The Willmore flow of Hopf-tori in the 3-sphere

Abstract

In this article, the author investigates flow lines of the classical Willmore flow, which start to move in a smooth parametrization of a Hopf-torus in $\textbf{S}^3$. We prove that any such flow line of the Willmore flow exists globally, in particular does not develop any singularities, and subconverges to some smooth Willmore-Hopf-torus in every $C^{m}$-norm. Moreover, if in addition the Willmore energy of the initial immersion $F_0$ is required to be smaller than or equal to the threshold $\frac{8\pi ^2}{\sqrt{2}}$, then the unique flow line of the Willmore flow, starting to move in $F_0$, converges fully to a conformally transformed Clifford torus in every $C^{m}$-norm, up to time dependent, smooth reparametrizations. Key instruments for the proofs are the equivariance of the Hopf-fibration $\pi :\textbf{S}^3 \longrightarrow \textbf{S}^2$ w.r.t. the effect of the $L^2$-gradient of the Willmore energy applied to smooth Hopf-tori in $\textbf{S}^3$ and to smooth closed regular curves in $\textbf{S}^2$, a particular version of the Lojasiewicz–Simon gradient inequality, and a well-known classification and description of smooth, arc-length parametrized solutions of the Euler–Lagrange equation of the elastic energy functional in terms of Jacobi elliptic functions and elliptic integrals, dating back to the 80s.

5 Appendix

In this appendix, we firstly prove existence of local, horizontal and smooth lifts of some arbitrary smooth, closed path $\gamma :\textbf{S}^1 \longrightarrow \textbf{S}^2$ w.r.t. fibrations of the type $\pi \circ F$ for “simple” parametrizations $F:\Sigma \longrightarrow \pi ^{-1}(\text {trace}(\gamma ))\subset \textbf{S}^3$, in the sense of Definition 3. Since $\gamma$ should be allowed to have self-intersections in view of our Theorem 1, we cannot blindly apply the general theory of smooth fiber bundles over smooth base manifolds. Instead, we have to construct here such local lifts elementarily, using Lemmata 1 and 2 and the theory of “ODEs.”

Lemma 3

Let $\gamma : \textbf{S}^1 \longrightarrow \textbf{S}^2$ be a smooth, closed and regular path in $\textbf{S}^2$, and let $F:\Sigma \longrightarrow \textbf{S}^3$ be a smooth immersion, which maps a smooth compact torus $\Sigma$ simply and smoothly onto the Hopf-torus $\pi ^{-1}(\text {trace}(\gamma )) \subset \textbf{S}^3$.

• (1) For every fixed $s^* \in \textbf{S}^1$ and $q^* \in \pi ^{-1}(\gamma (s^*))\subset \textbf{S}^3$, there is a unique horizontal, smooth lift $\eta ^{(s^*,q^*)}: \text {dom}(\eta ^{(s^*,q^*)}) \longrightarrow \pi ^{-1}(\text {trace}(\gamma ))$, defined on a non-empty, open interval $\text {dom}(\eta ^{(s^*,q^*)}) \subset \textbf{S}^1$, of $\gamma :\textbf{S}^1 \longrightarrow \textbf{S}^2$ w.r.t. the Hopf-fibration $\pi$, such that $\text {dom}(\eta ^{(s^*,q^*)})$ contains the point $s^*$ and such that $\eta ^{(s^*,q^*)}$ attains the value $q^*$ in $s^*$; i.e., $\eta ^{(s^*,q^*)}$ is a smooth path in the torus $\pi ^{-1}(\text {trace}(\gamma ))$, which intersects the fibers of $\pi$ perpendicularly and satisfies:

  $$\begin{aligned} (\pi \circ \eta ^{(s^*,q^*)})(s) = \gamma (s) \quad \forall \,s \in \text {dom}(\eta ^{(s^*,q^*)}) \,\,\, \text {and} \,\,\, \eta ^{(s^*,q^*)}(s^*) = q^*, \end{aligned}$$

  (86)

  and there is only one such function $\eta ^{(s^*,q^*)}$ mapping the open interval $\text {dom}(\eta ^{(s^*,q^*)}) \subset \textbf{S}^1$ into $\pi ^{-1}(\text {trace}(\gamma ))$.

• (2) There is some $\epsilon =\epsilon (F,\gamma )>0$, such that for every fixed $s^* \in \textbf{S}^1$ and every $x^*\in (\pi \circ F)^{-1}(\gamma (s^*))$ there is a horizontal smooth lift $\eta _F^{(s^*,x^*)}$ of $\gamma \lfloor _{\textbf{S}^1 \cap B_{\epsilon }(s^*)}$ w.r.t. the fibration $\pi \circ F:\Sigma \longrightarrow \text {trace}(\gamma ) \subset \textbf{S}^2$, attaining the value $x^*$ in $s^*$, i.e., $\eta _F^{(s^*,x^*)}$ is a smooth path in the torus $\Sigma$ which intersects the fibers of $\pi \circ F$ perpendicularly and satisfies:

  $$\begin{aligned} (\pi \circ F \circ \eta _F^{(s^*,x^*)})(s) = \gamma (s) \quad \forall \,s \in \textbf{S}^1 \cap B_{\epsilon }(s^*) \,\,\, \text {and} \,\,\, \eta _F^{(s^*,x^*)}(s^*)=x^*. \end{aligned}$$

  (87)

  In particular, for the above $\epsilon =\epsilon (F,\gamma )>0$ the function $\eta _F \mapsto F \circ \eta _F$ maps the set $\mathcal {L}(\gamma \lfloor _{\textbf{S}^1 \cap B_{\epsilon }(s^*)},\pi \circ F)$ of horizontal smooth lifts of $\gamma \lfloor _{\textbf{S}^1 \cap B_{\epsilon }(s^*)}$ w.r.t. $\pi \circ F$ surjectively onto the set $\mathcal {L}(\gamma \lfloor _{\textbf{S}^1 \cap B_{\epsilon }(s^*)},\pi )$ of horizontal smooth lifts of $\gamma \lfloor _{\textbf{S}^1 \cap B_{\epsilon }(s^*)}$ w.r.t. $\pi$.

