the equation one gets in the case of the Möbius energy the long time behavior of this equation should be much richer. Numerical experiments yield to the conjecture that for example the gradient flow of the Möbius energy can develop singularities for certain initial data. It would make even less sense to consider the negative gradient flow of E 2 + λL for a positive constant λ. As E 2 is scaling invariant the length term would force the flow to produce singularities in finite time.
Lin and Schwetlick showed similar results for the elastic energy plus some positive multiple of the Möbius energy and the length [20]. They succeeded in treating the term in the L 2 -gradient coming from the Möbius energy as a lower order perturbation of the gradient of the elastic energy of curves. This allowed them to carry over the analysis due to Dziuk, Kuwert and Schätzle of the latter flow [12]. They proved long time existence for their flow and sub-convergence to a critical point up to re-parameterizations and translations.
The situation is quite different in the case we treat in this article. We have to understand the gradient of O'Hara's energies in a much more detailed way and have to use sharper estimates than in the work of Lin and Schwetlick. Furthermore, in contrast to Lin and Schwetlick we show that the complete flow, without going to a subsequence and applying suitable translations, converges to a critical point of our energy.
We proceed with an outline of the proof of Theorem 1.1 and thus with an outline of the paper. In Sect. 2, we prove short time existence results of this flow for initial data in little Hölder spaces and smooth dependence on the initial data. To do that, we show that the gradient of E α is the normal part of an abstract quasilinear differential operator of fractional order (cf. Theorem 2.3). Combining Banach's fixed-point theorem with a maximal regularity result for the linearized equation, we get existence for a short amount of time. In order to keep this article as easily accessible as possible, we give a detailed proof of the necessary maximal regularity result.
The most important ingredient to the proof of long time existence in Sect. 3.4 is a strengthening of the classification of curves of finite energy E α in [5] using fractional Sobolev spaces. For s ∈ (0, 1), p ∈ [1, ∞) and k ∈ N 0 the space W k+s, For a thorough discussion of the subject of fractional Sobolev space we point the reader to the monograph of Triebel [26]. Chapter 7 of [2] and the very nicely written and easy accessible introduction to the subject [13].
where κ denotes the curvature of the curve c and ∂ s := ∂ x |c (x)| is the derivative with respect to the arc length parameter s. Note that in contrast to previous works like the work due to Dziuk, Kuwert, and Schätzle on the elastic flow or the work due to Lin and Schwetlick on the gradient flow of the elastic energy plus a positive multiple of the Möbius energy, we do not consider the normal derivatives of the curvature but the full derivatives with respect to arc length.
Using Gagliardo-Nirenberg-Sobolev inequalities for Besov spaces, which quite naturally appear during the calculation, together with a fractional Leibniz rule, we can then show that also the E k are bounded uniformly in time. By standard arguments this will lead to long time existence and smooth subconvergence to a critical point after suitable translations and re-parameterization of the curves.
To get the full statement, we study the behavior of solutions near such critical points using a Łojasiewicz-Simon gradient estimate in Sect. 4. This allows us to show that flows starting close enough to a critical point and remaining above this critical point in the sense of the energy, exist for all time and converge to critical points. More precisely we have. Theorem 1.2 (Long time existence above critical points) Let c M ∈ C ∞ (R/Z, R n ) be a critical point of the energy E = E α + λL, α ∈ (2, 3), λ > 0, let k ∈ N, δ > 0, and β > α. Then there is a constant ε > 0 such that the following is true: Suppose that (c t ) t∈[0,T ) is a maximal solution of the gradient flow of the energy E with smooth initial data satisfying In contrast to Theorem 1.1, the analogue to Theorem 1.2 holds for the gradient flow of the Möbius energy alone [6,Theorem 5.1]. Note, that though the proof also works for the gradient flow of E 2 + λL, λ = 0, the statement would be empty as there are no critical points of this energy. This is due to the fact that E 2 is scaling invariant but L is not. The same is true for the energies E α − λL, λ ≥ 0. Theorem 1.2 shows that in the situation of Theorem 3.1 the complete solution converges to a critical point of E α + λL-even without applying any translations or re-parameterizations.

