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

Marius Müller; Fabian Rupp

doi:10.1515/acv-2021-0014

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

Marius Müller und Fabian Rupp

Veröffentlicht/Copyright: 28. Juli 2021

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Aus der Zeitschrift Advances in Calculus of Variations Band 16 Heft 2

Abstract

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

Keywords: Li–Yau inequality; Willmore functional; elastic energy; embeddedness

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 W 2 , 2 -curves.

Theorem A.1 (Fenchel’s theorem).

Let γ ∈ C 2 ⁢ ( S 1 ; R 2 ) be an immersed curve. Then K ⁢ ( γ ) ≥ 2 ⁢ π 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 γ ∈ W 2 , 2 ⁢ ( S 1 ; R 2 ) , there exists θ ∈ W 1 , 2 ⁢ ( ( 0 , 1 ) ; R ) such that

(A.1) γ ′ ⁢ ( x ) | γ ′ ⁢ ( x ) | = ( cos ⁡ θ ⁢ ( x ) sin ⁡ θ ⁢ ( x ) ) ⁢ for all ⁢ x ∈ 𝕊 1 .

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

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 γ 1 ′ ⁢ ( x ) > 0 , one has locally

(A.2) θ ⁢ ( x ) := arctan ⁡ ( γ 2 ′ ⁢ ( x ) γ 1 ′ ⁢ ( x ) ) + 2 ⁢ π ⁢ ℓ for some ⁢ ℓ ∈ ℤ .

Hence, θ ∈ W 1 , 2 ⁢ ( 𝕊 1 ; ℝ 2 ) follows from (A.2) and the chain rule for Sobolev functions. ∎

Definition A.3 (Winding number).

Let γ ∈ W 2 , 2 ⁢ ( 𝕊 1 ; ℝ 2 ) be an immersion with corresponding angle function θ ∈ W 1 , 2 ⁢ ( ( 0 , 1 ) ; ℝ ) . We define the winding number of γ as T ⁢ [ γ ] := 1 2 ⁢ π ⁢ ( 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 γ ∈ W 2 , 2 ⁢ ( S 1 ; R 2 ) be an immersed curve. Then

T ⁢ [ γ ] = 1 2 ⁢ π ⁢ ∫ 𝕊 1 κ ⁢ d s .

Proof.

Let θ ∈ W 1 , 2 ⁢ ( ( 0 , 1 ) ; ℝ ) 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

κ ⁢ n → = κ → = ∂ s 2 ⁡ γ = θ ′ | γ ′ | ⁢ ( - sin ⁡ θ cos ⁡ θ ) = θ ′ | γ ′ | ⁢ ( - γ 2 ′ γ 1 ′ ) = θ ′ | γ ′ | ⁢ n → .

Consequently,

(A.3) κ ⁢ | γ ′ | = θ ′ almost everywhere.

Moreover, by the fundamental theorem of calculus for W 1 , 2 -functions, we find

T ⁢ [ γ ] = 1 2 ⁢ π ⁢ ( θ ⁢ ( 1 ) - θ ⁢ ( 0 ) ) = 1 2 ⁢ π ⁢ ∫ 0 1 θ ′ ⁢ d x = 1 2 ⁢ π ⁢ ∫ 0 1 κ ⁢ | γ ′ | ⁢ d x = 1 2 ⁢ π ⁢ ∫ 𝕊 1 κ ⁢ d s . ∎

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

Suppose γ ∈ W 2 , 2 ⁢ ( S 1 ; R 2 ) is an embedding. Then T ⁢ [ γ ] = ± 1 .

Proof.

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

Lemma A.6.

Let γ ∈ W 2 , 2 ⁢ ( S 1 ; R 2 ) be an immersion. Then there exists a constant speed reparametrization γ ~ of γ such that γ ~ ∈ W 2 , 2 ⁢ ( S 1 ; R 2 ) .

Proof.

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

W 2 , 2 ⁢ ( 𝕊 1 ; ℝ 2 ) ↪ C 1 ⁢ ( 𝕊 1 ; ℝ 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 function am ⁢ ( ⋅ , m ) : ℝ → ℝ with modulus m to be the inverse function of

ℝ ∋ z ↦ ∫ 0 z 1 1 - m ⁢ sin 2 ⁡ ( θ ) ⁢ d θ ∈ ℝ .

