Convergence for global curve diffusion flows

In this note we establish exponentially fast smooth convergence for global curve diffusion flows, and discuss open problems relating embeddedness to global existence (Giga's conjecture) and the shape of Type I singularities (Chou's conjecture).


Introduction
The curve diffusion flow is the one-parameter family of immersed curves γ : S 1 × [0, T ) → R 2 with normal velocity equal to −grad H −1 (L(γ)), that is (CD) ∂ ⊥ t γ = −k ss ν. In [W], the author proves that if the isoperimetric ratio and L 2 -normalised oscillation of curvature are close to their value on any circle (with an explicit estimate on the constant) then the flow exists globally and converges to a circle. However the rate of convergence was not established.
In this note we prove a general convergence principle that can be summarised as: T = ∞ =⇒ γ converges exponentially fast to an ω-circle in the smooth topology.
Theorem 1.1. Suppose γ : S × [0, ∞) → R 2 is a global curve diffusion flow with smooth initial data γ 0 that has winding number ω. Then γ converges exponentially fast to a round ω-circle in the smooth topology, with explicit estimates (here m ∈ N) k s m 2 2 ≤ c m e −k 4 0 t , with D(t) ≤ D 0 e −4k 4 0 t and K osc (t) ≤ c 0 e −2k 4 0 t . The constants c m depend only on the initial data γ 0 .
The quantities D and K osc are the isoperimetric defect and the L 2 -normalised oscillation of curvature: where L and A are the length and signed enclosed area of γ, and k, k are the curvature scalar and the average of k respectively. Section 2 contains the proof of Theorem 1.1.
Remark 1.2. Note that no smallness condition is required.
Remark 1.3. The regularity hypothesis is not optimal.
2000 Mathematics Subject Classification. 53C44 and 58J35. Remark 1.4. Theorem 1.1 is stated for flows of immersed multiply-covered circles which makes it applicable to the case considered by Miura-Okabe [MO]. However the curve diffusion flow starting from a multiply-covered circle is not expected to be stable in general -perturbations that unbalance the length of different leaves are should lead to finite-time singualrities. 1 Miura-Okabe use a rotational symmetry assumption so that only perturbations that preserve the balance of the different leaves are studied, and the power of this hypothesis can be seen in the isoperimetric inequality for immersed rotationally symmetric curves that they establish. That is, preservation of the smallness of the isoperimetric ratio (see Lemma 2.4) is proved in [MO] for rotationally symmetric curves without assuming global existence. This is the decisive new ingredient that allows their result on the curve diffusion flow to go through.
If this conjecture holds, our Theorem 1.1 implies: an embedded curve diffusion flow converges smoothly and exponentially fast to a round circle.
Note that in the case of the curve diffusion flow in homotopy classes outside ωcircles, shrinkers are expected and one explicit example is known (the Lemniscate of Bernoulli, see [EGBM + ]). Indeed, Chou [C] suggests that parabolic rescalings of Type I singularities (a definition also due to Chou) are self-similar solutions to the curve diffusion flow: Then a parabolic rescaling (we assume the centre of mass of γ is the origin) η(s, t) = (T − t) − 1 4 γ(s, t) about final time yields a self similar solution η to the curve diffusion flow, that is, η solves (2) η, ν η = 4k η ss . Curve diffusion flows satisfying (1) are said to be Type I. As suggested by Chou [C], this conjecture can be approached by studying all solutions to the shrinker equation (2), and classifying all such solutions is an interesting problem. Since area is constant along the curve diffusion flow, and shrinkers must eventually enclose zero area, all shrinkers have zero signed enclosed area. This means that all solutions to (2) are non-embedded (and have zero signed enclosed area).
The maximal time of smooth existence for the curve diffusion flow with ω = 0 is always finite. This is because along the flow the curvature scalar always has a zero, and so the Wirtinger inequality implies that the flow can not exist past time T = L 4 0 16π 4 . This is not a sharp estimate, as equality in the estimates used by the proof implies k = const, and there is no such curve that also has ω = 0. A reasonable conjecture is that the maximal existence time for a curve diffusion flow with winding number zero and initial length L 0 is bounded by that of the Lemniscate of Bernoulli starting with length L 0 .
