A new energy method for shortening and straightening complete curves
-
Tatsuya Miura
and
Fabian Rupp
Abstract
We introduce a novel energy method that reinterprets “curve shortening” as “tangent aligning”. This conceptual shift enables the variational study of infinite-length curves evolving by the curve shortening flow, as well as higher-order flows such as the elastic flow, which involves not only the curve shortening but also the curve straightening effect. For the curve shortening flow, we prove convergence to a straight line under mild assumptions on the ends of the initial curve. For the elastic flow, we establish a global well-posedness theory, and investigate the precise long-time behavior of solutions. In fact, our method applies to a more general class of geometric evolution equations including the surface diffusion flow, Chen’s flow, and the free elastic flow.
1 Introduction
The curve shortening flow is one of the most classical geometric flows (see, e.g., [14, 4]), defined by a one-parameter family of immersed curves
where
where
Identifying gradient flow structures is a fundamental technique in the analysis of parabolic PDEs. Such an underlying variational structure is however not unique – the same equation can be a gradient flow for different energies, possibly with respect to different metrics, as observed, for instance, in the seminal work of Jordan–Kinderlehrer–Otto [32]. In this spirit, this work develops a conceptually new energy method for the curve shortening flow, based on the direction energy; see (2.1) below. Our approach opens the possibility to apply energy methods to study the long-time behavior of curve-shortening-type flows for non-compact complete curves, avoiding the use of maximum principles, and thus provides a reliable and robust tool also in higher codimensions.
We first apply this method to the curve shortening flow. While the well-established theory based on maximum principles, monotonicity properties, and the classification of solitons provides some global existence results even for curves of infinite length, the asymptotic analysis remains challenging. As a direct application of our energy method, we obtain a new result on the convergence of the curve shortening flow to a straight line. Notably, this result applies to initial curves that are neither necessarily planar nor confined to a slab, which are typically difficult to handle using maximum principles.
More importantly, our method is directly applicable to higher-order flows, due to its independence from maximum principles. For higher-order flows, various energy methods have been well developed in the compact case, but the non-compact case remains significantly less explored due to the absence of a canonical finite energy.
Here we apply our direction energy method to a class of fourth-order flows of curve shortening type, including the classical surface diffusion flow [50],
as well as Chen’s flow introduced in [9],
to obtain results analogous to those for the curve shortening flow.
Here
Besides the direction energy method, we also develop a new technical tool for controlling higher-order derivatives, extending the interpolation method of Dziuk–Kuwert–Schätzle [20] from the compact to the non-compact setting. This tool is not only necessary for the analysis of curve-shortening-type flows but is particularly well-suited for analyzing curve-straightening-type flows, which decrease quantities involving the bending energy
In this paper, we also address the
which was introduced in [58, 20] and has since been extensively studied in the compact case (see also the survey [42]).
We establish a foundational global existence theory for this flow in the non-compact case and then investigate the precise long-time behavior of solutions.
In the purely straightening case
Our results also yield new rigidity theorems for solitons as direct corollaries.
This paper is organized as follows. In Section 2, we provide a more detailed explanation of our main results described above, while comparing them with previous works. Section 3 presents preliminaries, particularly basic properties of the direction energy. Section 4 provides the key interpretation of the flows as gradient flows involving the direction energy. In Section 5, we prove the crucial weighted interpolation estimate, and then in Section 6 apply it to deduce global curvature bounds of any order along the flows. Using these estimates, in Section 7, we obtain the main blow-up and convergence results for each flow. Finally, Section 8 discusses rigidity results for solitons. The paper is complemented by an appendix which includes a discussion of local well-posedness in the non-compact case (Appendix A).
2 Main results
2.1 Direction energy and interpolation method
For a curve 𝛾 immersed in
where, here and in the sequel,
and hence, if
In particular, if 𝛾 is closed, then we simply have
However, an important observation is that the integrand
Besides its conceptual interest, this idea is particularly useful in the case of non-compact complete curves, since
Remark 2.1 (Admissible ends)
The class of curves with finite direction energy comprises complete curves whose ends suitably converge to two parallel semi-lines with opposite directions. Not only that, this class also allows for slowly diverging ends with or without oscillation (Figure 1), such as the graphs of
An immersed planar curve with admissible ends.
