By the classical Li–Yau inequality, an immersion of a closed surface in with Willmore energy below has to be embedded. We discuss analogous results for curves in , involving Euler’s elastic energy and other possible curvature functionals. Additionally, we provide applications to associated gradient flows.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: 404870139
Funding statement: Marius Müller was supported by the LGFG Grant no. 1705 LGFG-E. Fabian Rupp is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Grant no. 404870139.
A Differential geometry in Sobolev spaces
In this section, we will review some standard results from elementary differential geometry in the setting of -curves.
Theorem A.1 (Fenchel’s theorem).
Let be an immersed curve. Then with equality if and only if γ is embedded and convex.
Lemma A.2 (Angle function).
For an immersed curve , there exists such that
We call θ an angle function for γ. Moreover, any two angle functions satisfying (A.1) can only differ by an integer multiple of .
The proof works exactly as in the case of smooth curves; see, for instance, [2, Lemma 2.2.5]. For the regularity of θ, we use local representations of θ. For instance, in the case , one has locally
Hence, follows from (A.2) and the chain rule for Sobolev functions. ∎
Definition A.3 (Winding number).
Let be an immersion with corresponding angle function . We define the winding number of γ as . Note that does not depend on the choice of θ and is always an integer.
Let be an immersed curve. Then
Let be an angle function for γ. Differentiating (A.1) and using the chain rule for Sobolev functions and the definition of the unit normal, we obtain
Moreover, by the fundamental theorem of calculus for -functions, we find
Proposition A.5 (Hopf’s Umlaufsatz for -embeddings).
Suppose is an embedding. Then .
Let be a sequence of smooth curves with in . By Theorem A.4, we can easily see that . Since the set of embeddings is open in by Lemma 4.3, we see that is an embedding for large enough, and hence for all by Hopf’s Umlaufsatz for smooth curves. However, since the sequence converges, it has to be eventually constant, say for all . But then . ∎
Let be an immersion. Then there exists a constant speed reparametrization of γ such that .
This lemma follows from the arguments in [2, Proposition 2.1.13], using the Sobolev embedding
and the chain rule for Sobolev functions. ∎
B Jacobi elliptic functions and Euler’s elastica
B.1 Elliptic functions
We provide some elementary properties of Jacobi elliptic functions, which can be found, for example, in [1, Chapter 16].
Definition B.1 (Amplitude function, complete elliptic integrals).
Fix . We define the Jacobi-amplitude function with modulusm to be the inverse function of
We define the complete elliptic integral of first and second kind by
and the incomplete elliptic integral of first and second kind by
Note that .
Definition B.2 (Elliptic functions).
For , the Jacobi elliptic functions are given by
The following proposition summarizes all relevant properties and identities for the elliptic functions. They can all be found in [1, Chapter 16].
(Derivatives and integrals of Jacobi elliptic functions.) For each and , we have
(Derivatives of complete elliptic integrals.) For , E and K are smooth and
(Trigonometric identities.) For each and , the Jacobi elliptic functions satisfy
(Periodicity.) All periods of the elliptic functions are given as follows, where and :
(Asymptotics of the complete elliptic integrals.)
B.2 Some computational lemmas involving elliptic functions
We will also need some more advanced identities for elliptic functions.
The map has a unique zero . Moreover, .
For we define
Note that f has the same zeroes as . By Proposition B.3, one has
and hence there has to exist a zero of f. To show that it is unique, we show that f is strictly decreasing, which follows immediately from the following computation:
It remains to show that . Indeed,
which implies that . Hence, by the monotonicity of f, we find . ∎
The expression is strictly positive for all .
Note note that is positive if and only if the expression in the statement is positive and . Further note that
To show the claim it suffices to prove that . To do so, it suffices to show that
We have already shown in (B.1) that
where the last inequality was obtained by estimating the square with zero. We will show that this estimate is always with strict inequality, i.e.
First, note again that
Now an easy computation yields
Therefore, we obtain that
Hence (B.2) is shown, and thus
By definition of f, we obtain . ∎
Let be the unique zero in Lemma B.4. Then the map
has exactly four zeroes in , namely , , and .
Let be the smooth function defined by . We show first that . Indeed, by Proposition B.3, one has for all ,
Next, we show that has four zeroes in . Indeed,
which is zero if and only if , which happens exactly four times in since by Lemma B.4. Assume now that there exists some apart from such that . We can now sort the set
with . Since
by the mean value theorem, for all there exists some such that . This however is a contradiction to the fact that has only 4 zeroes. As a consequence, there exists no as in the assumption. The claim follows. ∎
Lemma B.7 (cf. [21, Proposition B.5]).
For all , one has
The proof follows from [21, Proposition B.5]. Be aware that the authors there use a different notation of ; their definition of is actually in our notation. ∎
B.3 Explicit parametrization of Euler’s elasticae
In the sequel, we shall prove the following classification result.
Let be an interval and let be a smooth solution of (5.1) for some . Then, up to rescaling, reparametrization, and isometries of , γ is given by one of the following elastic prototypes:
(Linear elastica.) γ is a line, .
(Wavelike elastica.) There exists such that
(Orbitlike elastica.) There exists such that
(Circular elastica.) γ is a circle.
We give a proof in the rest of this section. Suppose that γ is parametrized by arc-length. We know that κ satisfies (5.1). The solutions of this equation are discussed explicitly [17, Proposition 3.3]:
(Constant curvature.) κ is constant.
In this case, .
In this case, .
In this case, .
We have to mention that in [17, Proposition 3.3] the solutions are described with a different parameter p instead of m. In our notation, there holds . In contrast to our list, the classification [17, Proposition 3.3] also allows a wider range of p, namely . However, it only distinguishes two cases – the wavelike case and the orbitlike case. The remaining cases in our list arise from the limit cases in [17, Proposition 3.3]. Indeed, the limit case in both the wavelike and the orbitlike case correspond to constant solutions. The limit case corresponds in both cases to the borderline elastica, as one can infer immediately from [1, (16.15.2) and (16.15.3)]. Once expressions for the curvature are known, we can find explicit parametrizations of all these elastica. Note that once we have parametrized the solutions for and ‘’ instead of ‘’, we can obtain all other solutions by reparametrization or reflection. Hence, we consider only the cases of ‘’ and .
for some . One can now compute that if γ is a solution of (B.3), then and for some constant . The last identity can be checked by taking the derivative of the expression and using the first line of (B.3). Following the lines of [4, Proposition 6.1], we also obtain that . We are also free to assume that as (B.3) is not affected by adding a constant to . From now on, the parameters α and m will be our main parameters. We will express in terms of them and use (B.3) to obtain an explicit parametrization.
Case 1: Constant curvature. This yields either lines or circles.
Case 2: Wavelike elastica. First, we show that and . Note that by point (ii) of the list of possible curvatures and , we have that . In particular, since , we have
and since , we obtain
Hence , which implies by (B.4) that
We can moreover improve the formula for μ to . Moreover, using in (B.5), we find
By integrating and by using , we obtain
Hence for fixed , one has , where is given by
Case 3: Orbitlike elastica. We proceed as in the wavelike case. We first show that and . From point (iii) in the list of curvatures and , we infer that . This leads to
and, by Proposition B.3,
Therefore, we find that , and from this follows that . We infer
Integrating, we obtain
We infer that for fixed one has that , where is given by
Case 4: Borderline elastica. One can proceed exactly as in the first two cases and obtain that for fixed one has that , where
Both authors would like to thank Anna Dall’Acqua for helpful discussions.
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