Short time existence
This section is devoted to an almost self-contained proof of short time existence for Eq. (1.4). We will show that for all c 0 ∈ C ∞ (R/Z) a solution exists for some time.
For any space X ⊂ C 1 (R/Z, R n ) we will denote by X ir the (open) subspace consisting of all injective (embedded) and regular curves in X .
We can strengthen the result above. In fact, we can reduce the regularity of the initial curve even below the level where our evolution equation makes sense. We still can prove existence and in a sense also uniqueness of a family of curves with normal velocity given by the gradient of E α + λL. For this purpose we will work in little Hölder spaces h β , β / ∈ N, which are the completion of C ∞ with respect to the C βnorm. We equip this space with the norm of the initial value problem This solution is unique in the sense that for each other solutioñ and all t ∈ (0, min(T,T )] there is a smooth diffeomorphism φ t ∈ C ∞ (R/Z, R/Z) such that c(t, ·) =c(t, φ t (·)).
As in the special case of the Möbius energy dealt with in [6], these results are based on the fact that the functional V α possesses a quasilinear structure-a statement that will be proven in the next subsection. After that we build a short time existence theory for the linearization of these equations from scratch and prove a maximal regularity result for this equation.
To get a solution of the evolution Eq. (1.4), we first have to break the symmetry that comes from the invariance of the equation under re-parameterizations. We do this by writing the time dependent family of curves c as a normal graph over some fixed smooth curve c 0 . Applying Banach's fixed-point theorem as done in [4] for nonlinear and quasilinear semiflows, we get short time existence for the evolution equation of the normal graphs and and continuous dependence of the solution on the initial data. A standard re-parameterization then gives Theorem 2.1 while Theorem 2.2 is obtained via an approximation argument.

Quasilinear structure of the gradient
(2.1) By exchanging every appearance of |c(x + w) − c(x)| and d c (x + w, x) by their first order Taylor expansion |c (x)||w| and |c (x + w)| by |c (x)| in the formula for V α , we are led to the conjecture that the leading order term of V α is the normal part is an operator of order α + 1 [24,Proposition 2.3], [17,Lemma 2.3]. This heuristic can be made rigorous using Taylor's expansions of the error terms and estimates for multilinear Hilbert transforms (cf. Lemma 6.2) leading to the next theorem. It will be essential later on that the remainder term is an analytic operator between certain function spaces-which we denote by C ω . We denote by H β , β > 2, the Bessel potential spaces.
Proof of Theorem 2.3 It is enough to show that there is a mapping Using Taylor's expansion up to first order, we get Together with the Taylor expansion for σ > 0 and x > −1, this leads to Furthermore, we will use the identities Plugging these formulas into the expressions for the terms R α 1 , R α 2 , R α 3 , R α 4 and factoring out, we see that they can be written as the sum of integrals as in the following Lemma 2.4. Hence, Lemma 2.4 completes the proof.
Proof The statement of the lemma for T 1 follows immediately from the boundedness of the multilinear Hilbert transform in Hölder spaces as stated in Remark 6.3 combined with the Lemmata 5.2 and 5.3. Using a similar argument, one deduces that c → 1 + τ X c is analytic, hence we get that for a given c 0 there is a neighborhood U such that for all c ∈ U where C does not depend on w and τ . Using that v → |v| σ is analytic away from 0, we deduce that using Lemma 5.2 and the fact that the composition of analytic functions is analytic. Hence, the integrands in the definitions of T 2 and by the same argument also ofT 2 satisfy the assumptions of Lemma 5.3. Hence, T 2 andT 2 are even analytic operators

Short time existence
Using the quasilinear form of V α , we derive short time existence results for the gradient flow of O'Hara's energies in this section. For this task, we will work with families of curves that are normal graphs over a fixed smooth curve c 0 and whose normal part belongs to a small neighborhood of 0 in h β , β > α.
To describe these neighborhoods, note that there is a strictly positive, lower semicontinuous function r : only contains regular embedded curves for all c ∈ C 2 i,r (R/Z, R n ) and Here, C β (R/Z, R n ) ⊥ c denotes the space of all vector fields N ∈ C β (R/Z, R n ) which are normal to c, i.e. for which c (u), N (u) = 0 for all u ∈ R/Z. Letting is an isomorphism from the normal space along c at u to the normal space along c + N . Otherwise there would be a v = 0 in the normal space of c at u such that For c ∈ C 1 ((0, T ), C 1 i,r (R/Z, R n )) we denote by the normal velocity of the family of curves. We prove the following strengthened version of the short time existence result mentioned at the beginning of Sect. 2. Theorem 2.5 (Short time existence for normal graphs) Let c 0 ∈ C ∞ (R/Z, R n ) be an embedded regular curve, α ∈ [2, 3), and β > α, β / ∈ N.
The proof of Theorem 2.5 consists of two steps. First we show that (2.5) can be transformed into an abstract quasilinear system of parabolic type. The second step is to establish short time existence results for the resulting equation.
The second step can be done using general results about analytic semigroups, regularity of pseudo-differential operators with rough symbols [7], and the short time existence results for quasilinear equations in [4] or [3]. Furthermore, we need continuous dependence of the solution on the data and smoothing effects in order to derive the long time existence results in Sect. 4.
For the convenience of the reader, we go a different way here and present a selfcontained proof of the short time existence that only relies on a characterization of the little Hölder spaces as trace spaces. In Sect. 2.2.1, we deduce a maximal regularity result for solutions of linear equations of type ∂ t u + a(t)Q s−1 u + b(t)u = f in little Hölder spaces using heat kernel estimates. Following ideas from [4], we then prove short time existence and differentiable dependence on the data for the quasilinear equation.