The key idea of the present work is to replace 𝐿 with 𝐷 to produce finite monotone quantities even for non-compact curves. This idea is inspired by previous works by the first author [44] and in collaboration with Wheeler [48] on the analysis of elastica, a stationary problem involving 𝐵 and 𝐿. In those papers, the direction energy is used to variationally handle the so-called borderline elastica (Figure 2), which has infinite length and horizontal ends. Our work here is the first to apply this idea to gradient flows.
The borderline elastica, parametrized as in (7.26).
After showing the energy decay, we need to handle higher-order quantities through interpolation estimates. In [20], Dziuk–Kuwert–Schätzle established the key Gagliardo–Nirenberg-type interpolation inequalities to apply energy-type methods to evolutions of closed curves. This has since been applied to a variety of flows in the compact case (even of more than fourth order, e.g., [56, 5, 38]), but does not directly extend to the non-compact case, since the inequalities involve the total length – which is infinite here. In this paper, we develop, for the first time, a variant of this interpolation method which is suitable to treat the non-compact situation. To that end, inspired by the localization procedure in [36], we first derive an interpolation estimate with a power-type weight function (Proposition 5.7). This approach is necessary, since we do not assume any a priori integrability and it is of particular importance to keep track of the relation between the exponents and the order of derivatives. We then establish higher-order estimates under localization, which are finally globalized in both time and space by a time cutoff argument in the spirit of Kuwert–Schätzle’s work [35] (Section 6).
In what follows, we discuss concrete geometric flows. Note that we only consider classical solutions in this work. However, using parabolic smoothing, our results easily transfer to appropriate weak solutions, e.g., in suitable Sobolev classes.
2.2 Curve shortening flow
The celebrated works of Gage–Hamilton [26] and Grayson [28] proved that the curve shortening flow of closed planar embedded curves must shrink to a round point in finite time.
The case of non-compact complete curves is more delicate. Following Polden [57] and Huisken [31], Chou–Zhu [13] established global-in-time existence for a wide class of complete embedded planar initial curves. Polden [57] and Huisken [31, Theorem 2.5] also proved convergence to self-similar solutions, assuming that the ends of the initial curve are asymptotic to semi-lines. The limit curve depends on the angle formed by the asymptotic semi-lines; in particular, if the angle is 𝜋, then the solution locally smoothly converges to a straight line parallel to the asymptotic semi-lines. Their proof relies on the fact that the solution remains inside an initial slab (see also Remark 7.7). Recently, Choi–Choi–Daskalopoulos [11] obtained a refined convergence to the grim reaper under convexity. See also [51, 52, 65, 66] for the planar stability of lines.
The above works strongly rely on several types of maximum principles, thus are based on planarity, since, in higher codimensions, some important maximum principles are not available; for example, embeddedness may be lost [2].
This makes the analysis much more complicated, and indeed, the authors are not aware of any relevant convergence results for infinite-length curves if
Our main result here extends Polden and Huisken’s result on convergence to the straight line, not only to more general ends (possibly not contained in any slab) but also to general codimensions by a purely energy-based approach.
We first formulate our result in the form of the following general dichotomy theorem.
Hereafter, we use the notation
Let
If
If
and moreover,
(2.2)for all
Here the translation is with respect to any base points; see (7.14) for the precise statement of convergence. The convergence of derivatives is global in space. In particular, this directly ensures that every global-in-time solution is eventually graphical; this is in stark contrast to the elastic flow discussed below. Even for the curve shortening flow, the curve itself may not converge uniformly (see Example 7.6). Also, translation is essentially needed, because the solution curve can keep oscillating or even escape to infinity (see Examples 7.5 and 7.6).
The
for all
Now we recall previous work providing sufficient conditions for global existence.
There are two typical cases: planar embedded curves, and (possibly non-planar) graphical curves.
We say that an immersed curve
Let
Proof
The planar embedded case directly follows by Huisken’s distance comparison argument for [31, Theorem 2.5] since
We can also give a new, simple blow-up criterion in the planar case.
2.3 Fourth-order flows of curve shortening type
As mentioned in the introduction, our argument also works even for a class of higher-order flows including the surface diffusion flow (SDF) and Chen’s flow (CF). These flows also decrease the length in the compact case, and some energy methods are already developed in previous work; see, e.g., [24, 20, 12, 67, 47] for the surface diffusion flow and [16] for Chen’s flow.
