Many closed K-magnetic geodesics on ${\mathbb {S}}^2$

Abstract

In this paper we adopt an alternative, analytical approach to Arnol’d problem [4] about the existence of closed and embedded K-magnetic geodesics in the round 2-sphere ${\mathbb {S}}^2$, where $K: {\mathbb {S}}^2 \rightarrow {\mathbb {R}}$ is a smooth scalar function. In particular, we use Lyapunov-Schmidt finite-dimensional reduction coupled with a local variational formulation in order to get some existence and multiplicity results bypassing the use of symplectic geometric tools such as the celebrated Viterbo’s theorem [21] and Bottkoll results [7].

1 Introduction

We deal with the motion $\gamma =\gamma (t)$ of a particle of unit mass and charge in ${\mathbb {R}}^3$, that experiences the Lorentz force $\mathbf{F}$ produced by a magnetostatic field $\mathbf{B}$. If the particle is constrained to the standard round sphere ${\mathbb {S}}^2\subset {\mathbb {R}}^3$, the motion law reads

$$\begin{aligned} \gamma ''+|\gamma '|^2\gamma =K(\gamma )~\!\gamma \wedge \gamma '~\!, \end{aligned}$$

(1.1)

where

$$\begin{aligned} K(p):=-\mathbf{B}(p)\cdot p~,\quad p\in {\mathbb {S}}^2~\!. \end{aligned}$$

A trajectory $\gamma (t)$ satisfying (1.1) is called K -magnetic geodesic.

Let us recall the elementary derivation of (1.1). We have $\mathbf{F}(\gamma )=\gamma '\wedge \mathbf{B}(\gamma )$; due to the constraint $|\gamma |\equiv 1$, the vectors $\gamma$ and $\gamma '$ are orthogonal along the motion. It follows that the projection of $\mathbf{F}$ on $T_\gamma {\mathbb {S}}^2=\langle \gamma \rangle ^\perp$ is proportional to $\gamma \wedge \gamma '$, and in fact $\mathbf{F}^{T\!}(\gamma )=-(\mathbf{B}(\gamma )\cdot \gamma )~\!\gamma \wedge \gamma ' =K(\gamma )~\!\gamma \wedge \gamma '$. Finally, by differentiating the identity $\gamma \cdot \gamma '\equiv 0$, we see that the tangent component of the acceleration vector is $\gamma ''-(\gamma ''\cdot \gamma )\gamma =\gamma ''+|\gamma '|^2\gamma$, and thus Newton’s law gives (1.1). Notice that $\gamma ''-(\gamma ''\cdot \gamma )\gamma =\nabla ^{{\mathbb {S}}^2}_{\!\gamma '}\gamma '$, where $\nabla ^{{\mathbb {S}}^2}$ is the Levi-Civita connection of ${\mathbb {S}}^2$.

Two remarkable facts immediately follow from (1.1). First, we have $2\gamma ''\cdot \gamma '=(|\gamma '|^2)'=0$. Thus the particle moves with constant scalar speed, say

$$\begin{aligned} |\gamma '|\equiv c~\!, \end{aligned}$$

for some $c>0$. In particular, $\gamma$ is a regular curve. Secondly, we learn from differential geometry that $\gamma$ has geodesic curvature

$$\begin{aligned} \kappa (\gamma )=\frac{\gamma ''\cdot \gamma \wedge \gamma '}{|\gamma '|^3}=\frac{K(\gamma )}{c}~\!. \end{aligned}$$

Next, let $c>0$ and $K:{\mathbb {S}}^2\rightarrow {\mathbb {R}}$ be given. In [4], see also [5, Problems 1988/30, 1994/14, 1996/18], Arnol’d proposed the following question (actually in a more general setting, where ${\mathbb {S}}^2$ is replaced by an oriented Riemannian surface $(\Sigma ,g)$):

Problem (${\mathcal {P}}_{K,c}$), together with its generalizations, attracted the attention of many authors and has been studied via different mathematical tools, such as symplectic geometric [4, 10, 11, 13, 17] and variational arguments for multivalued functionals [6, 15, 19, 20].

The relation between Problem (${\mathcal {P}}_{K,c}$) and symplectic geometry can be explained as follows. Let us consider on ${\mathbb {S}}^2$ the (restriction of the) two-form $\beta := i_\mathbf{B}(dx \wedge dy \wedge dz)$ and let us define on the cotangent bundle $T^* {\mathbb {S}}^2$ endowed with coordinates (q, p) the symplectic form

$$\begin{aligned} \Omega = c~ dq \wedge dp - \pi ^* \beta \end{aligned}$$

where $dq \wedge dp = \sum _{i=1}^2 dq_i \wedge dp_i$ denotes the standard symplectic form on $T^* {\mathbb {S}}^2$ and $\pi : T^* {\mathbb {S}}^2 \rightarrow {\mathbb {S}}^2$ is the canonical projection.

