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A Li–Yau inequality for the 1-dimensional Willmore energy

Marius Müller and Fabian Rupp ORCID logo

Abstract

By the classical Li–Yau inequality, an immersion of a closed surface in n with Willmore energy below 8π has to be embedded. We discuss analogous results for curves in 2, involving Euler’s elastic energy and other possible curvature functionals. Additionally, we provide applications to associated gradient flows.

MSC 2010: 53A04; 49Q10; 53E40

Communicated by Frank Duzaar


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 W2,2-curves.

Theorem A.1 (Fenchel’s theorem).

Let γC2(S1;R2) be an immersed curve. Then K(γ)2π with equality if and only if γ is embedded and convex.

Proof.

See [10, Satz 1]. An explicit characterization of the equality case can be deduced from [3, Theorem 3], for instance. ∎

Lemma A.2 (Angle function).

For an immersed curve γW2,2(S1;R2), there exists θW1,2((0,1);R) such that

(A.1)γ(x)|γ(x)|=(cosθ(x)sinθ(x)) for all x𝕊1.

We call θ an angle function for γ. Moreover, any two angle functions satisfying (A.1) can only differ by an integer multiple of 2π.

Proof.

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 γ1(x)>0, one has locally

(A.2)θ(x):=arctan(γ2(x)γ1(x))+2πfor some .

Hence, θW1,2(𝕊1;2) follows from (A.2) and the chain rule for Sobolev functions. ∎

Definition A.3 (Winding number).

Let γW2,2(𝕊1;2) be an immersion with corresponding angle function θW1,2((0,1);). We define the winding number of γ as T[γ]:=12π(θ(1)-θ(0)). Note that T[γ] does not depend on the choice of θ and is always an integer.

Proposition A.4.

Let γW2,2(S1;R2) be an immersed curve. Then

T[γ]=12π𝕊1κds.

Proof.

Let θW1,2((0,1);) 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

κn=κ=s2γ=θ|γ|(-sinθcosθ)=θ|γ|(-γ2γ1)=θ|γ|n.

Consequently,

(A.3)κ|γ|=θalmost everywhere.

Moreover, by the fundamental theorem of calculus for W1,2-functions, we find

T[γ]=12π(θ(1)-θ(0))=12π01θdx=12π01κ|γ|dx=12π𝕊1κds.

Proposition A.5 (Hopf’s Umlaufsatz for W2,2-embeddings).

Suppose γW2,2(S1;R2) is an embedding. Then T[γ]=±1.

Proof.

Let γ(n) be a sequence of smooth curves with γ(n)γ in W2,2(𝕊1;2). By Theorem A.4, we can easily see that T[γ(n)]T[γ]. Since the set of embeddings is open in C1(𝕊1;2) by Lemma 4.3, we see that γ(n) is an embedding for nN large enough, and hence T[γ(n)]=±1 for all nN by Hopf’s Umlaufsatz for smooth curves. However, since the sequence (T[γ(n)])n converges, it has to be eventually constant, say T[γ(n)]=τ{-1,1} for all nN. But then T[γ]=limnT[γ(n)]=τ{-1,1}. ∎

Lemma A.6.

Let γW2,2(S1;R2) be an immersion. Then there exists a constant speed reparametrization γ~ of γ such that γ~W2,2(S1;R2).

Proof.

This lemma follows from the arguments in [2, Proposition 2.1.13], using the Sobolev embedding

W2,2(𝕊1;2)C1(𝕊1;2)

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 m[0,1). We define the Jacobi-amplitude functionam(,m): with modulusm to be the inverse function of

z0z11-msin2(θ)dθ.

We define the complete elliptic integral of first and second kind by

K(m):=0π211-msin2(θ)dθ,E(m):=0π21-msin2(θ)dθ,

and the incomplete elliptic integral of first and second kind by

F(x,m):=0x11-msin2(θ)dθ,E(x,m):=0x1-msin2(θ)dθ.

Note that F(,m)=am(,m)-1.

Definition B.2 (Elliptic functions).