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

K ⁢ ( m ) := ∫ 0 π 2 1 1 - m ⁢ sin 2 ⁡ ( θ ) ⁢ d θ , E ⁢ ( m ) := ∫ 0 π 2 1 - m ⁢ sin 2 ⁡ ( θ ) ⁢ d θ ,

and the incomplete elliptic integral of first and second kind by

F ⁢ ( x , m ) := ∫ 0 x 1 1 - m ⁢ sin 2 ⁡ ( θ ) ⁢ d θ , E ⁢ ( x , m ) := ∫ 0 x 1 - m ⁢ sin 2 ⁡ ( θ ) ⁢ d θ .

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 ) : ℝ → ℝ , ⁢ cn ⁢ ( x , m ) := cos ⁡ ( am ⁢ ( x , m ) ) ,

sn ⁢ ( ⋅ , m ) : ℝ → ℝ , ⁢ sn ⁢ ( x , m ) := sin ⁡ ( am ⁢ ( x , m ) ) ,

dn ⁢ ( ⋅ , m ) : ℝ → ℝ , ⁢ dn ⁢ ( x , m ) := 1 - m ⁢ sin 2 ⁡ ( 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.

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

∂ ∂ ⁡ x ⁢ cn ⁢ ( x , m ) = - sn ⁢ ( x , m ) ⁢ dn ⁢ ( x , m ) ,

∂ ∂ ⁡ x ⁢ sn ⁢ ( x , m ) = cn ⁢ ( x , m ) ⁢ dn ⁢ ( x , m ) ,

∂ ∂ ⁡ x ⁢ dn ⁢ ( x , m ) = - m ⁢ cn ⁢ ( x , m ) ⁢ sn ⁢ ( x , m ) ,

∂ ∂ ⁡ x ⁢ am ⁢ ( x , m ) = dn ⁢ ( x , m ) .
(Derivatives of complete elliptic integrals.) For m ∈ ( 0 , 1 ) , E and K are smooth and

d d ⁢ m ⁢ E ⁢ ( m ) = E ⁢ ( m ) - K ⁢ ( m ) 2 ⁢ m , d d ⁢ m ⁢ K ⁢ ( m ) = ( m - 1 ) ⁢ K ⁢ ( m ) + E ⁢ ( m ) 2 ⁢ m ⁢ ( 1 - m ) .
(Trigonometric identities.) For each m ∈ [ 0 , 1 ) and x ∈ ℝ , the Jacobi elliptic functions satisfy

cn 2 ⁢ ( x , m ) + sn 2 ⁢ ( x , m ) = 1 , dn 2 ⁢ ( x , m ) + m ⁢ sn 2 ⁢ ( x , m ) = 1 .
(Periodicity.) All periods of the elliptic functions are given as follows, where l ∈ ℤ and x ∈ ℝ :

am ⁢ ( l ⁢ K ⁢ ( m ) , m ) = l ⁢ π 2 ,

cn ⁢ ( x + 4 ⁢ l ⁢ K ⁢ ( m ) , m ) = cn ⁢ ( x , m ) ,

sn ⁢ ( x + 4 ⁢ l ⁢ K ⁢ ( m ) , m ) = sn ⁢ ( x , m ) ,

dn ⁢ ( x + 2 ⁢ l ⁢ K ⁢ ( m ) , m ) = dn ⁢ ( x , m ) ,

F ⁢ ( l ⁢ π 2 , m ) = l ⁢ K ⁢ ( m ) ,

E ⁢ ( l ⁢ π 2 , m ) = l ⁢ E ⁢ ( m ) ,

am ⁢ ( x + 2 ⁢ l ⁢ K ⁢ ( m ) , m ) = l ⁢ π + am ⁢ ( x , m ) ,

F ⁢ ( x + l ⁢ π , m ) = F ⁢ ( x , m ) + 2 ⁢ l ⁢ K ⁢ ( m ) ,

E ⁢ ( x + l ⁢ π , m ) = E ⁢ ( x , m ) + 2 ⁢ l ⁢ E ⁢ ( m ) .
(Asymptotics of the complete elliptic integrals.)

lim m → 1 ⁡ K ⁢ ( m ) = ∞ ,

lim m → 0 ⁡ K ⁢ ( m ) = π 2 ,

lim m → 1 ⁡ E ⁢ ( m ) = 1 ,

lim m → 0 ⁡ E ⁢ ( m ) = π 2 .