The linear equation
We will derive a priori estimates and existence results for linear equations of the type For θ ∈ (0, 1), β > 0, and T > 0 we will consider solutions that lie in the space and equip this space with the norm The right hand side f of our equation should then belong to the space From the trace method in the theory of interpolation spaces (cf. [21, Section 1.2.2]), the following relation of the space X θ,β to the little Hölder space h β+sθ is well known if β + sθ is not an integer: On the other hand, for every Given u 0 we will see that the solution of the initial value problem will also be essential in the proof (cf. [21, p. 20].
The aim of this subsection is to prove the following theorem about the solvability of our linear equation: T . This will be enough to prove short time existence of a solution for some quasilinear equations later on using Banach's fixed-point theorem.
Equation (2.6) already guarantees that J is a bounded linear operator. So we only have to prove that it is onto for which we will use some a priori results (also called maximal regularity results in this context) together with the method of continuity.
To derive these estimates, we will freeze the coefficients and use a priori estimates for ∂ t u + λ(− ) s/2 u = f on R where λ > 0 is a constant. He observed in [16], that the fractional Laplacian can be expressed by for a c s > 0 for all u ∈ H s (R, R n ). We will use this identity together with a localization argument to get from (− ) s/2 living on R back to our operator Q s−1 which lives on the circle R/Z.
Note that the fractional Let us consider the heat kernel of the equation ∂ t u + (− ) s/2 u = 0 which is given by for all t > 0 and x ∈ R.
Since k → e −t|2π k| s is a Schwartz function, its inverse Fourier transform G t is a Schwartz function as well. Furthermore, one easily sees using the Fourier transformation that The most important property for us is the scaling Combining these relations with standard interpolation techniques, we get the following estimates for the heat kernel For l > m, l, m ∈ N 0 and β ∈ (0, 1), we deduce from R ∂ l−m x G t (y)dy = 0 and the fact that G 1 is a Schwartz function In a similar way we obtain for all l ≥ m Combining these three estimates, we get Furthermore, we will use that for 0 ≤ α ≤ β ≤ γ ≤ 1, α = γ and f ∈ C γ we have the interpolation inequality which can in the case of α > 0 be obtained from Forβ 2 ≥β 1 we get Forβ 1 >β 2 and hence k 1 < k 2 , we obtain To derive a representation formula for the solution of ∂ t u + (− ) s/2 u = f , we need the following simple fact Proof For g ∈ L 2 (R) letĝ denote the Fourier transform of g.
For t > 0 and f ∈ L 2 (R) we obtain from Lebesgue's theorem of dominated convergence Hence, Plancherel's formula shows To prove the second part, let f ∈ h β (R, R n ). From convergence results for smoothing kernels we get for allf ∈ C ∞ (R) Linking the heat kernel G t to the evolution equation ∂ t u + λ(− ) s/2 u = f for constant λ > 0 we derive the following a priori estimates.

