Convergence for global curve diffusion flows

1.   Introduction

The curve diffusion flow is the one-parameter family of immersed curves $\gamma: \mathbb{S}^1\times[0, T)\rightarrow \mathbb{R}^2$ with normal velocity equal to $-\text{grad}_{H^{-1}}(L(\gamma))$, that is

Here, $s$ denotes arclength, $k$ is the scalar curvature, and $\nu$ is the normal vector.

In ^[10], 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:

An $\omega$-circle is an immersion $\gamma: \mathbb{S}^1\rightarrow \mathbb{R}^2$ with winding number of the unit normal around the origin equal to $\omega$. Note that also $\omega = \frac1{2\pi}\int k\, ds$.

Theorem 1.1. Suppose $\gamma: \mathbb{S}\times[0, \infty)\rightarrow \mathbb{R}^2$ is a global curve diffusion flow with smooth initial data $\gamma_0$ that has winding number $\omega$. Then $\gamma$ converges exponentially fast to a round $\omega$-circle in the smooth topology, with explicit estimates (here $m\in \mathbb{N}$)

with

The constants $c_m$ depend only on the initial data $\gamma_0$.

The quantities $\mathcal{D}$ and $K_{\text{ $\mspace{-1mu}o\mspace{-1mu}s\mspace{-1mu}c$ }}$ are the isoperimetric deficit and the $L^2$-normalised oscillation of curvature:

where $L$ and $A$ are the length and signed enclosed area of $\gamma$, and $k$, $\overline{k}$ 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 that we impose (initial data of class $C^\infty$) is not optimal. There are in the literature well-posedness (and regularisation) results for curve and surface diffusion with initial data of class $C^{1, \alpha}$ ^[7] as well as initial data in a Besov space ^[4]. It remains an interesting problem to determine the weakest possible conditions on the initial data that allow generation of a unique, regularising solution to the flow.

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 ^[8]. 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.^* 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 ^[8] 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.

^* Rigorously establishing this is an interesting open problem.

Yoshikazu Giga famously conjectured around ten years ago that:

Conjecture (Giga's Conjecture). Suppose $\gamma: \mathbb{S}\times[0, T)\rightarrow \mathbb{R}^2$ is a curve diffusion flow with smooth initial data $\gamma_0$ that has the property:

Then $T = \infty$.

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 $\omega$-circles, shrinkers are expected and one explicit example is known (the Lemniscate of Bernoulli, see ^[5]). Indeed, Chou ^[2] suggests that parabolic rescalings of Type I singularities (a definition also due to Chou) are self-similar solutions to the curve diffusion flow:

Conjecture (Chou’s Conjecture). Suppose $\gamma: \mathbb{S}\times[0, T)\rightarrow \mathbb{R}^2$ is a curve diffusion flow with $T < \infty$ that satisfies the estimate

for some $C\in \mathbb{R}$, and $t\in[0, T)$.

Then a parabolic rescaling (we assume the centre of mass of $\gamma$ is the origin)

where $\tau = -\log(T-t)$ yields a self similar solution $\eta$ to the curve diffusion flow, that is, $\eta$ converges as $t\rightarrow\infty$ to a solution $\eta^\infty$ of

Curve diffusion flows satisfying (1.1) are said to be Type I. As suggested by Chou ^[2], this conjecture can be approached by studying all solutions to the shrinker Eq (1.2), 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 (1.2) are non-embedded (and have zero signed enclosed area).

The maximal time of smooth existence for the curve diffusion flow with $\omega = 0$ 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 $T = \frac{L^4_0}{16\pi^4}$. 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 $\omega = 0$. 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 ^[2]. It was later used in ^[10] 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 [10, Proposition 1.5] yields an estimate for the total waiting time before any global curve diffusion flow becomes uniformly convex.

2.   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 $t_j\rightarrow\infty$ such that $\lVert{k_s}\rVert_2^2(t_j)\rightarrow0$.