As far as we know the connection between non-convexity and maximal existence time was observed first by Chou [C]. It was later used in [W] to estimate the total waiting time before a curve diffusion flow with small L 2 -oscillation of curvature and isoperimetric ratio becomes uniformly convex. Theorem 1.1 here combined with the waiting time estimate [W,Proposition 1.5] yields an estimate for the total waiting time before any global curve diffusion flow becomes uniformly convex.

Acknowledgements
The author thanks his colleagues for discussions on the curve diffusion flow, especially relating to its convergence properties. In particular he would like to thank Tatsuya Miura for extensive discussion and encouraging the dissemination of this note. Without the encouragement of his peers this note would not have been written. This work was completed under the financial support of the Okinawa Institute of Science and Technology, hosted by James McCoy. He is grateful for their support and hospitality.

Proof of Theorem 1.1
We complete the proof in a sequence of lemmata. The hypotheses of Theorem 1.1 are not restated, but assumed throughout.
Lemma 2.1. There exists a subsequence of times t j → ∞ such that k s 2 2 (t j ) → 0.
Proof. Note that L ′ = − k s 2 2 . Integration and the monotonicity of length implies the result.
As far as we are aware, Lemma 2.1 was first observed by Chou [C]. Now we observe that the signed enclosed area of the curve under the flow can not be non-positive.
Lemma 2.2. The signed enclosed area A of γ is strictly positive.
Proof. Our convention is that the area of an ω-circle is positive (and equal to ωπr 2 , where r is its radius). If the area is not strictly positive, then the flow can exist for at most a finite amount of time. Since area is constant under the flow, if it is non-positive at any time, it must be so for all time. Then, since A(t) < 0 implies that k(s t , t) = 0 for some point s t , we use the Wirtinger and Poincaré inequalities to conclude L 3 which implies that the flow can exist at most for a finite time (in fact T ≤ L0 16π 4 ). This is a contradiction.
Since length is monotone decreasing and signed enclosed area is constant in time, we immediate have: By combining Lemma 2.1, Lemma 2.2 and Lemma 2.3 we find the following.
Corollary 2.4. Let ε > 0 be arbitrary. There exists a t j such that for all t > t j , The estimate (3) follows using an elementary argument (see the Appendix). Note that length is uniformly bounded below by the measure-theoretic isoperimetric inequality L 2 (t) ≥ 4πA + where A + is the area of γ changed so that the orientation of every loop is positive (no segments of negative area). Clearly we always have A + ≥ A and so (using the constancy of area along the flow) (0) follows. This is not a sharp estimate for ω = 1 but it is enough to give a uniform lower bound on evolving area for any winding number. Note that the shrinking Lemniscare of Bernoulli has A(0) = 0 and T < ∞. Lemma 2.2 implies that A(0) > 0 and so in particular length is uniformly bounded from below by a universal constant.
Therefore the estimate (4) implies that D(t j ) → 0, which implies I(t j ) → 1. Now take j sufficiently large that I(t j ) ≤ 1 + ε, where ε > 0 is as given by hypothesis. Then Lemma 2.3 implies that I(t) ≤ 1 + ε for all t > 0, as required.
Lemma 2.5. There are uniform bounds on all derivatives of curvature.
These estimates imply that the isoperimetric defect and K osc converge to zero at times t j as j → ∞. Therefore for j sufficiently large, the hypotheses of [W,Proposition 3.7] are satisfied at time t j . Given the eventual smallness of the isoperimetric ratio (Corollary 2.4), the argument in [W] (see also [MO]) applies to yield uniform estimates on K osc . This then implies bounds on all derivatives of curvature by the evolutionary interpolation inequalities of Dziuk-Kuwert-Schätzle [DKS]. 2 Lemma 2.6. The isoperimetric defect decays exponentially fast: Proof. Since Remark 2.7. One may also attempt the same strategy with the isoperimetric ratio in place of the isoperimetric defect. However, the resultant sharp estimate looks like This results in linear decay of the isoperimetric ratio -not exponential.
Lemma 2.8. All derivatives of curvature decay exponentially fast, with the explicit estimate k s m 2 2 ≤ c m e −k 4 0 t .