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 ) ∋ m ↦ 2 ⁢ E ⁢ ( m ) - K ⁢ ( m ) has a unique zero m * ∈ ( 0 , 1 ) . Moreover, m * > 1 2 .

Proof.

For m ∈ ( 0 , 1 ) we define

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

Note that f has the same zeroes as m ↦ 2 ⁢ E ⁢ ( m ) - K ⁢ ( m ) . By Proposition B.3, one has

lim m → 0 ⁡ f ⁢ ( m ) = 1 , lim m → 1 ⁡ f ⁢ ( 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:

d d ⁢ m ⁢ E ⁢ ( m ) K ⁢ ( m ) = 1 K ⁢ ( m ) 2 ⁢ ( E ⁢ ( m ) - K ⁢ ( m ) 2 ⁢ m ⁢ K ⁢ ( m ) - E ⁢ ( m ) ⁢ ( m - 1 ) ⁢ K ⁢ ( m ) + E ⁢ ( m ) 2 ⁢ m ⁢ ( 1 - m ) )

= 1 2 ⁢ m ⁢ ( 1 - m ) ⁢ K ⁢ ( m ) 2 ⁢ ( 2 ⁢ ( 1 - m ) ⁢ E ⁢ ( m ) ⁢ K ⁢ ( m ) - ( 1 - m ) ⁢ K ⁢ ( m ) 2 - E ⁢ ( m ) 2 )

= 1 2 ⁢ m ⁢ ( 1 - m ) ⁢ K ⁢ ( m ) 2 ⁢ ( 2 ⁢ E ⁢ ( m ) ⁢ ( 1 - m ) ⁢ K ⁢ ( m ) - ( 1 - m ) 2 ⁢ K ⁢ ( m ) 2 - E ⁢ ( m ) 2 )

+ 1 2 ⁢ m ⁢ ( 1 - m ) ⁢ K ⁢ ( m ) 2 ⁢ ( ( ( 1 - m ) 2 - ( 1 - m ) ) ⁢ K ⁢ ( m ) 2 )

= 1 2 ⁢ m ⁢ ( 1 - m ) ⁢ K ⁢ ( m ) 2 ⁢ ( - ( E ⁢ ( m ) - ( 1 - m ) ⁢ K ⁢ ( m ) ) 2 - m ⁢ ( 1 - m ) ⁢ K ⁢ ( m ) 2 )

(B.1) ≤ - 1 2 .

It remains to show that m * > 1 2 . Indeed,

2 ⁢ E ⁢ ( 1 2 ) - K ⁢ ( 1 2 ) = ∫ 0 π 2 ( 2 ⁢ 1 - 1 2 ⁢ sin 2 ⁡ ( θ ) - 1 1 - 1 2 ⁢ sin 2 ⁡ ( θ ) ) ⁢ d θ

= ∫ 0 π 2 cos 2 ⁡ ( θ ) 1 - 1 2 ⁢ sin 2 ⁡ ( θ ) ⁢ d θ > 0 ,

which implies that f ⁢ ( 1 2 ) > 0 . Hence, by the monotonicity of f, we find m * > 1 2 . ∎

Lemma B.5.

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

Proof.

Let

f ⁢ ( m ) := 2 ⁢ E ⁢ ( 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

lim m → 1 ⁡ f ⁢ ( m ) = 0 .

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

d d ⁢ m ⁢ E ⁢ ( m ) K ⁢ ( m ) < - 1 2 for all ⁢ m ∈ ( 0 , 1 ) .

We have already shown in (B.1) that

d d ⁢ m ⁢ E ⁢ ( m ) K ⁢ ( m ) = 1 2 ⁢ m ⁢ ( 1 - m ) ⁢ K ⁢ ( m ) 2 ⁢ ( - ( E ⁢ ( m ) - ( 1 - m ) ⁢ K ⁢ ( m ) ) 2 - m ⁢ ( 1 - m ) ⁢ K ⁢ ( m ) 2 ) ≤ - 1 2 ,

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 ) ≠ 0 for all ⁢ m ∈ ( 0 , 1 ) .