It is not hard to show, via a straight calculation, that K-magnetic geodesics on ${\mathbb {S}}^2$ having constant speed c are exactly the projections $\pi (\gamma )$ of the integral curves of the vector field on $T^* {\mathbb {S}}^2$ defined by

$$\begin{aligned} d H = i_X \Omega ~\!, \end{aligned}$$

(1.2)

where $H = \frac{1}{2} |p|^2$. In the language of symplectic geometry, X is the Hamiltonian vector field given by the Hamiltonian function H. Notice also that since $\gamma '$ as observed above has constant speed, then $H(\gamma )$ is constant and then by (1.2) we have $i_{\gamma '} \Omega = 0$, which by definition means that $\gamma$ is a characteristic of $\Omega$.

Now, for any smooth K and every $c > 0$ large enough the existence of a solution to (${\mathcal {P}}_{K,c}$) can be deduced via this symplectic geometric approach by applying the celebrated Viterbo result [21] on the existence of closed characteristics on compact hypersurfaces of contact type. It is worth to notice that this result can be generalized to any closed oriented surface $\Sigma$, yielding the existence of a solution for high energies c in every free homotopy class that can be represented by a non-degenerate geodesic [11, Theorem 2.1 (ii)].

For the case of low energy levels we cite [11, Theorem 2.1 (i)] and [17], where the author proves the existence of contractible periodic solutions for almost all sufficiently small energy levels and for arbitrary smooth magnetic fields.

The existence of at least two distinct¹ solutions to (${\mathcal {P}}_{K,c}$) in the case of the round two-sphere follows, always for $c > 0$ large enough, from a general result of Bottkoll [7] (see also [1]) about the number of periodic orbits of the flow of a Hamiltonian vector field which is close to a flow generating a free circle action (in our case, the geodesic flow on the round two-sphere), which implies that such periodic orbits are at least as many as one plus the cup-length of ${\mathbb {S}}^2$, i.e. two.

For other available results for (${\mathcal {P}}_{K,c}$) showing the existence of at least two distinct solutions for arbitrary metrics on ${\mathbb {S}}^2$ let us mention [11, Theorem 2.1 (i) and Theorem 2.7], [16, 18]. Notice that all these results require that K has constant sign: indeed, in [11] the assumption $K>0$ guarantees that $\Omega = K d\sigma$ is a symplectic form on ${\mathbb {S}}^2$; in [18, 16] an index-based topological argument is used to prove the existence of two distinct solutions for any $c>0$, and the assumption $K>0$ is needed to prove some crucial a-priori bound on the length of simple and closed K-magnetic geodesics. Schneider’s multiplicity result is indeed sharp, that is, Problem (${\mathcal {P}}_{K,c}$) might have exactly two distinct solutions, see [18, Theorem 1.3].

Let us however notice that from the physical point of view it is important to include sign-changing functions K, unless the existence of magnetic monopoles is admitted. In fact, the Gauss law for magnetism in absence of magnetic monopoles implies that

$$\begin{aligned} \int \limits _{{\mathbb {S}}^2} K(p)~\!d\sigma _p=0~\!, \end{aligned}$$

see also [4, Problem 1996-17].

The aim of this paper is twofold. Firstly, we provide a more direct, self-contained and analytical approach to Viterbo’s and Bottkoll’s results, in the special case of the round sphere. Secondly, we provide sufficient conditions on K to obtain as many solutions as we wish, provided that c is large enough.

Our main results are stated in Sects. 4 and 5, see Theorems 4.1 and 5.2, respectively.

For the proofs we took inspiration from the breakthrough paper [2], where Ambrosetti and Badiale showed how merging the Lyapunov-Schmidt finite-dimensional reduction with variational arguments allows to obtain extremely powerful tools to get existence and multiplicity results. This idea has been applied to tackle quite a large number of variational problems arising from mathematical physics and differential geometry, see the exhaustive list of references in the monograph [3].

Notice however that Arnol’d problem on K-magnetic geodesics in ${\mathbb {S}}^2$ does not admit a (standard) variational formulation through a (non-multivalued) energy functional, due to obvious topological obstructions. To overcome this difficulty, we take advantage of a ”local” variational approach which is developed in Sect. 2.

Notation.

The Euclidean space ${\mathbb {R}}^3$ is endowed with Euclidean norm |p|, scalar product $p\cdot q$, and exterior product $p\wedge q$. The canonical basis of ${\mathbb {R}}^3$ is $\{e_h~,~h=1,2,3\}$.

We isometrically embed the unit sphere ${\mathbb {S}}^2$ into ${\mathbb {R}}^3$, so that the tangent space $T_z{\mathbb {S}}^2$ at $z\in {\mathbb {S}}^2$ is identified with $\langle z\rangle ^\perp =\{p\in {\mathbb {R}}^3~|~p\cdot z=0\}$. We denote by ${{\mathcal {D}}}_{\!\rho }(z)\subset {\mathbb {S}}^2$ the geodesic disk of radius $\rho \in (0,\frac{\pi }{2}]$ about $z\in {\mathbb {S}}^2$.