For these flows, we obtain quite analogous results to (CSF); see Theorem 7.8 for dichotomy and Corollary 7.9 for blow-up. To the authors’ knowledge, the latter is the first rigorous blow-up result for the non-compact surface diffusion as well as Chen’s flow. Although no results corresponding to Corollary 2.5 are known due to the lack of maximum principles, global existence holds at least for suitable small initial data (cf. [34]).
In addition, the above results directly imply new classification results for solitons to (CSF), (SDF), and (CF) in a unified manner.
For (CSF), comprehensive classification results for solitons have been obtained in
2.4 Fourth-order flows of curve straightening type
Now we turn to flows involving the curve straightening effect, namely (𝜆-EF) with
We first address the case
which is also equivalent to the Willmore flow with initial surfaces
Let
In addition,
holds for all
Finally, we discuss the case
There are a number of known results about global existence and convergence in the compact case; see, e.g., [58, 20, 18, 17, 27, 39, 43, 54, 53, 19, 40].
To the authors’ knowledge, the only known result for the infinite-length elastic flow is a pioneering work by Novaga–Okabe [53].
Roughly speaking, they show that if the initial curve is planar and if the ends are asymptotic to the
In this paper, we apply our energy method to improve the result of Novaga–Okabe in several aspects: we prove uniqueness, extend the analysis to general codimension, cover more general ends (possibly unbounded), and ensure the horizontality of the convergence limits. To this end, we apply our direction energy method and regard the elastic flow as the gradient flow of the adapted energy
The finiteness of 𝐸 is equivalent to having graphical ends 𝑢 with
Our main result can be summarized as follows.
Theorem 2.8 (Theorems 7.10–7.12)
Let
In addition, after any reparametrization by arclength and some translation, the solution sub-converges to an elastica
In stark contrast to the previous flows, global-in-space convergence does not hold, in general. Indeed, we can construct a peculiar example with loops “escaping” to infinity (see Example 7.17 and Figure 3).
Two loops moving apart along the elastic flow.
Furthermore, we find several conditions for initial data such that we can identify the type of the limit elastica, which in particular implies full (locally smooth) convergence without taking a time subsequence. Among other results (see Corollary 7.15), we obtain the optimal energy threshold below which the same convergence holds as in the case of the curve shortening flow.
Let
Then the elastic flow (EF) starting from
for all
The energy threshold is optimal.
Indeed, a borderline elastica
We finally mention that, although the curve shortening flow preserves many positivity properties such as planar embeddedness or graphicality (cf. Corollary 2.5), a wide class of higher-order flows does not due to the lack of maximum principles [10, 25]. Recently, optimal energy thresholds for preserving embeddedness of elastic flows of closed curves have been obtained in [46]. Our new energy method is also useful in this direction; we can obtain optimal thresholds in terms of 𝐸 for preserving planar embeddedness (resp. graphicality) in the non-compact case. This will be addressed in our future work.
3 Preliminaries
3.1 Notation
Let 𝐍 denote the set of positive integers and
where
In similar fashion, we define
If the codomain is clear from the context, we also just write
3.2 Basic properties of the direction energy
The bending energy 𝐵 is invariant with respect to isometries of
Let
and hence 𝛾 is proper.
If in addition
Proof
The first assertion follows by taking the limits
since the right-hand side is bounded thanks to the assumption
The second assertion follows since the integrability of
Let
We have
of
We have
Proof
We first prove the “only if” part of (i).
Suppose
Since
with
Now we turn to (ii).
Suppose
This implies the “only if” part of (ii).
The “if” part is again easier since
4 Gradient flow structures
In this section, we obtain key energy-decay properties for the flows under consideration, through a cutoff argument.
From this section onward, we will primarily use
Finally, we define the norm
which we may also express as
Throughout this section, we use 𝜉 and 𝑉 to denote the tangential and normal velocity of a flow 𝛾, respectively; namely,
4.1 Evolution of localized energy functionals
We first compute general time-derivative formulae for the direction and bending energy.