Lemma 2.9 (Maximal regularity for constant coefficients) For all
, one sees that it is enough to prove the lemma for λ = 1 To this end, we first show that Duhamel's formula As our assumptions imply that u(t) ∈ H s (R, R n ), we get, comparing the Fourier transform of both sides, in C β as ε 0. This proves Equation (2.19). From Lemma 2.7 we get Then the definition of · Y θ,β T and the estimates for the heat kernel in Lemma 2.7 lead to (2.22) As for ξ > 0 and η ∈ (0, 1), we get Hence, by the estimate (2.6) for X (2.23) From (2.19), (2.21), (2.22), and (2.23) we obtain the desired estimate for u C s+β .
The estimate for ∂ t u then follows from ∂ t u = f − (− ) s/2 u and the triangle inequality. Lemma 2.10 (Maximal regularity) Let , T > 0, n ∈ N, and β > 0, θ ∈ (0, 1) with β + sθ / ∈ N be given. Then there is a constant C = C( , β, θ, n, T ), < ∞ such that the following holds: For all Proof Note that it is enough to prove the statement for small T . Let us fix T 0 > 0 and assume that T ≤ T 0 . Furthermore, we use the embedding h β (R/Z, R n ) → h β (R, R n ) and extend the definition of Q s−1 to functions f defined on R by setting Step 1 β ∈ (0, 1) and b = 0 Let φ, ψ ∈ C ∞ (R) be two cutoff functions satisfying For r < 1/8 we set a 0 = a(0, 0) and calculate From Lemma 2.9 we get Using Lemma 7.1, we obtain This leads to Furthermore, and hence Summing up, we thus get where C 1 does not depend on r . Choosing r and T small enough and absorbing the first term on the right hand side, leads to Of course, the same inequality holds for all balls of radius r/4. Thus, covering [0, 1] with balls of radius r/4 we obtain Using the interpolation inequality for Hölder spaces and absorbing, this leads to we can absorb the first term for T > 0 small enough to obtain Step 2 General β but b = 0 Let k ∈ N 0 ,β ∈ (0, 1) and let the lemma be true for β = k +β. We deduce the statement for β = k + 1 +β.
and we obtain by applying the induction hypothesis to get Step 3 General β and b From Step 2 we get As and by (2.24) we get, absorbing the first term for T > 0 small enough, where for some positive constants c s . For u 0 , f in C ∞ a smooth solution of the equation can be given by Duhamel's formula This formula can easily be checked comparing the Fourier coefficients.
Using the a priori estimate of Lemma 2.10, one deduces that the sequence {u n,k } n∈N is a Cauchy sequence in X θ,β T −ε for every ε > 0. The limit u n solves the equation Using the a priori estimates again, one sees that ∈ (0, 1), ε, δ > 0, we can assume, after going to a subsequence, that there is a u ∞ ∈ X θ,β T such that Hence we get for all ε > 0 which implies that u ∞ solves

The quasilinear equation
Now we are in position to prove short time existence for quasilinear equations and C 1 -dependence on the initial data.
Then for every Proof Let us first prove the existence. We setX σ,β Let v be the solution of (2.26) Using Lemma 2.10, we estimate As Hence, is a contraction on B r (v) if T and r are small enough. Similarly, we deduce from Then there is a neighborhood U of u 0 in h β+sθ such that for all x ∈ U there is a solution u x of ∂ t u + a(u)Q s−1 u = 0 Then the Fréchet derivative of φ with respect to u reads as

Setting a(t) = a(u(t)) and b(t)(h) = a (u)h Q s−1 u, Theorem 2.6 tells us that this is an isomorphism between
T . Hence, the statement of the lemma follows from the implicit function theorem on Banach spaces.

So we have brought the evolution equation into the form
for all β > 0. Now the statement follows from Proposition 2.11, Proposition 2.12, and a standard bootstrapping argument.

Proof of Theorem 2.1
From Theorem 2.5 we get a smooth solution of

Proof of Theorem 2.2
The proof of Theorem 2.2 is an immediate consequence of Theorem 2.5 and the following approximation argument

) be a lower semi-continuous function. Then for everycamma
there is an open neighborhood U of the set c(R/Z) and an ε 0 > 0 such that the nearest neighborhood retract r ε : U → R/Z onto c ε is defined on U simultaneously for all 0 ≤ ε < ε 0 . Note, that these retracts r ε are smooth as the curves c ε are smooth and for ε 0 small enough. We will show that c := c ε , N := N ε , and ψ := ψ ε satisfy the statement of the lemma if ε is small enough.
From ψ ε (x) := r ε (c(x)) we deduce that ψ ε is in fact a C 2+β diffeomorphism, as r ε is smooth. Since and N 0 = 0, we furthermore get Since r is lower semi-continuous and r (c 0 ) > 0, we hence get N ε C 1 < r (c ε ) for small ε. As N ε ∈ h 2+β we deduce that N ε ∈ V r (c ε ) if ε > 0 is small enough.
Proof Letc 0 ∈ h β i,r (R/Z). By Lemma 2.13 there is a curve c 0 ∈ C ∞ i,r , an N 0 ∈ V r,β (c) and a diffeomorphism ψ ∈ C β such that The family of curves c t • ψ −1 is a solution we were looking for.
If on the other handc t is a solution as in the theorem for the initial datac 0 , then for some time there is a N t and a smooth family of reparametrizationsψ t such that c t •ψ t =c 0 + N t and The uniqueness now follows from the uniqueness of the solution to the above equation in Theorem 2.5.

Long time existence
The aim of this section is to prove the following long time existence result.