Proof. Note that

Integration and the monotonicity of length implies the result.

As far as we are aware, Lemma 2.1 was first observed by Chou ^[2].

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 $\gamma$ is strictly positive.

Proof. Our convention is that the area of an $\omega$-circle is positive (and equal to $\omega\pi r^2$, 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

which implies that the flow can exist at most for a finite time (in fact $T \le \frac{L_0}{16\pi^4}$). This is a contradiction.

Now set

Recall that $\mathcal{D}$ is the isoperimetric defect. Note that $\mathcal{D}\ge0$ with equality if and only if $\gamma$ is an $\omega$-circle. Since length is monotone decreasing and signed enclosed area is constant in time, we immediately have:

Lemma 2.3. The isoperimetric defect satisfies

By combining Lemma 2.1, Lemma 2.2 and Lemma 2.3 we find the following.

Corollary 2.4. Let $\varepsilon > 0$ be arbitrary. There exists a $t_j$ such that for all $t > t_j$,

Proof. The estimate

(note that $\overline{k} = 2\omega\pi/L$ is the average of $k$) together with the Poincaré inequality implies

The estimate (2.1) follows using an elementary argument (see the Appendix).

Note that length is uniformly bounded below by the measure-theoretic isoperimetric inequality

where $A^+$ is the area of $\gamma$ changed so that the orientation of every loop is positive (no segments of negative area). Clearly we always have $A^+ \ge A$ and so (using the constancy of area along the flow)

follows. This is not a sharp estimate for $\omega \ne 1$ 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 $T < \infty$. 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 (2.2) implies that $\mathcal{D}(t_j) \rightarrow 0$, which implies $\mathcal{I}(t_j) \rightarrow 1$. Now take $j$ sufficiently large that $\mathcal{I}(t_j) \le 1 + \varepsilon$, where $\varepsilon > 0$ is as given by hypothesis. Then Lemma 2.3 implies that $\mathcal{I}(t) \le 1+\varepsilon$ for all $t > 0$, as required.

Lemma 2.5. There are uniform bounds on all derivatives of curvature.

Proof. We recall the estimate (2.2) and

which follows from the Poincaré inequality. Each of the estimates (2.1), (2.3) are uniform (from the uniform boundedness of length).

These estimates imply that the isoperimetric defect and $K_{\text{ $\mspace{-1mu}o\mspace{-1mu}s\mspace{-1mu}c$ }}$ converge to zero at times $t_j$ as $j\rightarrow\infty$. Therefore for $j$ sufficiently large, the hypotheses of [10, Proposition 3.7] are satisfied at time $t_j$. Given the eventual smallness of the isoperimetric ratio (Corollary 2.4), the argument in ^[10] (see also ^[8]) applies to yield uniform estimates on $K_{\text{ $\mspace{-1mu}o\mspace{-1mu}s\mspace{-1mu}c$ }}$. This then implies bounds on all derivatives of curvature by the evolutionary interpolation inequalities of Dziuk-Kuwert-Schätzle ^[3].^†

^† The argument is by now standard in the literature, see for instance [1,Theorem 6.4] for a recent example.

Lemma 2.6. The isoperimetric defect decays exponentially fast:

Proof. Since

the estimate (2.1) implies

Therefore

where $\overline{k}_0$ is the initial average of the curvature scalar. Then

as required.

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

Here $k_{s^m} = \partial^m_sk$.

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 ^[1]; we give an explanation in the appendix).^‡

^‡ The author recently learned that similar Fourier series arguments (and inequalities that imply the below) also appear in ^[9].

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_{\text{ $\mspace{-1mu}o\mspace{-1mu}s\mspace{-1mu}c$ }}$ satisfies

Integration by parts and the uniform estimates imply that for any $m$ we have

as required.