Let
Proof
Standard geometric evolution formulae (cf. [20, Lemma 2.1]) yield
We thus compute
Now we differentiate
After rearranging,
Integrating by parts in the first and second term of the right-hand side, and in particular using
Let
Then
Proof
In addition to (4.1), recalling from [20, Lemma 2.1] that
we argue along the lines of the proof of Lemma 4.1 to obtain
Integrating by parts in the first term yields
so that, after some rearranging,
Integration by parts for the first term yields the assertion. ∎
4.2 Energy identity
Now we turn to the proof of the energy identity.
We first address the flows having the structure of an
Let
such that
where
Proof
Suppose
Take a cutoff function
let
and also
Using (4.3), we can bound the integrands on the right-hand side of (4.4) by
Since this is bounded by 𝐶 thanks to (4.5) and (4.6), we conclude
Integrating over
In particular, it follows that
We have
Therefore, integrating (4.4) in time and sending
which converges to zero thanks to (4.5), (4.6), (4.7), and (4.8).
Finally, if
Our method also works for gradient flows which are not of
Let
for some constant
If
Proof
We use Lemma 4.1 with the same cutoff function
where we again used
which implies
Integrating (4.9) over
Using Hölder’s inequality and the fact that, by (4.5), for any
as
5 Localized interpolation estimates
The goal of this section is to prove the fundamental Gagliardo–Nirenberg-type estimate, Proposition 5.7 below.
Throughout this section, we fix a smooth immersion
where the arclength derivative is along 𝛾, i.e.,
We begin with establishing
Let
Proof
By integration by parts and Young’s inequality,
The claim follows after absorbing and using
Let
such that, for any smooth vector field 𝑋 normal along 𝛾 and any
Proof
We proceed by induction on 𝑘.
First, for
Suppose the statement is true for
is a constant that is allowed to change from line to line.
We first consider the case
for all
for any
plugging in (5.2) and using the choice of 𝛿. We now take
and observe that
It remains to consider the case
Taking
Since
We now establish a multiplicative version. For 𝑋 normal along 𝛾, we define the seminorms (using the weight 𝜁)
Let
Proof
Without loss of generality, we may assume
If
the statement follows. ∎
This will enable us to estimate nonlinearities arising in higher-order energy evolutions.
In order to apply
Let
for
Defining, for
with the interpretation
By taking 𝛾 in Lemma 5.4 to be an arclength parametrization of a straight line in
Let
for
Proof of Lemma 5.4
If
This yields the statement for
The statement follows. ∎
Following the notation in [18] (cf. [20]) for
Let
with
Proof
If suffices to consider the case
We first prove
We may assume
For proving (5.4), we first assume
we obtain
We now estimate the factors in (5.5) by using the Gagliardo–Nirenberg-type inequality.
Indeed, Lemma 5.4 with
where
using that
Noting that
On the other hand, suppose
where
As
Lastly, we estimate the first term of the right-hand side of (5.4) using Young’s inequality and
Using the same argument with 𝛿 replaced by
where we used that
6 Global curvature control
In this section, we prove global curvature estimates for a flow with uniformly bounded bending energy 𝐵. Having established the key interpolation estimate, Proposition 5.7, we now consider a rather general evolution law and present a unified approach to treat all the flows we consider in this work simultaneously by a combination of the strategies in [20, 35].
6.1 Evolution of localized curvature integrals
We consider a general class of geometric flows in this section, including (CSF), (SDF), (CF), (𝜆-EF), i.e., we assume that
is a smooth family of proper immersions, satisfying
Here
Let
Proof
We first show that the derivatives of the curvature satisfy
Indeed, for
For
so (6.2) follows by induction on
We now compute the time derivative of the localized integral.
Note that, since
By (6.2), the leading order term arising from the first term satisfies
using integration by parts twice. Similarly, by integration by parts,
Moreover, for the second term, we have
Using that
Inserting (6.2) into (6.3), using (6.4), (6.5), (6.6), (6.7), and noting that the terms of the form
6.2 Curvature control for fourth-order flows
Now fix
satisfies
In particular, 𝜁 satisfies the assumptions in (5.1), so that Proposition 5.7 is applicable. Then, for the flow in (6.1), we obtain the following key energy estimate.