It is convenient to regard at ${\mathbb {S}}^1$ as the unit circle in the complex plane.

Function spaces. Let $m\ge 0$, $n\ge 1$ be integer numbers. We endow $C^m({\mathbb {S}}^1,{\mathbb {R}}^n)$ with the standard Banach space structure. If $f\in C^1({\mathbb {S}}^1,{\mathbb {R}}^n)$, we identify $f'(x)\equiv f'(x)(ix)$, so that $f':{\mathbb {S}}^1\rightarrow {\mathbb {R}}^n$.

We write $C^m({\mathbb {S}}^1)$ instead of $C^m({\mathbb {S}}^1,{\mathbb {R}})$ and $C^m$ instead of $C^m({\mathbb {S}}^1,{\mathbb {R}}^3)$. For $U\subseteq {\mathbb {S}}^2$ we put

$$\begin{aligned} C^m_{U}:=C^m({\mathbb {S}}^1,U)= \{u\in C^m~|~ u(x)\in U~~\hbox { for any }\ x\in {\mathbb {S}}^1\}~\!. \end{aligned}$$

We identify U with the set of constant functions in $C^2_{U}$, so that $C^2_{U}\setminus U=C^2_{U}\setminus {\mathbb {S}}^2$ contains only nonconstant curves.

The Hilbertian norm in $L^2=L^2({\mathbb {S}}^1,{\mathbb {R}}^3)$ is

$$\begin{aligned} \displaystyle {\Vert u\Vert ^2_{L^2}=\fint \limits _{{\mathbb {S}}^1}|u(x)|^2~\!dx~\!=\frac{1}{2\pi } \int \limits _{{\mathbb {S}}^1}\!\!|u(x)|^2~\!dx}~\!, \end{aligned}$$

and the orthogonal to $T\subseteq C^0$ with respect to the $L^2$ scalar product is given by

$$\begin{aligned} T^\perp =\{\varphi \in C^0~|~\fint \limits _{{\mathbb {S}}^1}u\cdot \varphi ~\!dx=0~~\hbox { for any}\ u\in T~\}. \end{aligned}$$

We regard at $C^2_{{\mathbb {S}}^2}$ as a smooth complete submanifold of $C^2$. If $u\in C^2_{{\mathbb {S}}^2}$, the tangent space to $C^2_{{\mathbb {S}}^2}$ at u is

$$\begin{aligned} T_uC^2_{{\mathbb {S}}^2}=\{\varphi \in C^2~|~ u\cdot \varphi \equiv 0~\hbox { on}\ {\mathbb {S}}^1~\}. \end{aligned}$$

If u is regular, that means $u'(x)\ne 0$ for any $x\in {\mathbb {S}}^1$, then

$$\begin{aligned} T_uC^2_{{\mathbb {S}}^2}=\{g_1 u'+g_2 ~\!u\wedge u'~|~ g=(g_1,g_2)\in C^2({\mathbb {S}}^1,{\mathbb {R}}^2)~\}. \end{aligned}$$

Rotations. Any complex number ${\mathbb {S}}^1$ is identified with the rotation $x\mapsto \xi x$. Recall that $\det (R) = +1$ and $R^{-1} = ^{t}{}{\!R}$ for any $R\in SO(3)$, where SO(3) is the group of rotations of ${\mathbb {R}}^3$ and $^{t}{}{\!R}$ is the transpose of R.

It is well-known that SO(3) is a connected three-dimensional manifold. More precisely, it is a Lie group whose Lie algebra is given by the skew-symmetric matrices, and the tangent space $T_{\text {Id}_3}SO(3)$ at the identity matrix is spanned by

$$\begin{aligned} T_1=\left( \begin{array}{ccc} 0&{}0&{}0\\ 0&{}0&{}-1\\ 0&{}1&{}0 \end{array} \right) ,~ {T_2=\left( \begin{array}{ccc} 0&{}0&{}1\\ 0&{}0&{}0\\ -1&{}0&{}0 \end{array} \right) },~ T_3=\left( \begin{array}{ccc} 0&{}-1&{}0\\ 1&{}0&{}0\\ 0&{}0&{}0 \end{array} \right) ~\!. \end{aligned}$$