E α (c(t)) + λL(c(t)) ≤ E α (c(0)) + λL(c(0))
for all t ∈ [0, T ). So both, the energy and the length of the curve, is uniformly bounded in time. As Abrams et al. [1] have shown that for all closed curves c of unit length, we get from the scaling of the energy uniformly in t.
We will show that the energies E α are coercive in W α+1 2 ,2 (cf. Theorem 3.2) in Sect. 3.1. Together with the above inequalities this implies that the W α−1 2 ,2 norm of the unit tangents of the curve is uniformly bounded.
To get higher order estimates, we calculate the evolution equations of the terms 3) in Sect. 3.2 and show that the resulting terms can be estimated using Gagliardo-Nirenberg-Sobolev inequalities for fractional Sobolev spaces and Besov spaces. In the Sect. 3.4 we put all these pieces together to show that the solution to the flow exists for all time and subconverges after suitable translations and re-parameterizations if necessary to a critical point.

Coercivity of the energy
Theorem 1.1 in [5] states that for curves parameterized by arc length, the energy E α is finite if and only if the curve is injective and belongs to W Using Fubini and successively substituting u by u + τ 1 w and then w by (

Evolution equations of higher order energies
As for most of our estimates the precise algebraic form of the terms does not matter, we will use the following notation to describe the essential structure of the terms. For two Euclidean vectors v, w, the term v * w stands for a bilinear operator in v and w into another Euclidean vector space. For a regular curve c, let ∂ s = ∂ x |c | denote the derivative with respect to arc length denoted by s and let τ = ∂ s c = c |c | be the unit tagent vector along c. For μ, ν ∈ N, a regular curve c ∈ C ∞ (R/Z, R n ) and a function f : R/Z → R k we let P μ ν ( f ) be a linear combination of terms of the form ∂ j 1 s f * · · · * ∂ j ν s f , j 1 + · · · + j ν = μ. Furthermore, given a second function g : R/Z → R k the expression P μ ν (g, f ) denotes a linear combination of terms of the form ∂ Let c t be a smooth family of smooth closed curves moving only in normal direction, i.e., V := ∂ t c t is normal along c t . For the convenience of the reader let us derive some basic evolution equations which in similar forms can be found in (cf. [12]. Using that V is pointing in a normal direction we get Hence, and thus Furthermore, Using these equations, we inductively deduce the following evolution equations of arbitrary derivatives of the curvature. Proof By (3.4) the statement is true for k = 0. If the statement was true for some k then Hence, induction gives the assertion.
An immediate corollary of Lemma 3.3 and (3.2) is the following.

Corollary 3.4 Let c be a family of curves moving with normal speed V . Then
Together with Lemma 3.3 and (3.2) we get Partial integration for the second term on the left hand side yields the claim.

Interpolation estimates
In this section we will prove several estimates that will be needed later in the proof of the long time existence result. In a natural way the Besov spaces B s, p q appear during our calculations. Lemma 3.5 (Gagliardo-Nirenberg-Sobolev type estimates for a typical term) For α ∈ [2, 3) let j 1 , . . . , j ν+2 ∈ N, j 1 , . . . , j ν ≥ 2, and s > 3 2 be such that there are Proof Using Hölder's inequality for the integration with respect to u, we get Substituting w by (τ 1 − τ 2 )w we can estimate this further by Scaling the Gagliardo-Nirenberg-Sobolev estimates (Theorem 8.1), the last term can be estimated from above by for i = ν + 1, ν + 2. Thus the assertion of the theorem follows.
We will now use the lemma above to estimate a typical term that repeatedly will appear in the later calculations.

Lemma 3.6 (Estimates for terms containing the energy integrand) For all
< ∞ there is a C( ) < ∞ such that the following holds: Let −1 ≤ l ≤ and c ∈ C ∞ (R/lZ, R n ) be a curve parameterized by arc length satisfying the bi-Lipschitz estimate Then for β ≥ 0 the functions Proof Note that ∂ s = ∂ x since c is parametrized by arc-length. For l 2 > ε > 0 and and h β (s, w) := |w| β |c(s + w) − c(s)| α+β − |w| β |w| α+β for all s ∈ R/lZ and w ∈ [−l/2, l/2], w = 0. Then due to the bi-Lipschitz estimate for c we have g (ε) β ∈ C ∞ (R/lZ, R n ) and Using the product rule together with Fáa di Bruno's formula for higher derivatives of composite functions, we conclude that is a linear combination of terms T k 1 ,...,k m+2 of the form Using (3.7) and the fundamental theorem of calculus, the absolute value of such terms is bounded by Hence, From this we deduce that for ε 1 > ε 2 > 0 we have As α < 3, g (ε) β converges smoothly to a smooth representative of g β and Using that ∂ k s h β (s, w) is a linear combination of terms like (3.8) together with Lemma 3.5, we obtain s. This proves inequality (3.5) from which one obtains (3.6) using Cauchy-Schwartz.