Let
holds for all
Proof
We estimate the terms on the right-hand side of the identity in Lemma 6.1 for
where the sum runs over the finitely many terms with
where
the statement follows. ∎
A smooth initial datum with finite bending energy 𝐵 only needs to satisfy
Let
Proof
Fix any
where
We further define
Now Lemma 6.2, (6.10), and
We now show that, for all
We prove (6.12) by induction on 𝑗.
For
Let
where we also used the induction hypothesis in the last step.
This proves (6.12).
Evaluating (6.12) at
To get the same for
Absorbing, evaluating at
Renaming
Let
Proof
We first prove (6.15).
For any
and such that
If
Suppose
For 𝑅 sufficiently large, Lemma 6.2 and Gronwall’s inequality imply
using (6.17) and
For (6.16), by Lemma 5.4 (with
where we used (6.15) in the last step.
Sending
We next establish bounds on the derivatives
Under the assumptions of Lemma 6.4, for
Proof
We recall from [20, Lemma 2.6] that
Here
Estimate (6.18) thus follows by induction from (6.16) and (6.20).
For (6.19), one may also proceed by induction using (6.15), (6.18), and (6.20), as well as
6.3 Curvature control for the curve shortening flow
We now consider the case
Let
holds for all
Proof
For this choice of parameters, Lemma 6.1 with
where
Applying Proposition 5.7 (also with
yields the statement. ∎
We have the following analogue of Lemma 6.3.
Let
Proof
We proceed exactly as in the proof of Lemma 6.3, redefining
where
Arguing exactly as in Lemmas 6.4 and 6.5, we obtain the following.
Let
7 Long-time behavior
In this section, we discuss the long-time behavior, proving the main theorems in Section 2. Throughout this section, we will repeatedly use, without explicitly mentioning, the fact that, under the finiteness of the direction energy, we always have properness (Lemma 3.1) and hence we can use the results from Section 6.
7.1 Blow-up rates
Consider a general flow (6.1) with
Of course, not all flows as in (6.1) will develop a singularity. However, our curvature estimates enable us to characterize the behavior of the bending energy whenever this happens in finite time.
Suppose
To prove this, we first show the continuity of the bending energy along the flow.
Let
Proof
Let
Estimating
and rearranging, we deduce
Using that the derivatives of 𝛾 are uniformly bounded by
Dropping the term involving
and sending
We now integrate (7.1) in time over
where
On the other hand, integrating (7.3) on
which implies that
Proof of Theorem 7.1
We first note that (although the finiteness of the bending energy does not imply properness), since here we assume the properness of
By Lemma 7.2, we have
Indeed, if this was not the case, then there exists
From (6.1), we conclude that, for
and further deduce
In addition, by the differential equation
Differentiation and induction as in the proof of [20, Theorem 3.1] yield that, for
Combining the above estimates, we finally deduce that, for all
For any
This together with (7.5) implies that
as
Hence there exist
where the sum is finite.
The indices satisfy
where
Integrating and taking
In the case of the curve shortening flow (
Let
As in the fourth-order case, we first show the continuity of
Let
Proof
Using Lemma 4.2 with
Let
Dropping the second term on the left-hand side of (7.8) and using that the derivatives of 𝛾 are bounded by
Integrating over
At this moment,
Now going back to (7.9), using Gronwall’s lemma for the smooth function
This allows us to use the
and in particular the desired Lipschitz continuity
Proof of Lemma 7.3
We first establish an integral inequality for the bending energy (independent of the finiteness of 𝑇).
With
for some universal
Now, suppose that
Hereafter, we examine precise convergence properties for the flows discussed in Section 2.
Since in general the parametrization speed
For a smooth family of immersions
we define the canonical reparametrization
where
7.2 Convergence for the curve shortening flow
We first discuss convergence for the curve shortening flow, completing the proof of Theorem 2.2.
Proof of Theorem 2.2
We have already shown the blow-up rate (Lemma 7.3) and the decreasing property of the direction energy (Proposition 4.3).
Hereafter, we show the remaining convergence properties, so suppose that
We first prove that
and thus there exists
For smooth convergence, Lemma 7.4 and the above convergence
locally smoothly converges to an arclength parametrized curve
By Fatou’s lemma and
so the line is uniquely given by
Moreover, since
In order to complement our result, we mention two nontrivial examples. Both are given in the framework of planar graphs, and thus we use the fact that a planar graph remains so under the curve shortening flow.