A simple explanation of this elementary fact follows by introducing the matrices

$$\begin{aligned} {R}_1^\xi =\left( \begin{array}{ccc} 1&{}0&{}0\\ 0&{}\xi _1&{}-\xi _2\\ 0&{}\xi _2&{}\xi _1 \end{array} \right) ,~ {{R}_2^\xi =\left( \begin{array}{ccc} \xi _1&{}0&{}-\xi _2\\ 0&{}1&{}0\\ \xi _2&{}0&{}\xi _1 \end{array} \right) },~ {R}_3^\xi =\left( \begin{array}{ccc} \xi _1&{}-\xi _2&{}0\\ \xi _2&{}\xi _1&{}0\\ 0&{}0&{}1 \end{array} \right) \end{aligned}$$

for $\xi =\xi _1+i\xi _2\in {\mathbb {S}}^1$. Clearly $R_h^\xi$ is a rotation about the $\langle ~\!e_h~\!\rangle$ axis. By differentiating $R_h^\xi$ with respect to $\xi \in {\mathbb {S}}^1$ at $\xi =1$ one gets $T_h=d{{R}^\xi _h}_{\big |\xi =1}$, and thus infers that $\{T_h\}$ is a basis for $T_{\text {Id}_3}SO(3)$. In accordance with the Lie group structure of SO(3), the tangent space to SO(3) at $R\in SO(3)$ is obtained by rotating $T_{\text {Id}_3}SO(3)$. Hence

$$\begin{aligned} T_{R}SO(3)=\langle {R}T_1,{R}T_2,{R}T_3\rangle . \end{aligned}$$

Finally, for any $q\in {\mathbb {S}}^2$ we denote by $d_R$ the differential of the function $SO(3)\rightarrow {\mathbb {S}}^2$, $R\mapsto Rq$, so that $d_R(Rq)\tau \in T_{Rq}{\mathbb {S}}^2$ for any $\tau \in T_RSO(3)$. We have the formula

$$\begin{aligned} d_R(Rq)(RT_h)=R(e_h\wedge q)=Re_h\wedge Rq. \end{aligned}$$

(1.3)

2 A “local” variational approach

We put $\varepsilon =c^{-1}$ and study Problem (${\mathcal {P}}_{K,\varepsilon ^{-1}}$) for $\varepsilon$ close to 0. We take advantage of its geometrical interpretation to rewrite it in an equivalent way. Let $\gamma$ be a solution to (${\mathcal {P}}_{K,\varepsilon ^{-1}}$), and let ${\mathcal {L}}_\gamma$ be its length. Extend $\gamma$ to an $\varepsilon {\mathcal {L}}_\gamma$-periodic function on ${\mathbb {R}}$ and consider the curve $u\in C^2_{{\mathbb {S}}^2}$, $u(e^{i\theta })=\gamma \big (\frac{\varepsilon {\mathcal {L}}_\gamma }{2\pi }~\!\theta \big )$. Evidently u and $\gamma$ have the same length ${\mathcal {L}}_\gamma$ and curvature $\varepsilon K$. Moreover $|u'|\equiv {\mathcal {L}}_\gamma /2\pi$ and u solves the system

$$\begin{aligned} u''+|u'|^2u=|u'|\varepsilon K(u)~\!u\wedge u'~\qquad \hbox { on}\ {\mathbb {S}}^1, \end{aligned}$$

(2.1)

because $\gamma$ solves (1.1). Conversely, any solution $u\in C^2_{{\mathbb {S}}^2}\setminus {\mathbb {S}}^2$ to (2.1) has constant speed $|u'|$, curvature $\varepsilon K(u)$ and gives rise to a solution to (${\mathcal {P}}_{K,\varepsilon ^{-1}}$).

The main goal of the present section is to show that for any point $p\in {\mathbb {S}}^2$, the problem of finding solutions to (2.1) in $C^2_{{\mathbb {S}}^2\setminus \{p\}}$, that is an open subset of $C^2_{{\mathbb {S}}^2}$, can be faced by using variational methods. First, we need to introduce the functional

$$\begin{aligned} L(u)=\Big (\fint \limits _{{\mathbb {S}}^1}|u'|^2~\!dx\Big )^\frac{1}{2}~,\quad L:C^2_{{\mathbb {S}}^2}\setminus {\mathbb {S}}^2\rightarrow {\mathbb {R}}. \end{aligned}$$

(2.2)

Notice that the Cauchy-Schwarz inequality gives $\displaystyle {{\mathcal {L}}_u\le 2\pi L(u)}$, and equality holds if and only if $|u'|$ is constant. Moreover, it holds that

$$\begin{aligned} L({R} u\circ \xi )=L(u)\quad \text {for any } \xi \in {\mathbb {S}}^1,~{R}\in SO(3). \end{aligned}$$

(2.3)

Finally, we notice that L is Fréchet differentiable at any $u\in C^2_{{\mathbb {S}}^2}\setminus {\mathbb {S}}^2$, with differential

$$\begin{aligned} L'(u)\varphi =\frac{1}{L(u)}~\!\fint \limits _{{\mathbb {S}}^1}u'\cdot \ \varphi '~\!dx =\frac{1}{L(u)}~\!\fint \limits _{{\mathbb {S}}^1}(-u''-|u'|^2u)\cdot \varphi ~\!dx \quad \text {for any } \varphi \in T_uC^2_{{\mathbb {S}}^2}. \nonumber \\ \end{aligned}$$

(2.4)

In the next lemma we provide a variational reading of the right-hand side of (2.1), see also [15] and [11, Remark 2.2].