First, note again that

lim m → 0 ⁡ ( E ⁢ ( m ) - ( 1 - m ) ⁢ K ⁢ ( m ) ) = 0 .

Now an easy computation yields

d d ⁢ m ⁢ ( E ⁢ ( m ) - ( 1 - m ) ⁢ K ⁢ ( m ) ) = 1 2 ⁢ K ⁢ ( m ) > 0 .

Therefore, we obtain that

E ⁢ ( m ) - ( 1 - m ) ⁢ K ⁢ ( m ) > 0 for all ⁢ m ∈ ( 0 , 1 ) .

Hence (B.2) is shown, and thus

d d ⁢ m ⁢ E ⁢ ( m ) K ⁢ ( m ) < - 1 2 for 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 , 2 ⁢ π ) ∋ x ↦ 2 ⁢ E ⁢ ( x , m * ) - F ⁢ ( x , m * )

has exactly four zeroes in [ 0 , 2 ⁢ π ) , namely x 1 = 0 , x 2 = π 2 , x 3 = π and x 4 = 3 ⁢ π 2 .

Proof.

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

f ⁢ ( l ⁢ π 2 ) = l ⁢ ( 2 ⁢ E ⁢ ( m * ) - K ⁢ ( m * ) ) = 0 .

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

f ′ ⁢ ( x ) = 1 - 2 ⁢ m * ⁢ sin 2 ⁡ ( x ) 1 - m * ⁢ sin 2 ⁡ ( x ) ,

which is zero if and only if sin 2 ⁡ ( x ) = 1 2 ⁢ m * , which happens exactly four times in [ 0 , 2 ⁢ π ] since m * > 1 2 by Lemma B.4. Assume now that there exists some x 0 ∈ ( 0 , 2 ⁢ π ) apart from 0 , π 2 , π , 3 ⁢ π 2 , 2 ⁢ π such that f ⁢ ( x 0 ) = 0 . We can now sort the set

{ 0 , π 2 , π , 3 ⁢ π 2 , 2 ⁢ π , x 0 } = { y 1 , y 2 , y 3 , y 4 , y 5 , y 6 }

with 0 = y 1 < ⋯ < y 6 = 2 ⁢ π . Since

f ⁢ ( y 1 ) = ⋯ = f ⁢ ( y 6 ) = 0 ,

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

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

For all m ∈ ( 0 , 1 ) , one has

E ⁢ ( m ) ≤ π 2 ⁢ 2 ⁢ 2 - m .

Proof.

The proof follows from [21, Proposition B.5]. Be aware that the authors there use a different notation of m = p 2 ; their definition of E ⁢ ( p ) is actually E ⁢ ( p 2 ) 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 I ⊂ R be an interval and let γ : I → R be a smooth solution of (5.1) for some λ ∈ R . Then, up to rescaling, reparametrization, and isometries of R 2 , γ is given by one of the following elastic prototypes:

(Linear elastica.) γ is a line, κ ⁢ [ γ ] = 0 .
(Wavelike elastica.) There exists m ∈ ( 0 , 1 ) such that

γ ⁢ ( s ) = ( 2 ⁢ E ⁢ ( am ⁢ ( s , m ) , m ) - s - 2 ⁢ m ⁢ cn ⁢ ( s , m ) ) .

Moreover, κ ⁢ [ γ ] = 2 ⁢ m ⁢ cn ⁢ ( s , m ) .
(Borderline elastica.)

γ ⁢ ( s ) = ( 2 ⁢ tanh ⁡ ( s ) - s - 2 ⁢ sech ⁡ ( s ) ) .

Moreover, κ ⁢ [ γ ] = 2 ⁢ sech ⁡ ( s ) .
(Orbitlike elastica.) There exists m ∈ ( 0 , 1 ) such that

γ ⁢ ( s ) = 1 m ⁢ ( 2 ⁢ E ⁢ ( am ⁢ ( s , m ) , m ) + ( m - 2 ) ⁢ s - 2 ⁢ d ⁢ n ⁢ ( s , m ) )

Moreover, κ ⁢ [ γ ] = 2 ⁢ d ⁢ n ⁢ ( s , m ) .
(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.
(Wavelike elastica.)