The exponential decay implies convergence of the position vector $\gamma$ to an $\omega$-circle with a standard argument (for example this was used by Huisken ^[6] in his seminal work on mean curvature flow). Briefly, this is because we then have estimates on $\gamma$ by simply integrating the evolution equation in time. We may convert from arclength derivatives to arbitrary ones in a manner analogous to [3,proof of Theorem 3.1]. Note that this integration in time can not be done with only linear decay estimates.

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. The author would additionally like to thank the anonymous referee for their comments and suggestions that led to an improvement in this article.

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 occurred 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.

Conflict of interest

The author declares no conflict of interest.

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 $\gamma: \mathbb{S}\rightarrow \mathbb{R}^2$ 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 $\gamma$ to $\tilde\gamma$ so that

and the estimate

holds. The lower estimate in (6.1) is invariant under translation (this is true of both sides), so if we can prove the estimate for $\tilde\gamma$ it will be true for $\gamma$ also.

Recall that

Then integration by parts using $\left < {\tilde\gamma}, {\tilde\gamma_s}\right > _s = 1 + k\left < {\tilde\gamma}, {\nu}\right >$ implies

Since $\overline{k} = \frac{1}{L}\int k\, ds = \frac{2\omega\pi}{L}$, we have $\overline{k}-k = \frac1L(-kL + 2\omega\pi)$. Therefore

The lower estimate in (A.1) follows now from Hölder's inequality.

Remark A.2. The estimate

is false in general. Take for example a circle with unit radius centred at $(P, 0)$ in the plane. Then,

Since $P$ is arbitrary, this quantity is unbounded. This estimate is used in ^[9], although it is not a major issue (translation invariance as used here can also be used there).

A.2.   Upper estimate

As in [1,Proof of Theorem 6.1] we work in the complex plane. Let us identify $\gamma(s) = (x(s), y(s))$ with $s\mapsto x(s) + iy(s)$. The idea is to use a Fourier series decomposition for $\gamma$ to reduce the question of proving geometric inequalities to properties of infinite series of integers. Here we have $\gamma$ smooth, but in order to express $\gamma$ as a convergent Fourier series we only require $\gamma\in W^{1, 2}$. Our manipulations of the series below require significantly more regularity; for instance, the highest order one needs $\gamma\in W^{4, 2}$.

We write

where the coefficients $c_p$ are given by

with projection $\hat\gamma$ defined via

Curvature arises by differentiating $\gamma$, which is the same as differentiating the Fourier decomposition. The basic relation is:

Lemma A.3. For $q\ge2$ we have

where $Q_q = (\partial_s^{q-1}\gamma)\overline{\partial_s\gamma}$. For the $q = 1$ case we have

Proof. We provide a sketch with the essential steps. First, observe that for $q\ge2$ we have

This is because $Q_{q} = ikQ_{q-1} + \partial_sQ_{q-1}$ (this equality can be proved by a straightforward 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 $q\ge2$ we combine this with (A.2) 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

Note that $\partial_s \mathcal{T} = 1 + k \mathcal{N}$ and $\partial_s \mathcal{N} = -k \mathcal{T}$. We calculate

In fact, $Q_2 = (\partial_s\gamma)(\overline{\partial_s\gamma}) = 1$. We continue to calculate:

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 (A.3)). Here are the next two:

This last identity can be used trivially to deduce the estimate

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

and

hold.

Proof. We calculate

Using the expressions for $q = 2$ and $q = 1$, this shows

as required. Now

which, using the expressions for $q = 4$ and $q = 2$, gives

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 the Cauchy-Schwartz inequality in $\ell^2(\mathbb{Z})$ to conclude essentially the upper estimate.

Lemma A.5. We have

Proof. We use the Cauchy-Schwartz inequality to estimate

Now Lemma A.4 implies

The expressions for $q = 5$ and $q = 6$ now imply

as required.

Now we finish the proof of the upper estimate. Since by hypothesis we have

Lemma A.5 implies

as required.