Proof

Without loss of generality, we may assume here that $\gamma$ performs only one loop through its closed trace, and that it is defined on $\textbf{R}/L\textbf{Z}$ with $|\gamma '|_{g_{\textbf{S}^2}} \equiv 2$, where 2L is the length of the trace of $\gamma$. We consider the unique smooth vector field $V_{\gamma } \in \Gamma (T(\pi ^{-1}(\text {trace}(\gamma ))))\subset \Gamma (T\textbf{S}^3)$, which intersects the fibers of the Hopf-fibration perpendicularly, satisfies $|V_{\gamma }|_{g_{\textbf{S}^3}} \equiv 1$ throughout on $\pi ^{-1}(\text {trace}(\gamma ))$ and

$$\begin{aligned} D\pi _{q^*}(V_{\gamma }(q^*)) = c^* \, \gamma '(0), \,\, \text {for some} \,\, q^* \in \pi ^{-1}(\gamma (0)) \,\, \text {and some} \,\, c^* > 0. \end{aligned}$$

(88)

Now, the unique flow $\Psi$ on $\pi ^{-1}(\text {trace}(\gamma ))$ generated by the initial value problem

$$\begin{aligned} \frac{\hbox {d}v}{\hbox {d}\tau }(\tau ) = V_{\gamma }(v(\tau )) \qquad \text {and} \quad v(0)=q, \end{aligned}$$

(89)

for any fixed point $q \in \pi ^{-1}(\text {trace}(\gamma ))$, exists eternally, because the Hopf-torus $\pi ^{-1}(\text {trace}(\gamma ))$ is a compact subset of $\textbf{S}^3$, which can locally be parametrized by smooth charts, similarly to a smooth compact 2-manifold without any boundary points. We can readily infer from the requirements on $V_{\gamma }$, that the flow lines of the resulting eternal flow $\Psi :\textbf{R} \times \pi ^{-1}(\text {trace}(\gamma )) \longrightarrow \pi ^{-1}(\text {trace}(\gamma ))$ intersect the fibers of $\pi$ perpendicularly, have constant speed 1 and are mapped by $\pi$ onto $\text {trace}(\gamma )$. Now we choose some arbitrary initial point $q_0 \in \pi ^{-1}(\text {trace}(\gamma ))$, and we derive from the properties

$$\begin{aligned} |\Psi _{\tau }(q_0)|_{g_{\textbf{S}^3}}\equiv 1 \quad \text {and} \quad |\frac{\hbox {d}}{\hbox {d}\tau }\Psi _{\tau }(q_0)|_{g_{\textbf{S}^3}} = |V_{\gamma }(\Psi _{\tau }(q_0))|_{g_{\textbf{S}^3}} \equiv 1 \end{aligned}$$

as in Lemma 2 that there is some unit vector field $\tau \mapsto u(\tau )=u_{\gamma ,q_0}(\tau ) \in \text {Span}\{j,k\} \subset \textbf{H}$, satisfying $V_{\gamma }(\Psi _{\tau }(q_0)) = u(\tau ) \cdot \Psi _{\tau }(q_0) \,\, \text {in} \,\,\textbf{H}$, for every $\tau \in \textbf{R}$, and we can consequently compute exactly as in formula (12):

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}\tau }(\pi \circ \Psi _{\tau }(q_0)) = (I(V_{\gamma }(\Psi _{\tau }(q_0))) \cdot \Psi _{\tau }(q_0)) +(I(\Psi _{\tau }(q_0)) \cdot V_{\gamma }(\Psi _{\tau }(q_0))) \nonumber \\ = 2 \, I(\Psi _{\tau }(q_0)) \cdot u(\tau ) \cdot \Psi _{\tau }(q_0) \equiv 2 \,I(\Psi _{\tau }(q_0)) \cdot V_{\gamma }(\Psi _{\tau }(q_0)), \,\,\, \end{aligned}$$

(90)

for every $\tau \in \textbf{R}$. Now, since $\pi$ maps the Hopf-torus $\pi ^{-1}(\text {trace}(\gamma ))$ onto the trace of $\gamma$, we know a-priori that the trace of the path $[\tau \mapsto \pi \circ \Psi _{\tau }(q_0)]$ is contained in the trace of the curve $\gamma$. Moreover, Eq. (90) shows us:

$$\begin{aligned} |\frac{d}{d\tau }(\pi \circ \Psi _{\tau }(q_0))| \equiv 2 \equiv |\gamma '(\tau )|, \quad \forall \,\tau \in \textbf{R}, \end{aligned}$$

(91)

and for every initial point $q_0 \in \pi ^{-1}(\text {trace}(\gamma ))$. Hence, taking especially $q_0 \in \pi ^{-1}(\gamma (0))$ we have $\pi \circ \Psi _{0}(q_0) = \gamma (0)$, and therefore assumption (88) and Eq. (91) prove that there has to hold:

$$\begin{aligned} \pi \circ \Psi _{\tau }(q_0) = \gamma (\tau ), \quad \forall \, \tau \in \textbf{R}, \end{aligned}$$