κ ⁢ ( s ) = ± 2 ⁢ α ⁢ m ⁢ cn ⁢ ( α ⁢ ( s - s 0 ) , m ) for some ⁢ m ∈ [ 0 , 1 ) , α > 0 , s 0 ∈ ℝ .

In this case, λ = α 2 ⁢ ( 2 ⁢ m - 1 ) .
(Orbitlike elastica.)

κ ⁢ ( s ) = ± 2 ⁢ α ⁢ dn ⁢ ( α ⁢ ( s - s 0 ) , m ) for some ⁢ m ∈ [ 0 , 1 ) , α > 0 , s 0 ∈ ℝ .

In this case, λ = α 2 ⁢ ( 2 - m ) .
(Borderline elastica.)

κ ⁢ ( s ) = ± 2 ⁢ α ⁢ sech ⁡ ( α ⁢ ( s - s 0 ) ) for some ⁢ α > 0 , s 0 ∈ ℝ .

In this case, λ = α 2 .

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 = p 2 ∈ ( 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 s 0 = 0 and ‘ + ’ instead of ‘ ± ’, we can obtain all other solutions by reparametrization or reflection. Hence, we consider only the cases of ‘ + ’ and s 0 = 0 .

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

(B.3) { γ 1 ′′ = σ ⁢ γ 2 ⁢ γ 2 ′ , γ 2 ′′ = - σ ⁢ γ 2 ⁢ γ 1 ′ , γ 1 ′ ⁣ 2 + γ 2 ′ ⁣ 2 = 1 ,

for some σ > 0 . One can now compute that if γ is a solution of (B.3), then κ = γ 2 ′′ ⁢ γ 1 ′ - γ 1 ′′ ⁢ γ 2 ′ = - σ ⁢ γ 2 and γ 1 ′ - σ 2 ⁢ γ 2 2 ≡ μ for some constant μ ∈ ℝ . 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 = - σ ⁢ μ . We are also free to assume that γ 1 ⁢ ( 0 ) = 0 as (B.3) is not affected by adding a constant to γ 1 . 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 = α 2 and μ = 1 - 2 ⁢ m . Note that by point (ii) of the list of possible curvatures and λ = - σ ⁢ μ , we have that μ = α 2 σ ⁢ ( 1 - 2 ⁢ m ) . In particular, since κ = - σ ⁢ γ 2 , we have

(B.4) γ 2 ⁢ ( s ) = - 2 σ ⁢ α ⁢ m ⁢ cn ⁢ ( α ⁢ s , m )

and since γ 1 ′ = σ 2 ⁢ γ 2 2 + μ , we obtain

(B.5) γ 1 ′ ⁢ ( s ) = σ 2 ⁢ γ 2 ⁢ ( s ) 2 + α 2 σ ⁢ ( 1 - 2 ⁢ m ) = α 2 σ ⁢ ( 2 ⁢ m ⁢ cn 2 ⁢ ( α ⁢ s , m ) + 1 - 2 ⁢ m ) .

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

1 = γ 1 ′ ⁢ ( s ) 2 + γ 2 ′ ⁢ ( s ) 2

= α 4 σ 2 ⁢ ( ( 2 ⁢ m ⁢ cn 2 ⁢ ( α ⁢ s , m ) + 1 - 2 ⁢ m ) 2 + 4 ⁢ m ⁢ sn 2 ⁢ ( α ⁢ s , m ) ⁢ dn 2 ⁢ ( α ⁢ s , m ) )

= α 4 σ 2 ⁢ ( ( 1 - 2 ⁢ m ⁢ sn 2 ⁢ ( α ⁢ s , m ) ) 2 + 4 ⁢ m ⁢ sn 2 ⁢ ( α ⁢ s , m ) ⁢ ( 1 - m ⁢ sn 2 ⁢ ( α ⁢ s , m ) ) )

= α 4 σ 2

Hence σ = α 2 , which implies by (B.4) that

γ 2 ⁢ ( s ) = - 2 α ⁢ m ⁢ cn ⁢ ( α ⁢ s , m ) .

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

γ 1 ′ ⁢ ( s ) = 2 ⁢ m ⁢ cn 2 ⁢ ( α ⁢ s , m ) + ( 1 - 2 ⁢ m ) = 1 - 2 ⁢ m ⁢ sn 2 ⁢ ( α ⁢ s , m ) .