Proof. Since the previous lemma establishes uniform bounds for all derivatives of curvature, a Fourier series argument (taught to us by Ben Andrews many years ago and used in for example [AMWW]; we give an explanation in the appendix). 3 K osc ≤ c √ D L where c depends on the estimates for k and k s . Now the isoperimetric inequality (and constancy of area) implies that L is uniformly bounded from below, so the above implies that K osc satisfies K osc ≤ c 0 e −2k 4 0 t . Integration by parts and the uniform estimates imply that for any m we have k s m 2 2 ≤ c m e −k 4 0 t , as required. The argument is by now standard in the literature, see for instance [AMWW,Theorem 6.4] for a recent example.
3 The author recently learned that similar Fourier series arguments (and inequalities that imply the below) also appear in [NN].
The exponential decay implies convergence of the position vector γ to an ω-circle with a standard argument (for example this was used by Huisken [H] in his seminal work on mean curvature flow). Briefly, this is because we then have estimates on γ by simply integrating the evolution equation in time. We may convert from arclength derivatives to arbitrary ones in a manner analogous to [DKS,proof of Theorem 3.1]. Note that this integration in time can not be done with only linear decay estimates.
Appendix A. Bounding oscillation of curvature by the isoperimetric defect Both estimates used in this article are contained in the following statement.
Lemma A.1. Let γ : S → R 2 be a smooth immersed curve. Then where c is such that Note that when the upper estimate is used, curvature and its derivative(s) are uniformly bounded, so the hypothesis involving c is satisfied.
A.1. Lower estimate. Let us prove first the lower estimate. To begin, we translate the curve γ toγ so that γ ds = 0 and the estimate |γ| ≤ L 4 holds. The lower estimate in (6) is invariant under translation (this is true of both sides), so if we can prove the estimate forγ it will be true for γ also.
The lower estimate in (6) follows now from Hölder's inequality.
Since P is arbitrary, this quantity is unbounded. This estimate is used in [NN], although it is not a major issue (translation invariance as used here can also be used there).
Acknowledgement. We would like to express our great gratitiude to Ben Andrews, who taught us the technique we are about to use (and many, many other things). In particular Ben demonstrated how to prove a great variety of inequalities for curves, and how they can be applied to many different kinds of curvature flow. This occured while the author was his guest at the Mathematical Sciences Center at Tsinghua University in Beijing, 2011. These results should be thought of as due to Ben Andrews and not the author.
As in [AMWW, Proof of Theorem 6.1] we work in the complex plane. Let us identify γ(s) = (x(s), y(s)) with s → x(s) + iy(s). The idea is to use a Fourier series decomposition for γ to reduce the question of proving geometric inequalities to properties of infinite series of integers.
We write γ(s) = Curvature arises by differentiating γ, which is the same as differentiating the Fourier decomposition. The basic relation is: Proof. We provide a sketch with the essential steps. First, observe that for q ≥ 2 we have (7) Q q ds = i kQ q−1 ds .
This is because Q q = ikQ q−1 + ∂ s Q q−1 (this equality can be proved by a striaghtforward induction argument). But now we use the series decomposition with orthonormality of c p to conclude In the q = 1 case this finishes the proof; for q ≥ 2 we combine this with (7) to finish the proof.
Lemma A.3 allows us to simply express the isoperimetric defect and the L 2 oscillation of curvature in terms of infinite series. To see this, we consider the cases of q = 1, 2, 3, 4, 5, 6 in Lemma A.3. For each of computation we make the notation T = γ, ∂ s γ , and N = γ, ν .
Although we won't need it here, one can continue to calculate more of these identities for greater values of q. They can be used to establish higher order inequalities -we include an example (inequality (8)). Here are the next two: L 7 (2ωπ) 7 k 5 − 6ik 3 k s − 4k 2 k ss − 3kk 2 s + ikk sss ds = L 7 (2ωπ) 7 k 5 + 5kk 2 s ds .
This estimate does not seem to follow from using more common methods. Now we return to our aim of proving the upper estimate. First, we write key quantities in terms of infinite series.
Lemma A.4. The equalities Proof. We calculate Using the expressions for q = 2 and q = 1, this shows D = 4ω 2 π 2 L p∈Z (p 2 − p)|γ(p)| 2 , as required. Now For the second term we could have used the equality for q = 3 instead, which gives the last equality and finishes the proof.
Finally, we use an inequality for infinite series (in this case it is simply the discrete Hölder inequality) to conclude essentially the upper estimate.
Lemma A.5. We have