For m[0,1), the Jacobi elliptic functions are given by

cn(,m):,cn(x,m):=cos(am(x,m)),
sn(,m):,sn(x,m):=sin(am(x,m)),
dn(,m):,dn(x,m):=1-msin2(am(x,m)).

The following proposition summarizes all relevant properties and identities for the elliptic functions. They can all be found in [1, Chapter 16].

Proposition B.3.

  1. (i)

    (Derivatives and integrals of Jacobi elliptic functions.) For each x and m(0,1), we have

    xcn(x,m)=-sn(x,m)dn(x,m),
    xsn(x,m)=cn(x,m)dn(x,m),
    xdn(x,m)=-mcn(x,m)sn(x,m),
    xam(x,m)=dn(x,m).

  2. (ii)

    (Derivatives of complete elliptic integrals.) For m(0,1), E and K are smooth and

    ddmE(m)=E(m)-K(m)2m,ddmK(m)=(m-1)K(m)+E(m)2m(1-m).
  3. (iii)

    (Trigonometric identities.) For each m[0,1) and x, the Jacobi elliptic functions satisfy

    cn2(x,m)+sn2(x,m)=1,dn2(x,m)+msn2(x,m)=1.
  4. (iv)

    (Periodicity.) All periods of the elliptic functions are given as follows, where l and x:

    am(lK(m),m)=lπ2,
    cn(x+4lK(m),m)=cn(x,m),
    sn(x+4lK(m),m)=sn(x,m),
    dn(x+2lK(m),m)=dn(x,m),
    F(lπ2,m)=lK(m),
    E(lπ2,m)=lE(m),
    am(x+2lK(m),m)=lπ+am(x,m),
    F(x+lπ,m)=F(x,m)+2lK(m),
    E(x+lπ,m)=E(x,m)+2lE(m).

  5. (v)

    (Asymptotics of the complete elliptic integrals.)

    limm1K(m)=,
    limm0K(m)=π2,
    limm1E(m)=1,
    limm0E(m)=π2.

B.2 Some computational lemmas involving elliptic functions

We will also need some more advanced identities for elliptic functions.

Lemma B.4.

The map (0,1)m2E(m)-K(m) has a unique zero m*(0,1). Moreover, m*>12.

Proof.

For m(0,1) we define

f(m):=2E(m)K(m)-1.

Note that f has the same zeroes as m2E(m)-K(m). By Proposition B.3, one has

limm0f(m)=1,limm1f(m)=-1,

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:

ddmE(m)K(m)=1K(m)2(E(m)-K(m)2mK(m)-E(m)(m-1)K(m)+E(m)2m(1-m))
=12m(1-m)K(m)2(2(1-m)E(m)K(m)-(1-m)K(m)2-E(m)2)
=12m(1-m)K(m)2(2E(m)(1-m)K(m)-(1-m)2K(m)2-E(m)2)
+12m(1-m)K(m)2(((1-m)2-(1-m))K(m)2)
=12m(1-m)K(m)2(-(E(m)-(1-m)K(m))2-m(1-m)K(m)2)
(B.1)-12.

It remains to show that m*>12. Indeed,

2E(12)-K(12)=0π2(21-12sin2(θ)-11-12sin2(θ))dθ
=0π2cos2(θ)1-12sin2(θ)dθ>0,

which implies that f(12)>0. Hence, by the monotonicity of f, we find m*>12. ∎

Lemma B.5.

The expression 2E(m)-K(m)+mK(m) is strictly positive for all m(0,1).

Proof.

Let

f(m):=2E(m)K(m)-1+m.

Note note that f(m) is positive if and only if the expression in the statement is positive and K(m)>0. Further note that

limm1f(m)=0.

To show the claim it suffices to prove that f<0. To do so, it suffices to show that

ddmE(m)K(m)<-12for all m(0,1).

We have already shown in (B.1) that

ddmE(m)K(m)=12m(1-m)K(m)2(-(E(m)-(1-m)K(m))2-m(1-m)K(m)2)-12,

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.

(B.2)E(m)-(1-m)K(m)0for all m(0,1).

First, note again that

limm0(E(m)-(1-m)K(m))=0.

Now an easy computation yields

ddm(E(m)-(1-m)K(m))=12K(m)>0.