Example 7.5 (Oscillation)
There exists a graphical planar initial curve
and
This follows by Nara–Taniguchi’s result [52, Theorem 1.4] on a certain equivalence of the asymptotic behaviors of solutions to the heat equation and to the graphical curve shortening equation, combined with oscillating solutions to the heat equation by Collet–Eckmann [15, Lemma 8.6].
The function
In this case, it is also easy to show that
Example 7.6 (Escaping to infinity)
There exists an unbounded graphical planar initial curve
In fact, we may take any embedded curve
For the first assertion, given any
For the second assertion, let
Remark 7.7 (Convergence proof using maximum principles)
Here, for comparison purposes, we sketch how maximum principles ensure convergence of the planar curve shortening flow to a line.
Consider an initial planar embedded curve with ends
7.3 Convergence for the surface diffusion and Chen’s flow
We now turn to fourth-order flows of curve shortening type including (SDF) and (CF), and obtain an analogous result to Theorem 2.2.
Let
with initial datum
If
If
and moreover,
for all
Proof
The decay property for 𝐷 is already shown in Proposition 4.4.
Case i is a direct consequence of Theorem 7.1.
We prove case ii, so suppose that
then the remaining argument is completely parallel to the proof of Theorem 2.2 ii, where we use Lemma 7.2 instead of Lemma 7.4, as well as Lemma 6.5 instead of Lemma 6.8.
We prove (7.16).
By Proposition 4.4, we have
Moreover, since 𝜂 has compact support, we have
Combining (7.17) and (7.18), and using Hölder, we find
For almost all
As
Analogously to Corollary 2.6, we obtain the following result by the same proof.
7.4 Global existence and convergence for the 𝜆-elastic flow
Now we consider the 𝜆-elastic flow (𝜆-EF), which may be regarded as a gradient flow of
Let
Proof
By Theorem 7.1, it suffices to show that the unique maximal solution
from
We now prove that any global solution to the 𝜆-elastic flow converges to a stationary solution, thus being an elastica.
Recall that a curve is called an elastica if it solves the equation
Let
For any sequence of times
The above limit satisfies the lower semi-continuity
Proof
By Proposition 4.3, we have
In particular,
We now prove property i.
Fix any sequences
Indeed, by Fatou’s lemma, this would then imply
Taking any cutoff function
Integrating (7.25), taking absolute values, sending
for all
Property ii then easily follows from (7.21), the invariance of the energy
We now examine the precise convergence behavior of the elastic flow with
Proof of Theorem 2.7
Thanks to Theorems 7.10 and 7.11, we only need to discuss the asymptotic behavior.
By Theorem 7.11 ii, the free elastica
From now on, we discuss the case
In this case, the stationary solution may not be a line even in the finite energy regime – it may be a so-called borderline elastica. We recall basic facts on the (planar) borderline elastica. A prototypical arclength parametrization is given by
cf.
Figure 2.
The tangential angle of
which increases from 0 to
Let
The limit curve
for some
so that
If
If
Proof
Property i is shown by using the known classification of elasticae [37, 45] as follows.
The only infinite-length elasticae with bounded bending energy are a straight line and a borderline elastica, because otherwise the squared curvature
Property ii follows from smooth convergence and the fact that the tangent direction of our borderline elastica is never rightward, i.e.,
Finally, property iii follows by direct computation; cf. (7.27). ∎
In the elastic flow case
On the other hand, it is not clear whether the limit also depends on the choice of the time sequence
Next we give several conditions on initial data for full convergence as
Proof of Corollary 2.10
Theorem 7.11 ii and Theorem 7.12 i, iii imply that the only limit
We also exhibit some conditions on initial data such that, for specific choices of base points, we obtain full convergence results, locally in space.
Let
If
Suppose that
Suppose that
Proof
We first prove case i.
By unique solvability, the point symmetry of the initial curve is preserved under the flow.
Hence, in particular,
We prove case ii.
By symmetry,
We finally prove case iii.
We only discuss the case
Then, for each
Novaga–Okabe [53, Theorem 3.3] also claimed convergence to a borderline elastica assuming nonzero rotation number, but their argument seems not ruling out the possibility that the limit curve is a non-rightward straight line. Our argument not only improves their result, but also clearly rules out non-rightward lines using the finiteness of the direction energy.