Lemma 2.1

Let $K \in C^0({\mathbb {S}}^2)$ and let U, V be open and contractible subsets of ${\mathbb {S}}^2$.

i):

There exists a unique $C^1$ functional ${\mathcal {A}}^U_K: C^2_U \rightarrow {\mathbb {R}}$, such that ${\mathcal {A}}^U_K( u) = 0$ if u is constant, and

$$\begin{aligned} ({\mathcal {A}}^U_K)'( u) \phi = \fint _{S^1} K(u) \phi \cdot u \wedge u' ~\!dx\quad \text {for any } u\in C^2_U,\, \phi \in T_u C^2_{{\mathbb {S}}^2}; \end{aligned}$$

(2.5)

ii):

If $R\in SO(3)$, $\xi \in {\mathbb {S}}^1$ and $u\in C^2_U$, then ${{\mathcal {A}}^{RU}_{K\circ ^{t}{}{\!R}}(Ru\circ \xi )={\mathcal {A}}^U_K(u)}$;

iii):

If $U\cap V$ is nonempty and contractible, then ${\mathcal {A}}^U_K(u)={\mathcal {A}}^V_K(u)$ for any $u\in C^2_{U\cap V}$;

iv):

Let $u\in C^2_{{\mathbb {S}}^2}$. The function $p \mapsto {\mathcal {A}}_K^{{\mathbb {S}}^2\setminus \{p\}}(u)$ is constant on each connected component of ${\mathbb {S}}^2 \setminus u({\mathbb {S}}^1)$;

v):

Let $u\in C^2_U$ be a positively oriented parametrization of the boundary of a regular open set $\Omega _u \subset U$. Then

$$\begin{aligned} {\mathcal {A}}_K^U(u) = -\frac{1}{2 \pi } \int \limits _{\Omega _u} K(q)~\!d\sigma _{\!q}~\!. \end{aligned}$$

Proof

Take a 1-form $\beta _K^U$ on U, such that

$$\begin{aligned} d \beta _K^U ={-} K(q) ~\!d\sigma _{\!q}~\!, \end{aligned}$$

(2.6)

where $d\sigma _{\!q}$ is the restriction of the volume form on the sphere. We put

$$\begin{aligned} {\mathcal {A}}^U_K( u) = \fint \limits _{{\mathbb {S}}^1}u^*\beta _K^U = \fint \limits _{{\mathbb {S}}^1}\beta _K^U(u) u'~\! dx~,\quad u\in C^2_U~\!. \end{aligned}$$

It is evident that ${\mathcal {A}}^U_K( u)=0$ if u is constant. Formula (2.5) can be derived by using Lie differential calculus or local coordinates, like in the proof of [6, Lemma 3]. Elementary arguments and (2.5) give the $C^1$ differentiability of the functional ${\mathcal {A}}^U_K$. Uniqueness is trivial, because $C^2_U$ is a connected manifold. In particular, for $u\in C^2_U$ the real number ${\mathcal {A}}^U_K(u)$ does not depend on the choice of $\beta _K^U$.

To prove ii) take a 1-form $\beta$ in the domain RU such that $d\beta =-(K\circ ^{t}{}{\!R}) ~\!d\sigma _{\!q}$. Clearly $R^*\beta$ is a 1-form in U, and $d(R^*\beta )=R^*(d\beta )= - K(q)d\sigma _{\!q}$. Thus we can take $\beta _K^U=R^*\beta$ in formula (2.6) and we obtain

$$\begin{aligned} {{\mathcal {A}}^{RU}_{K\circ ^{t}{}{\!R}}(Ru)= \fint \limits _{{\mathbb {S}}^1}(Ru)^*\beta = \fint \limits _{{\mathbb {S}}^1}u^* (R^*\beta ) = {\mathcal {A}}^U_K(u)} \end{aligned}$$

for any $u\in C^2_U$. The invariance of the area functional with respect to composition with rotations of ${\mathbb {S}}^1$ is immediate.

Now we prove iii). If $V\subset U$ and $u\in C^2_V$, then the restriction of $\beta ^U_K$ to V can be used to compute ${\mathcal {A}}_K^V(u)$. Thus ${\mathcal {A}}_K^V(u)={\mathcal {A}}_K^U(u)$. It follows that if two open, connected sets U, V have contractible intersection and $u\in C^2_{U\cap V}$, then ${\mathcal {A}}_K^{U\cap V}(u)={\mathcal {A}}_K^U(u)$ and ${\mathcal {A}}_K^{U\cap V}(u)={\mathcal {A}}_K^V(u)$.