By integrating and by using γ 1 ⁢ ( 0 ) = 0 , we obtain

γ 1 ⁢ ( s ) = s - 2 ⁢ m ⁢ ∫ 0 s sn 2 ⁢ ( α ⁢ s , m ) ⁢ d s

= s - 2 α ⁢ ∫ 0 am ⁢ ( α ⁢ s , m ) m ⁢ sin 2 ⁡ θ 1 - m ⁢ sin 2 ⁡ θ ⁢ 𝑑 θ

= s - 2 α ⁢ ∫ 0 am ⁢ ( α ⁢ s , m ) ( 1 1 - m ⁢ sin 2 ⁡ θ - 1 - m ⁢ sin 2 ⁡ θ ) ⁢ 𝑑 θ

= s - 2 α ⁢ F ⁢ ( am ⁢ ( α ⁢ s , m ) , m ) + 2 α ⁢ E ⁢ ( am ⁢ ( α ⁢ s , m ) , m )

= 1 α ⁢ ( 2 ⁢ E ⁢ ( am ⁢ ( α ⁢ s , m ) , m ) - α ⁢ s ) .

Hence for fixed α > 0 , one has γ ⁢ ( s ) = 1 α ⁢ γ wave ⁢ ( α ⁢ s ) , where γ wave is given by

γ wave ⁢ ( s ) = ( 2 ⁢ E ⁢ ( am ⁢ ( s , m ) , m ) - s - 2 ⁢ m ⁢ cn ⁢ ( s , m ) ) .

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

γ 2 ⁢ ( s ) = - 2 σ ⁢ α ⁢ dn ⁢ ( α ⁢ s , m )

and, by Proposition B.3,

γ 1 ′ ⁢ ( s ) = σ 2 ⁢ γ 2 ⁢ ( s ) 2 + α 2 ⁢ ( m - 2 ) σ = α 2 ⁢ m σ ⁢ ( 1 - 2 ⁢ s ⁢ n 2 ⁢ ( α ⁢ s , m ) ) .

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

1 = γ 1 ′ ⁢ ( s ) 2 + γ 2 ′ ⁢ ( s ) 2

= α 4 ⁢ m 2 σ 2 ⁢ ( ( 1 - 2 ⁢ s ⁢ n 2 ⁢ ( α ⁢ s , m ) ) 2 + 4 ⁢ s ⁢ n 2 ⁢ ( α ⁢ s , m ) ⁢ cn 2 ⁢ ( α ⁢ s , m ) )

= α 4 ⁢ m 2 σ 2 ( ( 1 - 2 s n 2 ( α s , m ) ) 2 + 4 s n 2 ( α s , m ) ( 1 - sn 2 ( α s , m ) )

= α 4 ⁢ m 2 σ 2 .

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

γ 2 ⁢ ( s ) = - 2 m ⁢ α ⁢ dn ⁢ ( α ⁢ s , m )

and

γ 1 ′ ⁢ ( s ) = 1 - 2 ⁢ s ⁢ n 2 ⁢ ( α ⁢ s , m ) .

Integrating, we obtain

γ 1 ⁢ ( s ) = s - 2 ⁢ ∫ 0 s sn 2 ⁢ ( α ⁢ s , m ) ⁢ d s = s ⁢ ( 1 - 2 m ) + 2 α ⁢ m ⁢ E ⁢ ( am ⁢ ( α ⁢ s , m ) , m ) .

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

γ orbit = 1 m ⁢ ( 2 ⁢ E ⁢ ( am ⁢ ( s , m ) , m ) + ( m - 2 ) ⁢ s - 2 ⁢ d ⁢ n ⁢ ( 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 = 1 α ⁢ γ border ⁢ ( α ⁢ s ) , where

γ border ⁢ ( s ) = ( 2 ⁢ tanh ⁡ ( s ) - s - 2 ⁢ sech ⁡ ( 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

Published in Print: 2023-04-01

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Artikel in diesem Heft

https://doi.org/10.1515/acv-2021-0014

Schlagwörter für diesen Artikel

Li–Yau inequality; Willmore functional; elastic energy; embeddedness