(92)

and for every $q_0 \in \pi ^{-1}(\gamma (0))$. Moreover, combining identity (92) with the group property “$\Psi _{\tau _1 +\tau _2} = \Psi _{\tau _1} \circ \Psi _{\tau _2}$” of the flow $\Psi$, it follows that for any initial point $q_0 \in \pi ^{-1}(\text {trace}(\gamma ))$—not only for $q_0 \in \pi ^{-1}(\gamma (0))$—the path $[\tau \mapsto \pi \circ \Psi _{\tau }(q_0)]$ parametrizes the trace of $\gamma$ exactly once, as $\tau$ increases from 0 to L, which means that any such flow line $\{\Psi _{\tau }(q_0)\}_{\tau \in [0,L]}$ intersects each fiber of $\pi$ over the trace of $\gamma$ exactly once. Now, for any $s^* \in (0,L)$ and $q^* \in \pi ^{-1}(\gamma (s^*))$ we consider the unique flow line of $\Psi$, which starts moving in the point $\Psi _{s^*}^{-1}(q^*) = \Psi _{-s^*}(q^*)$ at time $\tau = 0$. Applying now statement (92) to the initial point $q_0:= \Psi _{s^*}^{-1}(q^*)\in \pi ^{-1}(\gamma (0))$ we can easily infer, that the function

$$\begin{aligned} \eta ^{s^*,q^*}(\tau ):= \Psi (\tau ,\Psi _{s^*}^{-1}(q^*)) \quad \text {for} \,\,\, \tau \in (0,L) \end{aligned}$$

satisfies the first property in (86), where $\text {dom}(\eta ^{(s^*,q^*)})\subset \textbf{S}^1$ can be any open and connected subset $\not =\textbf{S}^1$, which contains the prescribed point $s^*$—here being identified with some open subinterval of $(\textbf{R}/L\textbf{Z}) \setminus \{0\}$—and we also verify that

$$\begin{aligned} \eta ^{(s^*,q^*)}(s^*)=\Psi (s^*,\Psi _{s^*}^{-1}(q^*)) =\Psi _{s^*-s^*}(q^*)=q^*, \end{aligned}$$

which is just the second property in (86). Uniqueness of such a local, horizontal smooth lift of $\gamma$ follows easily from the above construction.

2) First of all, since $F:\Sigma \longrightarrow \textbf{S}^3$ is required to be an immersion whose image is $\pi ^{-1}(\text {trace}(\gamma ))$ and since $\Sigma$ is compact, there is some $\delta =\delta (F)>0$ such that for an arbitrarily fixed $q \in \pi ^{-1}(\text {trace}(\gamma ))$ the preimage $F^{-1}(B_{\delta }(q))$ consists of finitely many disjoint open subsets of $\Sigma$, which are mapped diffeomorphically onto their images in $B_{\delta }(q) \cap \pi ^{-1}(\text {trace}(\gamma ))$ via F. Moreover, from the first part of the lemma we infer the existence of horizontal smooth lifts $\eta :\text {dom}(\eta ) \rightarrow \pi ^{-1}(\text {trace}(\gamma ))$ of $\gamma \lfloor _{\text {dom}(\eta )}$ w.r.t. $\pi$. Now, we infer from Lemma 2 and from the compactness of $\textbf{S}^1$ that there is some $C=C(\gamma )>0$ such that

$$\begin{aligned} |\eta '|^2(s)=\frac{|\gamma '|^2(s)}{4} \le C, \quad \forall \, s \in \text {dom}(\eta ), \end{aligned}$$

holds for every horizontal lift $\eta$ of $\gamma \lfloor _{\text {dom}(\eta )}$ w.r.t. $\pi$. Hence, there is some $\epsilon =\epsilon (F,\gamma )>0$, such that for every $\tilde{s} \in \textbf{S}^1$ there is some ${\tilde{q}} \in \textbf{S}^3$ such that $\text {trace}(\eta \lfloor _{\textbf{S}^1 \cap B_{\epsilon }({\tilde{s}})}) \subset B_{\delta }({\tilde{q}})$. Now, using the fact that F maps each connected component of $F^{-1}(B_{\delta }({\tilde{q}}))$ diffeomorphically onto its image in $B_{\delta }({\tilde{q}}) \cap \pi ^{-1}(\text {trace}(\gamma ))$, we obtain the existence of at least one smooth map $\eta _F:\textbf{S}^1 \cap B_{\epsilon }({\tilde{s}}) \longrightarrow F^{-1}(B_{\delta }(\tilde{q}))$ satisfying $F\circ \eta _F = \eta$ on $\textbf{S}^1 \cap B_{\epsilon }({\tilde{s}})$, and thus also $\pi \circ F\circ \eta _F = \gamma$ on $\textbf{S}^1 \cap B_{\epsilon }({\tilde{s}})$. Again in combination with the first part of the lemma, we finally infer from this construction that for every $s^* \in \textbf{S}^1$ and every $x^*\in (\pi \circ F)^{-1}(\gamma (s^*))$ there is a smooth map $\eta _F^{(s^*,x^*)}:\textbf{S}^1 \cap B_{\epsilon }(s^*) \rightarrow \Sigma$, which possesses the two desired properties in (87). Moreover, it immediately follows from the horizontal property of every constructed lift $\eta$ of $\gamma$ w.r.t. $\pi$ in the first part of this lemma that every constructed lift $\eta _F$ of $\gamma$ w.r.t. $\pi \circ F$ intersects the fibers of $\pi \circ F$ perpendicularly w.r.t. the induced pullback metric $F^{*}(g_{\text {euc}})$. Hence, the last assertion of the lemma now turns out to be evident. $\square$

Remark 6

It is important to understand that horizontal lifts w.r.t. $\pi$ of closed curves in $\textbf{S}^2$ would in general not close up in $\textbf{S}^3$. As explained in [25, p. 381], a horizontal lift $\eta$ w.r.t. $\pi$ of a simple, closed path $\gamma :\textbf{S}^1 \longrightarrow \textbf{S}^2$, which performs $k \ge 2$ loops and encloses the area A on $\textbf{S}^2$, closes up, if and only if there holds the relation $A=\frac{4\,\pi }{k}$. Consider, for example, the standard parametrization $p(\varphi ):= [\cos (2\varphi ) + j \sin (2\varphi )]$ of a great circle in $\textbf{S}^2$. Its preimage w.r.t. $\pi$ is the Clifford torus, and one can easily infer at first from Remark 3, that the Clifford torus is conformally equivalent to the special parallelogram D in $\textbf{C}$ with vertices (0, 0), $(2\pi ,0)$, $(\pi ,\pi )$ and $(3\pi ,\pi )$, and then secondly check that any horizontal lift $\eta$ of p corresponds to a certain diagonal in D, which indeed closes up exactly when the parameter $\varphi$ reaches the value $2\pi$.