Claim iv) readily follows from iii). In fact, take $p_0\in {\mathbb {S}}^2\setminus u({\mathbb {S}}^1)$ and a small disk ${\mathcal {D}}_\delta (p_0)\subset {\mathbb {S}}^2\setminus u({\mathbb {S}}^1)$. For any $p\in {\mathcal {D}}_\delta (p_0)$ we have

$$\begin{aligned} {\mathcal {A}}^{{\mathbb {S}}^2\setminus \{p\}}(u)={\mathcal {A}}^{{\mathbb {S}}^2\setminus {\mathcal {D}}_\delta (p_0)}(u)= {\mathcal {A}}^{{\mathbb {S}}^2\setminus \{p_0\}}(u)~\!. \end{aligned}$$

We proved that the function $p\mapsto {\mathcal {A}}^{{\mathbb {S}}^2\setminus \{p\}}(u)$ is locally constant on ${\mathbb {S}}^2\setminus u({\mathbb {S}}^1)$, and hence is constant on each connected component of ${\mathbb {S}}^2 \setminus u({\mathbb {S}}^1)$.

For the last claim we use Stokes’ theorem to get

$$\begin{aligned} 2\pi {\mathcal {A}}_K^U(u) = \int \limits _{{\mathbb {S}}^1} u^*\beta ^U_K =\int \limits _{\partial \Omega _u} \beta ^U_K= \int \limits _{\Omega _u} d \beta ^U_K = -\int \limits _{\Omega _u} K(q) d\sigma _{\!q} \end{aligned}$$

by (2.6). The lemma is completely proved. $\square$

From now on we write

$$\begin{aligned} A_K(p;u)= {\mathcal {A}}_K^{{\mathbb {S}}^2\setminus \{p\}}(u) ~,\quad p\in {\mathbb {S}}^2~,~~ u\in C^2_{{\mathbb {S}}^2\setminus \{p\}}. \end{aligned}$$

By Lemma 2.1, the functional $A_K$ enjoys the following properties,

A1):

The functional $A_K(p; \cdot )$ is of class $C^1$ on $C^2_{{\mathbb {S}}^2\setminus \{p\}}$, and

$$\begin{aligned} A_K'(p; u) \phi = \fint _{{\mathbb {S}}^1} K(u) \phi \cdot u \wedge u' \ dx \ \ \text {for any } u\in C^2_{{\mathbb {S}}^2\setminus \{p\}},\, \phi \in T_u C^2_{{\mathbb {S}}^2}. \end{aligned}$$

A2):

If $R\in SO(3)$, $\xi \in {\mathbb {S}}^1$, and $u\in C^2_{{\mathbb {S}}^2\setminus \{p\}}$, then ${A_{K\circ ^{t}{}{\!R}}(Rp;Ru\circ \xi )=A_K(p;u)}$.

A3):

Let $u \in C^2_{{\mathbb {S}}^2}$. The function $p \mapsto A_K(p; u)$ is locally constant on ${\mathbb {S}}^2 \setminus u({\mathbb {S}}^1)$.

A4):

Let $u\in C^2_{{\mathbb {S}}^2\setminus \{p\}}$ be a positively oriented parametrization of the boundary of a regular open set $\Omega _u \subset {\mathbb {S}}^2 \setminus \{p\}$. Then

$$\begin{aligned} A_K(p; u) = -\frac{1}{2 \pi } \int \limits _{\Omega _u} K(q)~\!d\sigma _{\!q}~\!. \end{aligned}$$

Remark 2.2

To find an explicit formula for $A_K(p;~\!\cdot ~\!)$ let $\Pi _p:{\mathbb {S}}^2\setminus \{p\}\rightarrow {\mathbb {R}}^2$ be the stereographic projection from the pole p. If $u\in C^2_{{\mathbb {S}}^2\setminus \{p\}}$, then $\Pi _p\circ u$ is a curve in ${\mathbb {R}}^2$ and $(\Pi _p^{-1})^*(Kd\sigma _{\!q})=(K\circ \Pi _p^{-1})\mathrm{det}J_{\Pi _p^{-1}}(z)dz$ is a 2-form on ${\mathbb {R}}^2$. Let ${\tilde{\beta }}_K^p$ be a 1-form on ${\mathbb {R}}^2$ such that $d{\tilde{\beta }}_K^p=(\Pi _p^{-1})^*(Kd\sigma _{\!q})$. Then

$$\begin{aligned} A_K(p;u)=\fint \limits _{{\mathbb {S}}^1}u^*(\Pi _p^*~\!{\tilde{\beta }}_K^p)~\!=\fint \limits _{{\mathbb {S}}^1}(\Pi _p\circ u)^*{\tilde{\beta }}_K^p~\!. \end{aligned}$$

For instance, if $K\equiv 1$ is constant one can take

$$\begin{aligned} A_1(p;u) =\fint \limits _{{\mathbb {S}}^1}\frac{p}{1-u\cdot p}~\cdot u\wedge u'~\!dx =2\fint \limits _{{\mathbb {S}}^1}\frac{p}{|u-p|^2}\cdot u\wedge u'~\!dx. \end{aligned}$$

The next lemma provides the predicted "local" variational approach to (2.1).