The above results already reveal interesting dynamical aspects of the non-compact elastic flow. In particular, we have the following peculiar example.
Example 7.17 (Escaping loops)
Consider the planar elastic flow 𝛾 starting from a planar initial curve
Then Corollary 7.15 ii implies that, around the origin, the curve
In addition, note that, in this case,
locally converges to a borderline elastica
In view of the compatibility of the above local convergences, we need to have
We also observe that the above solution directly shows that we cannot have global-in-space convergence.
Indeed, now we have
Motivated by the above observations, we finally pose a conjecture on the general behavior of planar elastic flows.
For a general planar curve
we can obtain the general lower bound of the energy 𝐸 in terms of the absolute rotation number (in the spirit of the phase transition aspect [44]),
Therefore, any planar elastic flow with
In fact, we conjecture that the energy of a general solution will be asymptotically “quantized” by the number of loops captured in Corollary 7.15 iii in the sense that equality is attained in (7.29).
If
In view of (7.27), equality in (7.28) is attained for the borderline elastica
8 Rigidity theorems for solitons
In this final section, we observe that simple applications of our convergence as well as energy decay results yield new rigidity theorems for solitons with finite direction energy.
Let
We first address the flows of curve shortening type.
Let
Proof
The expanding case is easily dealt with thanks to the energy decay results in Theorems 2.2 and 7.8, since an expanding dilation increases the direction energy.
In the non-scaling case, the convergence results there imply that
For (SDF), several non-stationary solitons are known in compact and non-compact cases, e.g., shrinking Bernoulli’s figure-eight [23] and a non-compact complete translator [55]. Also, in [34], an abstract existence theorem is obtained for expanding solitons of entire graphs with conical ends. See also [8, 33] for solitons with free boundary. For (CF), the only known non-stationary solitons would be the circles and the figure-eight [16]. It is not known whether there are shrinking solitons with finite direction energy for these flows.
As for the free elastic flow, we obtain a similar type of new rigidity result.
Let
Proof
The shrinking case can be treated by the energy decay in Theorem 2.7, while in the non-scaling case, the convergence result implies the assertion. ∎
Concerning expanding solitons to (FEF), the circles and the figure-eight are again explicit examples [49], and also the result in [34] implies existence of expanding entire graphs. It is unknown if the latter examples have finite bending energy.
Finally, we also obtain a new rigidity result for (EF), where we can rule out both expanding and shrinking solitons.
Let
Proof
The expanding case (with
For (CF), (FEF), and (EF), non-stationary non-scaling solitons (with infinite energy) are not known.
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: JP21H00990
Award Identifier / Grant number: JP23H00085
Award Identifier / Grant number: JP24K00532
Funding source: Austrian Science Fund
Award Identifier / Grant number: 10.55776/ESP557
Funding statement: The first author is supported by JSPS KAKENHI Grant Numbers JP21H00990, JP23H00085, JP24K00532. The second author is funded in whole, or in part, by the Austrian Science Fund (FWF), grant number 10.55776/ESP557. Part of this work was done when the second author was visiting Kyoto University supported by a Mobility Fellowship of the Strategic Partnership Program between the University of Vienna and Kyoto University.
A Local well-posedness
In this section, we discuss local well-posedness for a class of arbitrary order geometric flows of complete curves. The analogous results in the case of compact curves are standard. The usual parabolic Hölder space approach is however doomed to fail in the non-compact situation. For the convenience of the reader, we provide some details here.
A similar result for arbitrary order flows on compact manifolds has been proven by Mantegazza–Martinazzi [41].
For
Here
A.1 Transformation into a parabolic problem
Equation (A.1) is not parabolic due to its geometric invariance.
In the spirit of the “De Turck trick”, this issue can be overcome by adding a suitable tangential velocity to the evolution law.
We first make the following observation which is easily proven by induction on
For each
We now consider the operator of order
so that
We will see that sufficiently regular solutions to (A.2) can be reparametrized to give solutions of (A.1); see the proof of Theorem A.3 below. The leading order term on the right-hand side of (A.2) is now of the form
making (A.2) a quasilinear parabolic system of order
We now formulate our short-time well-posedness result using the notation of parabolic Hölder spaces
Let
and
which solves (A.2).