Proof of long time existence
First we will derive the following estimate from the evolution equation of the higher order energies in Corollary 3.4 using Lemma 3.6. For a periodic function f ∈ C ∞ (R/lZ, R n ) we use the shorthand for the fractional Laplacian. Lemma 3.7 For every k ∈ N and ε > 0 there are constants C ε < ∞ such that Proof Corollary 3.4 tells us that Let us now fix the time t and let us re-parameterize c t for this fixed time by arc length to estimate the right-hand side of this equation and let l denote the length of the curve at time t.
We decompose 2Ṽ Using the fundamental theorem of calculus, we rewrite (R α 2 c)(s) as and set Hence, Now we first estimate all the terms appearing in (3.9) after plugging into it the decomposition (3.11) except for the term using Hölder's inequality together with the standard Gagliardo-Nirenberg-Sobolev inequality or the version in Lemma 3.5. We get that for all ε > 0 there is a constant C ε < ∞ such that all these terms can be estimated from above by We will only give the details for some exemplary terms as all the other terms can be estimated following the same line of arguments. We start by estimating the terms containing the remainder R α . For the first term in Eq. (3.9) we calculate Using the definition of R α 1 and the Leibniz rule together with Lemma 3.6, we obtain Along the same lines we get Furthermore, we can estimate Using the same line of arguments one sees that also the terms For the terms containing λκ we use Hölder's inequality and standard Gagliardo-Nirenberg-Sobolev estimates together with Cauchy's inequality as above, to estimate these terms by To estimate the terms containing Q α , we will use the fact that Q α = c α D α+1 +R whereR is a bounded operator from W s+2, p to W s, p [24,Proposition 2.3]. For k 1 , k 2 ∈ N 0 with k 1 + k 2 = k we estimate using Hölder's inequality and the Gagliardo- and thus Similarly we obtain, using the Leibniz rule, Hence, Similarly, one gets and substituting firstx = x + τ w and thenw = −w shows The Leibniz rule now tells us that Setting g m, j = ∂ j+1 c ∂ m+1 c, ∂ k s κ and using the formula above for Q α we get Using the fractional Leibniz rule Lemma 8.2 and then the interpolation estimates, we can estimate this again by As furthermore Summing up these estimates proves Lemma 3.7.
A standard argument now concludes the proof of Theorem 3.1: Proof Theorem 3.1 Let us assume that [0, T ) is the maximal interval of existence and T < ∞. If we apply Lemma 3.7 with ε = c α 2 we get Together with the Poincare inequality for a constant c > 0. Hence, for a constant C(k) < ∞ depending on the inital data but not on T . Furthermore, Since V, κ is uniformly bounded there is a constant C > 0 depending on T such that It was shown in [12, end of p.7], that this estimate together with the uniform in time bounds on all derivatives with respect to arc-length yield ∂ k x c ≤ C k for constants C k that are independent of t.
Thus, there is a subsequence t i → T such that c t i converges smoothly to ac(T ) and we can use the short time existence result Theorem 2.1 to extend the flow beyond T .
Let us finally prove the subconvergence to a critical point up to translations and reparametrizations. Let the re-parametrizsations ψ t be such that the curves are parametrized proportional by arc-length. As the uniform in time estimates (3.13) hold, there is a subsequence t i → ∞ such that the curvesc t i converge smoothly to a curve c ∞ : Since our uniform estimates further imply d dt which shows that c ∞ is a critical point of E α + λL .