Lemma 2.3

Let $K\in C^0({\mathbb {S}}^2)$.

i) For any $p\in {\mathbb {S}}^2$, the functional

$$\begin{aligned} E_{\varepsilon K}(p;u)=L(u)+\varepsilon A_K(p;u)~\!, \quad E_{\varepsilon K}(p;~\!\cdot ~\!): C^2_{{\mathbb {S}}^2\setminus \{p\}}\setminus {\mathbb {S}}^2\rightarrow {\mathbb {R}}\end{aligned}$$

is of class $C^1$, with differential

$$\begin{aligned} L(u)E'_{\varepsilon K}(p;u)\varphi =\fint \limits _{{\mathbb {S}}^1}\big (-u''+ L(u)~\!\varepsilon K(u) u\wedge u'\big )\cdot \varphi ~\!dx,\quad \text {for any } \varphi \in T_u C^2_{{\mathbb {S}}^2}.\nonumber \\ \end{aligned}$$

(2.7)

In particular, any critical point $u\in C^2_{{\mathbb {S}}^2\setminus \{p\}}\setminus {\mathbb {S}}^2$ for $E_{\varepsilon K}(p;~\!\cdot ~\!)$ solves (2.1).

ii) If $R\in SO(3)$, $\xi \in {\mathbb {S}}^1$ and $p\in {\mathbb {S}}^2$, then $E_{{\varepsilon K}\circ ^{t}{}{\!R}}(Rp;Ru\circ \xi )=E_{\varepsilon K}(p;u)$ for any nonconstant curve $u\in C^2_{{\mathbb {S}}^2\setminus \{p\}}$, and thus

$$\begin{aligned} E'_{\varepsilon K}(p;u)u'=0\quad \text {for any } u\in C^2_{{\mathbb {S}}^2\setminus \{p\}}\setminus {\mathbb {S}}^2. \end{aligned}$$

(2.8)

iii) Let $u\in C^2_{{\mathbb {S}}^2}\setminus {\mathbb {S}}^2$. The function $E_{\varepsilon K}(~\!\cdot ~\!;u):{\mathbb {S}}^2\setminus u({\mathbb {S}}^1)\rightarrow {\mathbb {R}}$ is locally constant.

iv) If $K\in C^1({\mathbb {S}}^2)$ then the functional $E_{\varepsilon K}(p;~\!\cdot ~\!)$ is of class $C^2$ on its domain.

Proof

Formula (2.4) and the property A1) of the area functional give the $C^1$ regularity of $E_{\varepsilon K}(p;~\!\cdot ~\!)$ and (2.7). Let u be a critical point for $E_{\varepsilon K}(p;~\!\cdot ~\!)$. Take any $\varphi \in C^2$ and put $\varphi ^\top =\varphi -(\varphi \cdot u)u\in T_u C^2_{{\mathbb {S}}^2}$. We have $\varphi \cdot u\wedge u'= \varphi ^\top \cdot u\wedge u'$ on ${\mathbb {S}}^1$, and $u'\cdot (\varphi ^\top )'=u'\cdot \varphi '-(\varphi \cdot u)|u'|^2$ because $u'\cdot u\equiv 0$. Since

$$\begin{aligned} \begin{aligned} 0=L(u)E'_{\varepsilon K}(p;u)\varphi ^\top =&\fint \limits _{{\mathbb {S}}^1}\big (u'\cdot (\varphi ^\top )'+L(u)~\!\varepsilon K(u)\varphi ^\top \cdot u\wedge u'\big )~\!dx\\ =&\fint \limits _{{\mathbb {S}}^1}\big (u'\cdot \varphi '-(\varphi \cdot u)|u'|^2+L(u)~\!\varepsilon K(u)\varphi \cdot u\wedge u'\big )~\!dx~\!, \end{aligned} \end{aligned}$$

and therefore u solves $u''+|u'|^2u=L(u)~\!\varepsilon K(u)~\!u\wedge u'$ on ${\mathbb {S}}^1$. Since $u''\cdot u'\equiv 0$, we see that $|u'|\equiv L(u)$ is constant, and thus u solves (2.1).

Statements ii), iii) follow from (2.3), A2) and A3) (to check (2.8) take the derivative of the identity $E_{\varepsilon K}(p;u\circ \xi )=E_{\varepsilon K}(p;u)$ with respect to $\xi \in {\mathbb {S}}^1$ at $\xi =1$). Finally, iv) can be proved via elementary arguments, starting from (2.7). $\square$

3 Geodesics

For any rotation $R\in SO(3)$, the loop

$$\begin{aligned} \omega _{\!R}(x)=R\big (x_1,x_2,0)~,\quad x=x_1+ix_2\in {\mathbb {S}}^1~\!, \end{aligned}$$

is a parameterization of the boundary of ${{\mathcal {D}}}_{\!{\frac{\pi }{2}}}(Re_3)$ and solves

$$\begin{aligned} \omega _{\!R}''+|\omega _{\!R}'|^2\omega _{\!R}=0~,\quad L(\omega _{\!R})=|\omega _{\!R}'|={1}~\!. \end{aligned}$$