Finally, we are going to employ the complete integrability of the Euler–Lagrange equation of the elastic energy $\mathcal {E}$—see the right hand side of formula (21)—and the theory of elliptic integrals and Jacobi elliptic functions, in order to precisely compute the critical values of the elastic energy $\mathcal {E}$, particularly in order to exclude critical values of $\mathcal {E}$ in the surprisingly large interval $\big (2\pi , \frac{8\pi }{\sqrt{2}} \big ]$.

Proposition 6

• (1) Up to isometries of $\textbf{S}^2$, there are only countably many different smooth closed curves $\gamma :\textbf{S}^1 \longrightarrow \textbf{S}^2$, parametrized with constant speed, which are critical points of the elastic energy $\mathcal {E}$, called “closed elastic curves on $\textbf{S}^2$.” Vice versa, for each pair of positive integers (m, n) with $\text {gcd}(m,n)=1$ and $\frac{m}{n} \in (0,2-\sqrt{2})$ there is—up to isometric equivalence—a unique arc-length parametrized elastic curve $\gamma _{(m,n)}$ in $\textbf{S}^2$, which closes up after n periods and traverses some fixed great circle exactly m times.

• (2) There are no critical values of the elastic energy $\mathcal {E}$ in the interval $\big ( 2\pi , \frac{8\pi }{\sqrt{2}} \big ]$.

Proof of the first part of Proposition6: As explained in [17], every smooth closed stationary curve $\gamma :[a,b]/(a \sim b) \rightarrow \textbf{S}^2$ of the elastic energy $\mathcal {E}$ satisfies the differential equation

$$\begin{aligned} 2 \, \Big ( \nabla ^{\perp }_{\frac{\gamma '}{|\gamma '|}}\Big )^2(\mathbf {\kappa }_{\gamma }) + |\mathbf {\kappa }_{\gamma }|^2 \mathbf {\kappa }_{\gamma } + \mathbf {\kappa }_{\gamma } \equiv \nabla _{L^2} \mathcal {E}(\gamma ) \equiv 0, \quad \text {on} \,\,\,[a,b]. \end{aligned}$$

(93)

Now, Langer and Singer have pointed out in [17], that equation (93) holds for a non-geodesic closed curve $\gamma$, if and only if the signed curvature $\kappa _{\gamma }$ of its parametrization with speed $|\gamma '|\equiv 1$ satisfies the ordinary differential equation

$$\begin{aligned} \Big ( \frac{\hbox {d}\kappa }{\hbox {d}s} \Big )^2 = -\frac{1}{4} \, \kappa ^4 - \frac{1}{2} \kappa ^2 + A, \quad \text {on} \,\,\, \textbf{R}, \end{aligned}$$

(94)

for some integration constant $A \in \textbf{R}$, depending on the respective solution $\gamma$ of (93). Moreover, for such non-geodesic solutions $\gamma$ of (93), i.e., having non-constant curvature $\kappa _{\gamma } \not \equiv 0$, Eq. (94) is equivalent to the KdV-type-equation

$$\begin{aligned} \Big (\frac{\hbox {d}u}{\hbox {d}s} \Big )^2 = - \, u^3 \, - 2 \, u^2 + 4\,A\,u, \quad \text {on} \,\,\, \textbf{R}, \end{aligned}$$

(95)

to be satisfied by the square $\kappa ^2_{\gamma }$ of the curvature of the arc-length parametrized solution $\gamma$, which is simply obtained by pointwise multiplication of Eq. (94) with $4\, \kappa ^2_{\gamma }$. One can easily verify that for every non-geodesic closed solution $\gamma$ of (93), the polynomial $P(x):=x^3+2\,x^2-4A\,x$, occurring on the right hand side of Eq. (95), must have three different real roots $-\alpha _1< 0=\alpha _2<\alpha _3$ satisfying the algebraic relations

$$\begin{aligned} \alpha _1 - \alpha _3 = 2 \quad \text {and} \quad \alpha _1 \, \alpha _3 = 4 \,A. \end{aligned}$$

(96)

Now, as explained in Section 2 of [17] the solutions u of Eq. (95) are exactly given by the Jacobi elliptic functions² of the particular type³:

$$\begin{aligned} u(s)= \alpha _3 \,\text {cn}^2(r \cdot s;\,p), \quad \forall \, s \in \textbf{R}, \end{aligned}$$

(97)

with

$$\begin{aligned} r:=\frac{1}{2} \sqrt{\alpha _1 + \alpha _3} \equiv \frac{1}{\sqrt{2}} \sqrt{\alpha _3 + 1} \quad \text {and} \quad p:=\sqrt{\frac{\alpha _3}{\alpha _3 + \alpha _1}} \equiv \frac{1}{\sqrt{2}} \, \sqrt{\frac{\alpha _3}{\alpha _3+1}}. \end{aligned}$$

(98)