Therefore, we obtain that

E(m)-(1-m)K(m)>0for all m(0,1).

Hence (B.2) is shown, and thus

ddmE(m)K(m)<-12for all m(0,1).

By definition of f, we obtain f<0. ∎

Lemma B.6.

Let m* be the unique zero in Lemma B.4. Then the map

[0,2π)x2E(x,m*)-F(x,m*)

has exactly four zeroes in [0,2π), namely x1=0, x2=π2, x3=π and x4=3π2.

Proof.

Let f: be the smooth function defined by f(x):=2E(x,m*)-F(x,m*). We show first that f(0)=f(π2)=f(π)=f(3π2)=f(2π). Indeed, by Proposition B.3, one has for all l,

f(lπ2)=l(2E(m*)-K(m*))=0.

Next, we show that f has four zeroes in [0,2π]. Indeed,

f(x)=1-2m*sin2(x)1-m*sin2(x),

which is zero if and only if sin2(x)=12m*, which happens exactly four times in [0,2π] since m*>12 by Lemma B.4. Assume now that there exists some x0(0,2π) apart from 0,π2,π,3π2,2π such that f(x0)=0. We can now sort the set

{0,π2,π,3π2,2π,x0}={y1,y2,y3,y4,y5,y6}

with 0=y1<<y6=2π. Since

f(y1)==f(y6)=0,

by the mean value theorem, for all i{1,,5} there exists some zi(yi,yi+1) such that f(zi)=0. This however is a contradiction to the fact that f has only 4 zeroes. As a consequence, there exists no x0 as in the assumption. The claim follows. ∎

Lemma B.7 (cf. [21, Proposition B.5]).

For all m(0,1), one has

E(m)π222-m.

Proof.

The proof follows from [21, Proposition B.5]. Be aware that the authors there use a different notation of m=p2; their definition of E(p) is actually E(p2) in our notation. ∎

B.3 Explicit parametrization of Euler’s elasticae

In the sequel, we shall prove the following classification result.

Proposition B.8.

Let IR be an interval and let γ:IR be a smooth solution of (5.1) for some λR. Then, up to rescaling, reparametrization, and isometries of R2, γ is given by one of the following elastic prototypes:

  1. (i)

    (Linear elastica.) γ is a line, κ[γ]=0.

  2. (ii)

    (Wavelike elastica.) There exists m(0,1) such that

    γ(s)=(2E(am(s,m),m)-s-2mcn(s,m)).

    Moreover, κ[γ]=2mcn(s,m).

  3. (iii)

    (Borderline elastica.)

    γ(s)=(2tanh(s)-s-2sech(s)).

    Moreover, κ[γ]=2sech(s).

  4. (iv)

    (Orbitlike elastica.) There exists m(0,1) such that

    γ(s)=1m(2E(am(s,m),m)+(m-2)s-2dn(s,m))

    Moreover, κ[γ]=2dn(s,m).

  5. (v)

    (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]:

  1. (i)

    (Constant curvature.) κ is constant.

  2. (ii)

    (Wavelike elastica.)

    κ(s)=±2αmcn(α(s-s0),m)for some m[0,1),α>0,s0.

    In this case, λ=α2(2m-1).

  3. (iii)

    (Orbitlike elastica.)

    κ(s)=±2αdn(α(s-s0),m)for some m[0,1),α>0,s0.

    In this case, λ=α2(2-m).

  4. (iv)

    (Borderline elastica.)

    κ(s)=±2αsech(α(s-s0))for some α>0,s0.

    In this case, λ=α2.

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 m=p2(0,1). In contrast to our list, the classification [17, Proposition 3.3] also allows a wider range of p, namely p[0,1]. However, it only distinguishes two cases – the wavelike case and the orbitlike case. The remaining cases in our list arise from the limit cases p=0,1 in [17, Proposition 3.3]. Indeed, the limit case p=m=0 in both the wavelike and the orbitlike case correspond to constant solutions. The limit case p=m=1 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 s0=0 and ‘+’ instead of ‘±’, we can obtain all other solutions by reparametrization or reflection. Hence, we consider only the cases of ‘+’ and s0=0.