Łojasiewicz-Simon gradient estimate
In order to prove convergence to critical points of the complete flow without taking care of translations, we will prove a Łojawiewicz-Simon gradient estimate.
Proof After scaling the curve we can assume that c M is parameterized by arc length and that the length of the curve c M is 1.
We will first prove that there are constants θ ∈ [0, 1/2],σ > 0, and C < ∞, such that That is, we show that the func-tionalẼ satisfies a Łojasiewicz-Simon gradient estimate. By [10,Corollary 3.11] it suffices to show that E is analytic with values in L 2 and that E is a Fredholm operator from H α+1 (R/Z, R n ) ⊥ c M to L 2 (R/Z, R n ). It is easy to see that κ defines an analytic operator on a neighborhood of 0 from (H α+1 ) ⊥ c M to L 2 using the fact that H α+1 is embedded in C 2 . That the same is true for V α can be seen from Theorem 2.3, using the fact that H α+1 (R/Z, R n ) is embedded in C α+β for every 0 < β < 1/2 and C β is embedded in L 2 .
We calculate the second variation ofẼ at 0 and try to write it as a compact perturbation of a Fredholm operator of index 0. Using |c M | = 1 and the fact that V is the gradient of E α + λL, we get where L 1 h 1 = V α c M · c M , h 1 is a differential operator of order 1 in h 1 . We know from Theorem 2.3 that where F α ∈ C ω (C α+β , C β ) for all β > 0. Thus where L 3 ∈ C ω (C α+β , C β ) by the fractional Leibniz rule Lemma 7.1. From [24,Proposition 2.3] we know that there is a constant a (α) > 0 such that that L 4 := Q α − a α (− ) (α+1)/2 is a bounded linear operator from H 2 to L 2 . Combining (4.3), (4.4), and (4.5), we get where is a bounded operator from C α+ε to L 2 for all ε > 0. Since the linear mapping defines an homeomorphism between H s (R/Z, R n ) ⊥ c M and H s (R/Z, R n−1 ) for all s ≥ 0 and (− ) (α+1)/2 is a Fredholm operator of index zero from H α+1 (R/Z, R n−1 ) to L 2 (R/Z, R n−1 ), the operator 2 ) h 1 , ν i ν i + Lh 1 as a compact perturbation of A is a Fredholm operator as well. This implies that E is a Fredholm The proof of (4.2) is complete.
To prove the full estimate of Theorem 4.1, we use Lemma 9.1 to write curves close to c M as normal graphs over c M . More precisely, we can choose σ ∈ (0,σ ) such that for all c Making σ > 0 smaller if necessary, using that c M is parameterized by arc length and that H α+1 is embedded continuously in C 1 , we furthermore can achieve that Thus,

The flow above critical points
In this section we apply the Łojasiewich-Simon gradient estimate to reprove longtime existence for solutions that approach a critical point from above as stated in Theorem 1.2. Using the techniques from this section, we will show that even the complete flow converges to a critical point without applying any translations. Theorem 1.2 will follow easily from the following long time existence result for normal graphs over a critical point of the energies E = E α + λL: Theorem 4.2 (Long time existence and asymptotics for normal graphs) Let c M ∈ C ∞ (R/Z, R n ) be a critical point of E and let k ∈ N, δ > 0 and β > α. Then there is an open neighborhood U of 0 in C β ⊥ such that the following holds: is a maximal solution of the equation whenever N t C k ≤ δ. Then T = ∞ and N t converges smoothly to an N ∞ ∈ C ∞ (R/Z, R n ) ⊥ c M satisfying Furthermore c M + N ∞ is a critical point of the energy.
Proof Letk = max{4, k, β + 1 }, where β + 1 denotes the smallest integer larger or equal to β + 1. Then of course we still have under the stronger condition N t Ck ≤ δ. We first use Theorem 4.1 to get an ε 0 > 0, a θ ∈ (0, 1 2 ), and a constant C < ∞ such that Making ε 0 > 0 smaller if necessary, we can furthermore achieve that and on R/Z. It is now crucial to use the smoothing properties of our short time existence result Theorem 2.5. This theorem tells us that there is a radius r > 0 and a time T > 0 such that the solution N t = N N 0 ,t to . Furthermore, these solutions exist as long as they stay inside of U as one can then always continue the solution.
Since the mapping belongs to C ∞ (U × (0, T ), C ∞ (R/Z)) and hence is especially continuous and that N 0,t = 0, by making r smaller if necessary we can guarantue that for all t ∈ [T /4, T /2] and all initial data N 0 ∈ U . Let ε ∈ (0, min{r, ε 0 }) to be chosen later. Using again that the mapping in Eq. (4.10) is continuous together with N 0,t = 0 one sees that there is a δ ∈ (0, r ) such that for (4.12) We know from the considerations above that T max ≥ T , 13) and the solution exists at least as long as N T 4 C β < r . If the solution does not exist for all time, there hence is a t 0 > T 2 such that which leads to a contradiction for ε > 0 small enough.
We will now show that Applying (4.11) for all times t ∈ [ T 4 , t 0 ] we get and hence N t satisfies a Łojasiewich-Simon gradient estimate (4.7) and (4.8) and (4.9) ≤ Cε θσ . (4.14) So if ε > 0 is small enough we have for all t ∈ [T, t 0 ] which contradicts our choice of t 0 . Hence, we have shown that the flow exists for all time and satisfies for all t ≥ T . From Theorem 2.5 we get sup t≥ T 2 c t C l < ∞ for all l ∈ N and hence there is a subsequence t i → ∞ such that Since our uniform estimates further imply d dt Using the Łojasiewicz-Simon gradient inequality again we get To get convergence of the complete flow, we repeat the estimates that led to (4.14) with c ∞ in place of c M to get for all t ≥ t i . So the complete flow converges in C β and hence by interpolation in C ∞ to c ∞ Proof of Theorem 1.2 Due to Lemma 9.1 for all c ∈ C β (R/Z, R n ) with c−c M C β ≤ ε there is a diffeomorphism φ c and a vector field N c ∈ C β (R/Z, R n ) normal to c M such that c • φ c = c M + N c (4.15) and ThenT ≤ T and for all t ∈ [0,T ) there are diffeomorphisms φ t such that c t = (c M + N t )(φ t ). Hence N t satisfies all the assumptions of Theorem 4.2 if ε is small enough and thus ∞ =T . FromT ≤ T we deduce T = ∞.