Hence, the modulus p occurring in formula (97) has to be contained in the open interval $\big (0,\frac{1}{\sqrt{2}}\big )$, and combining formulae (96) and (98) one can express the roots $\alpha _3$ and $\alpha _1$ of the polynomial $P(x):=x^3+2\,x^2-4Ax$, occurring in equation (95), in terms of p:

$$\begin{aligned} \alpha _3 = \frac{2p^2}{1-2p^2} \quad \text {and} \quad \alpha _1 = \frac{2p^2}{1-2p^2}+2 = \frac{2-2p^2}{1-2p^2}. \end{aligned}$$

Moreover, combining these formulae again with formula (96) the modulus p in formula (97) automatically yields the integration constant A appearing in equations (94) and (95), and also the frequency r in (97):

$$\begin{aligned} A = \frac{1}{4} \, \alpha _1 \, \alpha _3 = \frac{p^2-p^4}{(1-2p^2)^2}>0, \quad r=\frac{1}{\sqrt{2-4\,p^2}} \in \big (\frac{1}{\sqrt{2}},\infty \big ). \end{aligned}$$

(99)

Formula (97) particularly implies that every arc-length parametrized solution $\gamma$ of equation (93) performs a periodic path on $\textbf{S}^2$ with

$$\begin{aligned} \text {one period of}\, \gamma =4\, \frac{K(p)}{r} = 4 \,\sqrt{2-4\,p^2} \,K(p) \end{aligned}$$

(100)

on account of formula (2.2.5) in [19] and formula (99), where

$$\begin{aligned} K(p):=\int _0^{\pi /2} \frac{1}{\sqrt{1-p^2 \sin ^2(\varphi )}} \, \textrm{d}\varphi \end{aligned}$$

(101)

denotes the complete elliptic integral of the first kind with parameter $p\in [0,\frac{1}{\sqrt{2}}]$; see [19], Sections 3.1 and 3.8, and [3], pp. 8–17. Another important consequence of equation (97) is that every arc-length parametrized solution $\gamma$ of (93) possesses a well-defined wavelength $\Lambda (\gamma ) \in \textbf{R}$ whose quotient $\frac{\Lambda }{2\pi }$ has to be rational, because $\gamma$ is supposed to be a closed curve on [0, L], with $L:=\text {length}(\gamma )$.⁴ Hence, for every non-geodesic, closed and arc-length parametrized solution $\gamma$ of equation (93) there is a unique pair $(m,n) \in \textbf{N} \times \textbf{N}$ of positive integers with $\text {gcd}(m,n)=1$, such that $\Lambda = \frac{m}{n} \, 2\pi$, which means geometrically that this particular path $\gamma$ closes up after n periods, respectively, “lobes”—whose common lengths are given by formula (100)—and traverses some fixed great circle in $\textbf{S}^2$  m times, while its arc-length parameter s runs from 0 to L.⁵ Hence, up to isometric equivalence we are able to count non-geodesic, closed and arc-length parametrized elasticae on $\textbf{S}^2$ systematically, which has already proved the first assertion of the first part of Proposition 6.

Now, in order to prove the entire classification of closed elastic curves on $\textbf{S}^2$, as asserted in the first part of Proposition 6, we have to understand vice versa, for which quotients $\frac{m}{n}$ of coprime, positive integers there are actually closed elastic curves “$\gamma _{(m,n)}$,” possessing the aforementioned two geometric properties, according to the respective pairs (m, n). To this end, we employ the second quantitative ingredient of the proof of Proposition 6, namely the following formula (102), taken directly from Section 4 of [18], p. 148, which expresses the wavelength $\Lambda$ of a non-geodesic, closed, arc-length parametrized solution $\gamma$ of equation (93) as a function of the above parameter $p \in \big (0,\frac{1}{\sqrt{2}}\big )$:

$$\begin{aligned} \Lambda (\gamma (p))= & {} 2 \pi \, \varepsilon \,\Lambda _0(\psi (p),p) - 2 \,(3-4\,p^2)\,\nonumber \\{} & {} \sqrt{1-(1- p^2) \sin ^2(\psi (p))}\, \sin (\psi (p))\, K(p) \end{aligned}$$

(102)

where $\psi (p):=\arcsin \Big ( \sqrt{8} \, \frac{\sqrt{1-2p^2}}{3-4p^2} \Big )$, $\varepsilon := \frac{4 \,p^2-1}{|4 \,p^2 - 1|}$, $\Lambda _0$ denotes the Heuman-Lambda function—see [3], pp. 35–37 and pp. 344–349—and K(p) had been introduced in (101). By means of the computations in Section 3.8 in [19], one can verify that the first derivative w.r.t. p of the function in (102) reads⁶:

$$\begin{aligned} \frac{\hbox {d}\Lambda (\gamma (p))}{\hbox {d}p} = \frac{2 \,\sqrt{8} \, [(1- p^2) \,K(p) - E(p)]}{p \, (1-p^2)\, \sqrt{1-2p^2} \,(3 - 4 p^2) \, \sqrt{1 - p^2 \,\sin ^2(\psi (p))}} \end{aligned}$$