From [4, Proposition 6.1] it is known that each smooth solution γ:I2 of (5.1) with some parameter λ corresponds up to isometries of 2 and reparametrization to a solution of

(B.3){γ1′′=σγ2γ2,γ2′′=-σγ2γ1,γ12+γ22=1,

for some σ>0. One can now compute that if γ is a solution of (B.3), then κ=γ2′′γ1-γ1′′γ2=-σγ2 and γ1-σ2γ22μ for some constant μ. 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 λ=-σμ. We are also free to assume that γ1(0)=0 as (B.3) is not affected by adding a constant to γ1. From now on, the parameters α and m will be our main parameters. We will express σ,μ 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 σ=α2 and μ=1-2m. Note that by point (ii) of the list of possible curvatures and λ=-σμ, we have that μ=α2σ(1-2m). In particular, since κ=-σγ2, we have

(B.4)γ2(s)=-2σαmcn(αs,m)

and since γ1=σ2γ22+μ, we obtain

(B.5)γ1(s)=σ2γ2(s)2+α2σ(1-2m)=α2σ(2mcn2(αs,m)+1-2m).

Therefore, using (B.3) and Proposition B.3, we obtain

1=γ1(s)2+γ2(s)2
=α4σ2((2mcn2(αs,m)+1-2m)2+4msn2(αs,m)dn2(αs,m))
=α4σ2((1-2msn2(αs,m))2+4msn2(αs,m)(1-msn2(αs,m)))
=α4σ2

Hence σ=α2, which implies by (B.4) that

γ2(s)=-2αmcn(αs,m).

We can moreover improve the formula for μ to μ=1-2m. Moreover, using σ=α2 in (B.5), we find

γ1(s)=2mcn2(αs,m)+(1-2m)=1-2msn2(αs,m).

By integrating and by using γ1(0)=0, we obtain

γ1(s)=s-2m0ssn2(αs,m)ds
=s-2α0am(αs,m)msin2θ1-msin2θ𝑑θ
=s-2α0am(αs,m)(11-msin2θ-1-msin2θ)𝑑θ
=s-2αF(am(αs,m),m)+2αE(am(αs,m),m)
=1α(2E(am(αs,m),m)-αs).

Hence for fixed α>0, one has γ(s)=1αγwave(αs), where γwave is given by

γwave(s)=(2E(am(s,m),m)-s-2mcn(s,m)).

Case 3: Orbitlike elastica. We proceed as in the wavelike case. We first show that σ=α2m and μ=m-2m. From point (iii) in the list of curvatures and λ=-σμ, we infer that μ=α2(m-2)σ. This leads to

γ2(s)=-2σαdn(αs,m)

and, by Proposition B.3,

γ1(s)=σ2γ2(s)2+α2(m-2)σ=α2mσ(1-2sn2(αs,m)).

Using (B.3) and Proposition B.3 we obtain

1=γ1(s)2+γ2(s)2
=α4m2σ2((1-2sn2(αs,m))2+4sn2(αs,m)cn2(αs,m))
=α4m2σ2((1-2sn2(αs,m))2+4sn2(αs,m)(1-sn2(αs,m))
=α4m2σ2.

Therefore, we find that σ=α2m, and from this follows that μ=m-2m. We infer

γ2(s)=-2mαdn(αs,m)

and

γ1(s)=1-2sn2(αs,m).

Integrating, we obtain

γ1(s)=s-20ssn2(αs,m)ds=s(1-2m)+2αmE(am(αs,m),m).

We infer that for fixed α>0 one has that γ(s)=1αγorbit(αs), where γorbit is given by

γorbit=1m(2E(am(s,m),m)+(m-2)s-2dn(s,m)).

Case 4: Borderline elastica. One can proceed exactly as in the first two cases and obtain that for fixed α>0 one has that γ=1αγborder(αs), where

γborder(s)=(2tanh(s)-s-2sech(s)).

Acknowledgements

Both authors would like to thank Anna Dall’Acqua for helpful discussions.

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Received: 2021-02-05
Revised: 2021-04-29
Accepted: 2021-05-05
Published Online: 2021-07-28

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