Completion of the Proof of Theorem 1.1
It is only left to show that we get convergence of the flow without applying translations from the smooth subconvergence of the re-parameterized and translated curves we get from Sect. 3.4. Letc t be the re-parameterizations of c t and let t i → ∞ and p i ∈ R n be such that the curvesc(t i )− p i converge smoothly to a curvec ∞ parameterized by arc length. Due to the smooth convergence, the datac t i − p i satisfies all the assumptions of Theorem 1.2 for i large enough and c M = c ∞ . Hence, the statement follows directly from the conclusions of Theorem 1.2.
As D m T (y)(h 1 , . . . , h m ) = D m g(y)(h 1 , . . . , h m ) (can easily be deduced from the Taylor expansion of g) and since C k,α (R/Z) is a Banach algebra, we get

Lemma 5.3
Let (X, · X ) and (Y, · Y ) be Banach spaces and assume that T t ∈ C ω (X, Y ) for t ∈ I is such that the functions t → T t are measurable and for all a ∈ X there is a neighborhood U of a in X such that I sup y∈U D m T t (y) dt ≤ C m m!. from which we get that T ∈ C ω (X, Y ) using the estimates (5.1). In fact this follows from well-known facts about differentiation of parameter dependent integrals.

Remark 5.4
In the case that Y = C k,α (R/Z, R n ) we get from the fact that for u ∈ R/Z the evalution mapping F : Y → R n y → y(u) is a bounded linear operator form Y to R n . Hence, the interchangability of the Bochner integral with linear operators (cf. for example [28, p. 134, Corollary 2]) tells us that Proof For u, v ∈ R/Z and a ∈ (0, 1 2 ) we get T (c 1 , . . . , c m )(u) − T (c 1 , . . . , c m )(v) Since |t i | ≤ 1, we obtain  Proof First, let us note that it is enough to prove the statement for β =β + n, n ∈ N 0 , 1 − α ≥β > 0. One then uses real interpolation to prove the full statement. For n = 0 the claim is the content of Lemma 6.1. So let us assume that the statement holds for β =β+n as above. Let (τ h f )(x) := f (x+h). Using the relation τ h (T (c 1 , . . . , c m )) = T (τ h (c 1 ), . . . , τ h (c m )) and the multilinearity of T , the difference quotient can be written as Since and T a bounded linear operator from C β to Cβ , we get in C β . Hence, T is a bounded multilinear mapping from (C α+β+1 ) m to C β+1 .

Remark 6.3
Let us state a simple extension of Lemma 6.2. Given a multilinear form M : R n × · · · × R n → R k , 1 > α > 0, β > 0, m ∈ N, and t i ∈ (0, 1) the singular integral Taylor expansion yields and hence the first term in the last row of Eq. (7.1) can be written as  It is an easy exercise to prove that the last term defines a bounded operator from C 2+β to C β for all β > 0. Using Lemma 6.2 to estimate the first two terms, we get Together with the fact that Q defines a bounded linear operator from C α+1+β to C β this proves the estimate.

Interpolation and Leibniz rule for fractional Sobolev spaces
For the convenience of the reader we present mutliplicative Gagliardo-Nirenberg-Sobolev inequalities and a Leibniz rule in fractional Sobolev spaces and nonhomogeneous Besov spaces that we used in this text. We use the notation . Furthermore, Proof Ehrling's lemma together with the estimate f C k ≤ C( f (k) L ∞ + f L ∞ ) tells us that for every ε > 0 there is an C(ε) < ∞ such that (B 1 (0)) .

Scaling this inequality yields
for all r > 0 and hence we get covering R by balls of radius r ∈ [0, 1/2] Hence, we get the first estimate by choosing r = Combining this with the first estimate gives

Normal graphs
The following lemma is used in the proofs of Lemma 4.1 and Theorem 3.1.