(103)

for $p\in \big (0,\frac{1}{\sqrt{2}}\big )$, where $E(p):=\int _0^{\pi /2}\sqrt{1-p^2 \sin ^2(\varphi )}\, d\varphi$ denotes the complete elliptic integral of the second kind. Since we have $(1-p^2)\,K(p)-E(p)<0$ for $p\in \big (0,\frac{1}{\sqrt{2}}\big )$, and furthermore $\sin (\psi (0))=\frac{\sqrt{8}}{3}$, $\Lambda _0\big (\arcsin \big ( \frac{\sqrt{8}}{3}\big ),0\big )=\frac{\sqrt{8}}{3}$, $\sin (\psi (\frac{1}{2}))=1$ and $\Lambda _0\big (\frac{\pi }{2},\frac{1}{2}\big )=1$, formula (103) shows us that the function $p \mapsto \Lambda (\gamma (p))$ in (102) decreases strictly monotonically from its initial value $\Lambda (0) = - 2 \pi \cdot \frac{\sqrt{8}}{3} - 2\, \frac{\sqrt{2}}{3} \, \pi = -\sqrt{2} \,(2 \pi )$ to its minimal value $\min _{p \in [0,\frac{1}{\sqrt{2}}]} \Lambda (p) = \lim _{p \nearrow \frac{1}{2}} \Lambda (p) = -2 \pi - 2\, K\big (\frac{1}{2}\big ) \approx -9,65469$ as p increases from 0 to $\frac{1}{2}$, then jumps at the point $p=\frac{1}{2}$ from its minimal value $\lim _{p \nearrow \frac{1}{2}} \Lambda (p)$ to its maximal value⁷$\max _{p \in [0,\frac{1}{\sqrt{2}}]} \Lambda (p) = \lim _{p \searrow \frac{1}{2}} \Lambda (p) = 2 \pi - 2 \, K\big (\frac{1}{2}\big ) \approx 2,91169$, and finally decreases strictly monotonically from this maximum to its final value $\Lambda \big (\frac{1}{\sqrt{2}}\big ) = 2\, \pi \Lambda _0\big (0,\frac{1}{\sqrt{2}}\big )=0$. Since we are allowed to shift the value of the wavelength $\Lambda (p)$ in the first interval $[-2 \pi -2 \,K\big (\frac{1}{2}\big ),-\sqrt{2} \,(2 \pi )]$ about the height of the jump of $\Lambda (p)$, i.e., about $4 \pi$, to the right in $\textbf{R}$ into the interval $[2 \pi - 2\,K\big (\frac{1}{2}\big ), 2 \pi \,(2-\sqrt{2})]$, we conclude that for each coprime pair $(m,n) \in \textbf{N} \times \textbf{N}$ satisfying

$$\begin{aligned} \frac{m}{n} \in \Big ( 0, 1-\frac{K\big (\frac{1}{2}\big )}{\pi } \Bigg ] \cup \Big [ 1-\frac{K\big (\frac{1}{2}\big )}{\pi }, 2-\sqrt{2} \Big )=(0,2-\sqrt{2}) \end{aligned}$$

(104)

there is a “unique” arc-length parametrized elastic curve $\gamma _{(m,n)}$, which closes up after n periods and traverses some fixed great circle C in $\textbf{S}^2$ exactly m times - “unique” only up to the action of all those isometries of $\textbf{S}^2$, which leave the great circle C invariant, just as claimed in the first part of the proposition. $\square$

Proof of the second part of Proposition6: Relying on the proof of the first part of Proposition 6, we attempt to prove its second part in a “straightforward manner,” which means that we fix a pair of coprime integers $(m,n) \in \textbf{N} \times \textbf{N}$ satisfying condition (104) and that we simply combine formulae (97) and (100) with the definition of the elastic energy in (15), in order to compute both length and elastic energy of the unique solution $\gamma _{(m,n)}$ of equation (93) directly. This is actually possible, because the given data “(m, n)” determine the value of the wavelength $\Lambda =\frac{m}{n} \,2\pi$ and thus also a unique value of the parameter $p=p(\gamma _{(m,n)})$ in (98), inverting the strictly monotonic wavelength function $p \mapsto \Lambda (\gamma (p))$ in (102). Hence, also recalling that we only work with arc-length parametrized elastic curves $\gamma _{(m,n)}$, we can firstly compute by means of formula (100), abbreviating here $p=p(\gamma _{(m,n)})$⁸:

$$\begin{aligned} \text {length}(\gamma _{(m,n)}) = n \,\text {periods of}\, \gamma _{(m,n)} = 4n\, \frac{K(p)}{r} = 4n \, \sqrt{2-4\,p^2} \,K(p), \end{aligned}$$

(105)

and together with formulae (15) and (97) and with formulae (3.4.15) and (3.4.27) in [19], we obtain furthermore:

$$\begin{aligned}{} & {} \mathcal {E}(\gamma _{(m,n)}) = \int _{0}^{4n \,\frac{K(p)}{r}} 1 + \kappa ^2_{\gamma _{(m,n)}}(s) \, ds = \int _0^{4n \,\frac{K(p)}{r}} 1+\alpha _3 \,\text {cn}^2(r \cdot s;\,p) \,ds \nonumber \\{} & {} \quad = 4n \,\sqrt{2-4\,p^2} \,K(p) + \frac{2p^2}{1-2p^2} \,\frac{4n}{r} \int _0^{K(p)} \text {cn}^2(u;\,p) \, du \nonumber \\{} & {} \quad = 4n \,\sqrt{2-4\,p^2} \,K(p) + \frac{2p^2}{1-2p^2} \, 4n \,\sqrt{2-4\,p^2} \,\,\frac{1}{p^2}\, \Big [ E(p) - (1-p^2) K(p)\, \big ] \nonumber \\{} & {} \quad = 4n \,\sqrt{2-4\,p^2} \,K(p) \Big ( 1+ \frac{2p^2}{1-2p^2} \Big ) + \frac{16n}{\sqrt{2-4p^2}} \, (E(p)-K(p)) \nonumber \\{} & {} \quad = \frac{8n}{\sqrt{2-4p^2}} \,(2E(p)-K(p)), \end{aligned}$$