Moreover, there exist
the corresponding solution
Further, if even
This result can be established with classical tools from the theory of quasilinear parabolic systems.
Since it is not the main focus of this work, we will use Theorem A.2 without proof, but instead, we give a rough sketch of the main idea.
It can be proven by applying the contraction principle and parabolic maximal regularity in Hölder spaces, in particular [62, Theorem 4.10].
A subtlety is of course that the curves we consider are unbounded, thus naturally not in the parabolic Hölder spaces
A.2 Local solutions to the geometric problem
By construction, Theorem A.2 gives a flow with a particular tangential velocity. We will now construct a solution to (A.1). It is worth pointing out that a subtlety of our construction based on an ODE argument is that it does not preserve a finite degree of smoothness; see [63, 64]. This is why we focus on the smooth category in the sequel.
Let
and
The solution is given by a time-dependent reparametrization of the solution constructed in Theorem A.2, and it is unique in the class of solutions satisfying (A.6) and (A.7).
Moreover, if
Proof
Let 𝛾 be the solution of (A.2) constructed in Theorem A.2. Then, setting
we have
Local existence and uniqueness for fixed
we have
so that
using (A.8) and the transformation of the geometric objects.
We conclude that
For the uniqueness part, let 𝛾 be any solution to (A.1) satisfying (A.6) and (A.7).
By the definition of 𝒜, if
As before, the composition with Φ is only in the spatial variable. By Faà di Bruno’s formula,
is a polynomial in its arguments. We now consider the PDE
The structure of (A.10) is similar to (A.2), and thus the same techniques as for proving Theorem A.2 can be used to show that there exist
satisfying
so that, defining
where
and therefore coincides (on
Hence, if
where again we used the transformation of the geometric objects in the last step.
It follows
If
Lastly, the properness of
The finite degree of smoothness in Theorem A.2 is natural in view of the order of the PDE (A.2) and is enough for short-time existence of (A.1) if one allows for a tangential velocity in the flow.
The class
Moreover, it is worth pointing out that, even if an arclength parametrized initial curve has derivatives of all orders, it may not be possible to have a short-time existence result.
An instructive example for (CSF) is a locally convex curve with infinitely many loops such that the enclosed area converges to zero at infinity, which would however have unbounded curvature.
This is in stark contrast to the compact case, where pointwise smoothness automatically yields
B Technical lemmas
B.1 A general blow-up and convergence lemma
Let
If
(B.2)If
Proof
Using the inequality
To prove i, fix
For 𝑡 sufficiently close to 𝑇, we have
choosing
Taking
As above, we conclude that
Hence, taking
Renaming
We prove part ii.
By assumption, there are
Fix any
is strictly positive, i.e.,
On the other hand, by (B.1), we have
Combining the above estimates gives
This yields that
B.2 Rotation number
We define the rotation number for planar complete curves and prove its preservation along smooth flows.
For a planar curve
since
where
Let
such that
Proof
Since the range of 𝑁 is discrete, it is sufficient to show that, at each
Acknowledgements
The authors would like to thank Ben Andrews for discussions about the curve shortening flow.
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Articles in the same Issue
- Frontmatter
- A new energy method for shortening and straightening complete curves
- Exceptional Tannaka groups only arise from cubic threefolds
- Gradient estimates for the conductivity problem with imperfect bonding interfaces
- On the structure of singularities of weak mean curvature flows with mean curvature bounds
- Second adjointness and cuspidal supports at the categorical level
- Transverse surfaces and pseudo-Anosov flows
- On the minimal number of closed geodesics on positively curved Finsler spheres
- On the geometry of spaces of filtrations on local rings
Articles in the same Issue
- Frontmatter
- A new energy method for shortening and straightening complete curves
- Exceptional Tannaka groups only arise from cubic threefolds
- Gradient estimates for the conductivity problem with imperfect bonding interfaces
- On the structure of singularities of weak mean curvature flows with mean curvature bounds
- Second adjointness and cuspidal supports at the categorical level
- Transverse surfaces and pseudo-Anosov flows
- On the minimal number of closed geodesics on positively curved Finsler spheres
- On the geometry of spaces of filtrations on local rings