(106)

a formula which has essentially already appeared in the proof of Corollary 6.4 in [23] for closed wavelike elasticae in the hyperbolic plane, being similar to the slightly simpler formula (46) in [21] for the elastic energy of closed orbitlike elasticae in the hyperbolic plane.⁹ Now, formula (106) is not only useful for numerical purposes, i.e., in order to compute elastic energies $\mathcal {E}(\gamma _{(m,n)})$ effectively, but it also meets the aim of the second part of Proposition 6, to rigorously determine a “rather accurate” lower bound for all possible elastic energies $\mathcal {E}(\gamma _{(m,n)})$ of non-geodesic elastic curves. The key observation in this situation is that the function $f(p):=\frac{1}{\sqrt{1-2p^2}} \,(2E(p)-K(p))$—appearing on the right hand side of formula (106)—is actually strictly monotonically increasing on the entire open interval $\big (0,\frac{1}{\sqrt{2}}\big )$. In order to prove this, we firstly compute by means of formulae (3.8.7) and (3.8.12) in [19]:

$$\begin{aligned}{} & {} \frac{\hbox {d}}{\hbox {d}p}\big (2E(p) - K(p)\big ) = 2 \,\frac{E(p)- K(p)}{p} - \frac{E(p) - (1-p^2)\,K(p)}{p\,(1-p^2)}\nonumber \\{} & {} \quad = \frac{1-2p^2}{p\,(1-p^2)} \,E(p) - \frac{1}{p}\, K(p)<\frac{1}{p} \, (E(p)-K(p))<0, \end{aligned}$$

(107)

for $p\in \big (0,\frac{1}{\sqrt{2}}\big )$, showing first of all that the function $p\mapsto 2E(p) - K(p)$ decreases monotonically from $\frac{\pi }{2}$ in $p=0$ to $2E\big (\frac{1}{\sqrt{2}}\big ) - K\big (\frac{1}{\sqrt{2}}\big )>0$ in $p=\frac{1}{\sqrt{2}}$, and we can continue by deriving the function f w.r.t. p:

$$\begin{aligned}{} & {} \frac{df(p)}{dp} = \frac{1}{\sqrt{1-2p^2}} \Big ( \frac{1-2p^2}{p(1-p^2)} E(p) - \frac{1}{p} \,K(p) \Big ) + \frac{2p}{(1-2p^2)^{3/2}} (2E(p)-K(p)) \\{} & {} \quad =\frac{\sqrt{1-2p^2}}{p\,(1-p^2)} E(p) - \frac{1}{p\,\sqrt{1-2p^2}} K(p) + \frac{4p}{(1-2p^2)^{3/2}} E(p) - \frac{2p}{(1-2p^2)^{3/2}} K(p) \\{} & {} \quad = \frac{1}{p\,(1-p^2)\,(1-2p^2)^{3/2}} E(p) - \frac{1}{p\,(1-2p^2)^{3/2}} K(p) \\{} & {} \quad = \frac{1}{p\, (1-2p^2)^{3/2}} \, \Big ( \frac{1}{1-p^2}\, E(p)-K(p) \Big ). \end{aligned}$$

Moreover, we see that the function $g(p):=\frac{1}{1-p^2}\, E(p) - K(p)$ satisfies $g(0)=0$, and that by formulae (3.8.7) and (3.8.12) in [19] its derivative is:

$$\begin{aligned} \frac{dg(p)}{dp}= & {} {} \frac{E(p)- K(p)}{(1-p^2)\,p} + \frac{2p}{(1-p^2)^2} \, E(p) - \frac{E(p) - (1-p^2)\,K(p)}{p\,(1-p^2)} \\= & {} {} \frac{2p}{(1-p^2)^2} \,E(p) - \frac{p}{1-p^2} \,K(p)> \frac{2p}{1-p^2} \,E(p) - \frac{p}{1-p^2} \,K(p) \\= & {} {} \frac{p}{1-p^2} \,(2 \,E(p) - K(p)) >0 \end{aligned}$$

for every $p\in \big (0,\frac{1}{\sqrt{2}}\big )$, using that here $(1-p^2) \in \big (\frac{1}{2},1\big )$ and $E(p)>0$, and furthermore that by (107) $\min _{p\in \big [0,\frac{1}{\sqrt{2}}\big ]} (2 \,E(p) - K(p)) = 2E\big (\frac{1}{\sqrt{2}}\big ) - K\big (\frac{1}{\sqrt{2}}\big )>0$. Hence, we can infer that $g(p)>0$ for $p\in \big (0,\frac{1}{\sqrt{2}}\big )$ and that therefore f increases strictly monotonically from $f(0)= \frac{\pi }{2}$ to $\infty$, as p runs from 0 to $\frac{1}{\sqrt{2}}$. Hence, without even guessing for which pair (m, n) of coprime integers—respectively, for which value of the modulus $p=p(\gamma _{(m,n)})\in \big (0,\frac{1}{\sqrt{2}}\big )$—the elastic energy $\mathcal {E}(\gamma _{(m,n)})$ in (106) might attain its minimal value among all non-geodesic elasticae on $\textbf{S}^2$, we can roughly, but rigorously estimate by means of formula (106) and $\inf _{p\in [0,\frac{1}{\sqrt{2}})} f(p)=f(0)=\frac{\pi }{2}$:

$$\begin{aligned} \mathcal {E}(\gamma _{(m,n)}) > \frac{16}{\sqrt{2}} f(0) = \frac{8\pi }{\sqrt{2}} \approx 17,771532 \end{aligned}$$

(108)

for coprime pairs $(m,n)\in \textbf{N} \times \textbf{N}$ satisfying condition (104), where we have also used the fact that we must have “$n\ge 2$” on account of condition (104) that $p(\gamma _{(m,n)}) \in \big (0,\frac{1}{\sqrt{2}}\big )$ and that f increases strictly monotonically from f(0) to $f(p(\gamma _{(m,n)}))$, for any fixed coprime pair (m, n) satisfying condition (104). $\square$

Remark 7

In order to assess the quality of both formula (106) and estimate (108), we mention here a totally different method yielding a precise formula for the elastic energy of any non-geodesic elastic curve $\gamma _{(m,n)}$ on $\textbf{S}^2$ and also a rigorous lower bound for all those elastic energies, which is $16 \,\sqrt{\frac{\pi }{3}}$.¹⁰ The starting point of this second method is the perfect match between equation (2.2) in [10] with equation (94), simply with $\nu =-A$, $G=1$ and $\mu =-\frac{1}{2}$ in equation (2.2) of [10]. Therefore, the formulae in Lemma 1 and Remark 2 of [10] yield here the coefficients $a_2 = \frac{1}{48} - \frac{A}{4}$ and $a_3 = \frac{1}{1728} + \frac{A}{48}$ of a concrete polynomial equation $y^2=4\,x^3-a_2 \,x-a_3$, whose set of solutions $[x:y:1] \in \textbf{CP}^2$ yields a particular elliptic curve $E_{(m,n)} \subset \textbf{CP}^2$, which turns out to carry all the relevant information about the elastic curve $\gamma _{(m,n)}$, for any fixed pair (m, n) of positive, coprime integers with $\frac{m}{n} \in (0,2-\sqrt{2})$. One can easily infer from the above concrete formulae for the coefficients $a_2$, $a_3$ that the discriminant $D(F):= a_2^3 - 27 \, a_3^2$ of the polynomial $F(x) = 4\,x^3-a_2\,x-a_3$ vanishes, if and only if the integration constant A satisfies $A=0$ or $A=-\frac{1}{4}$. However, formula (99) rules out these two possibilities. Hence, for any $p=p(\gamma _{(m,n)})\in \big (0,\frac{1}{\sqrt{2}}\big )$ the corresponding polynomial $F(x) = 4\, x^3 - a_2 \,x - a_3$ has a non-vanishing—here actually a negative—discriminant D(F). Hence, by the uniformization theorem—see, e.g., Theorem 2.9 in [1]—there exists a unique lattice $\Omega \equiv \Omega _{(m,n)}:=\omega _1 \textbf{Z} \oplus \omega _2 \textbf{Z}$ in $\textbf{C}$, with $\Im (\frac{\omega _2}{\omega _1})>0$, such that the corresponding Weierstrass-$\wp$-function $\wp (z,\Omega ):=\frac{1}{z^2} + \sum _{\omega \in \Omega {\setminus } \{0\}} \frac{1}{(\omega -z)^2} - \frac{1}{\omega ^2}$ solves the complex ordinary differential equation

$$\begin{aligned} (\wp '(z))^2= 4 \,(\wp (z))^3 - a_2 \, \wp (z) - a_3 \equiv F(\wp (z)), \quad \forall \, z\in \textbf{C}/\Omega , \end{aligned}$$

(109)

and therefore “parametrizes” the elliptic curve $E_{(m,n)}$.¹¹ Now, Eq. (109) and Lemmata 1 and 2 in [10] reveal the surprising relation between the elliptic curve $E_{(m,n)}$ and our elastic curve $\gamma _{(m,n)}$, because they yield the equation $\wp (x + x_0) = - i \frac{\kappa '(x)}{4}-\frac{\kappa ^2(x)}{8} -\frac{1}{24}$, for every $x \in \textbf{R}$, where $x_0 \in \textbf{C} {\setminus } \Big (\frac{1}{2} \, \Omega \oplus \textbf{R}\Big )$ is some suitably chosen point and $\kappa$ the signed curvature function of $\gamma _{(m,n)}$, as in Eq. (94). In combination with equations (95) and (97) this implies that the Weierstrass-$\wp$-function $\wp (z,\Omega )$ has a real primitive period, say $\omega _1$, namely the real primitive period of the Jacobi elliptic function $\text {cn}(r \,\cdot ;p)$, precisely: $\omega _1(\wp ) = 4\, \sqrt{2 - 4\,p^2} \,K(p)$ by formula (100). This identity converts the formula “$\mathcal {E}(\gamma _{(m,n)}) = 8 \cdot n \cdot \Re \big (\eta (\omega _1(\wp ),\Omega _{(m,n)})\big ) + \frac{2}{3} \cdot n \cdot \omega _1(\wp )$” from Theorem 5 in [10] into a computational tool—being comparable to formula (106), but much less effective—which can be used as in (108), in order to prove the lower bound $16 \,\sqrt{\frac{\pi }{3}}$ for the elastic energies of all non-geodesic elasticae $\gamma _{(m,n)}$ on $\textbf{S}^2$. $\square$

We should also check the accuracy of the second statement of our Proposition 6 by means of a comparison between several numerical values of energies—first table below—and lengths—second table—of elasticae $\gamma _{(m,n)}$ and our threshold “$\frac{8\pi }{\sqrt{2}}$” from formula (108). To this end, we apply formulae (105) and (106) for coprime integers $1 \le m \le 4$ and $1 \le n \le 7$, satisfying condition (104). The curious reader can compare these values with the corresponding ones for closed, orbitlike elasticae in the hyperbolic plane, collected in Table 1 of [21].

m/n	$m=1$	$m=2$	$m=3$	$m=4$
$n=1$	–	–	–	–
$n=2$	19.17	–	–	–
$n=3$	38.38	–	–	–
$n=4$	62.88	–	–	–
$n=5$	96.62	55.01	–	–
$n=6$	134.95	–	–	–
$n=7$	192.23	98.87	74.97	62.89

m/n	$m=1$	$m=2$	$m=3$	$m=4$
$n=1$	–	–	–	–
$n=2$	14.68	–	–	–
$n=3$	13.68	–	–	–
$n=4$	13.98	–	–	–
$n=5$	13.77	28.51	–	–
$n=6$	13.99	–	–	–
$n=7$	13.15	27.95	41.63	60.22