(3.1)

In order to simplify notations, from now on we write

$$\begin{aligned} \omega (x)=\omega _{\mathrm{Id}}(x)=\big (x_1,x_2,0)~,\quad x=x_1+ix_2\in {\mathbb {S}}^1~\!. \end{aligned}$$

The tangent space to the smooth 3-dimensional manifold

$$\begin{aligned} {\mathcal {S}}=\big \{\omega _{\!R}~|~{R}\in SO(3)~\!\big \}\subset C^2_{{\mathbb {S}}^2} \end{aligned}$$

at $\omega _{\!R}\in {\mathcal {S}}$ can be easily computed via formula (1.3). It turns out that

$$\begin{aligned} T_{\omega _{\!R}}{\mathcal {S}} =\{q\wedge \omega _{\!R}~|~ q\in {\mathbb {R}}^3\}= \langle {R}e_1\wedge \omega _{\!R}~,~ Re_2\wedge \omega _{\!R}~,~Re_3\wedge \omega _{\!R}~\rangle . \end{aligned}$$

We introduce the function

$$\begin{aligned} J_0(u):= -u''-|u'|^2u~,\quad J_0:C^2_{{\mathbb {S}}^2}\setminus {\mathbb {S}}^2\rightarrow C^0, \end{aligned}$$

so that ${\mathcal {S}}\subset \{J_0=0\}$. By (2.4) we have

$$\begin{aligned} L(u)L'(u)\varphi =\fint \limits _{{\mathbb {S}}^1}J_0(u)\cdot \varphi ~\!dx\quad \text {for any } u\in C^2_{{\mathbb {S}}^2}\setminus {\mathbb {S}}^2,\, \varphi \in T_uC^2_{{\mathbb {S}}^2}. \end{aligned}$$

(3.2)

Moreover, for $u\in C^2_{{\mathbb {S}}^2}\setminus {\mathbb {S}}^2$, $q\in {\mathbb {R}}^3$ and ${R}\in SO(3)$ it holds that

$$\begin{aligned} \fint \limits _{{\mathbb {S}}^1}J_0(u)\cdot q\wedge u~\!dx=0~,\quad J_0({R} u)=RJ_0(u)~\!. \end{aligned}$$

(3.3)

The first identity readily follows via integration by parts or can be obtained by differentiating the identity $L(Ru)=L(u)$ with respect to $R\in SO(3)$. The second one is immediate.

Clearly $J_0$ is of class $C^2$; for $R\in SO(3)$ and $\varphi$ in the tangent space

$$\begin{aligned} T_{\omega _{\!R}}C^2_{{\mathbb {S}}^2}=\{\varphi =g_1~\!\omega _{\!R}'+g_2 ~\!\omega _{\!R}\wedge \omega _{\!R}'~|~g=(g_1,g_2)\in C^2({\mathbb {S}}^1,{\mathbb {R}}^2)~\}, \end{aligned}$$

(3.4)

we have

$$\begin{aligned} J_0'(\omega _{\!R})\varphi =-\varphi ''-2 (\omega _{\!R}'\cdot \varphi ')\omega _{\!R}-\varphi \\ ~\!. \end{aligned}$$

Further, the operator $J'_0(\omega _R)$ is self adjoint in $L^2({\mathbb {S}}^1,{\mathbb {R}}^3)$, that is,

$$\begin{aligned} \fint \limits _{{\mathbb {S}}^1}J'_0(\omega _R)\varphi \cdot {{\tilde{\varphi }}}~\!dx=\fint \limits _{{\mathbb {S}}^1}J'_0(\omega _R){{\tilde{\varphi }}}\cdot \varphi ~\!dx \quad \text {for any } \varphi ,{\tilde{\varphi }}\in T_{\omega _R}C^2_{{\mathbb {S}}^2}. \end{aligned}$$

(3.5)

By differentiating the identity $J_0(\omega _{\!R})=0$ with respect to ${R}\in SO(3)$, we see that $T_{\omega _{\!R}}{\mathcal {S}}\subseteq \ker \! J_0'(\omega _{\!R})$. Actually, equality holds, as shown in the next crucial lemma.

Lemma 3.1

(Nondegeneracy) Let ${R}\in SO(3)$. Then

i):

$\displaystyle {\ker \! J'_0(\omega _{\!R})=T_{\omega _{\!R}}{\mathcal {S}}}$;

ii):

If $\varphi \in T_{\omega _{\!R}}C^2_{{\mathbb {S}}^2}$ and $J_0'(\omega _{\!R})\varphi \in T_{\omega _{\!R}}{\mathcal {S}}$, then $\varphi \in T_{\omega _{\!R}}{\mathcal {S}}$;

iii):

For any $u\in T_{\omega _{\!R}}{\mathcal {S}}^\perp$ there exists a unique $\varphi \in {T_{\omega _{\!R}}C^2_{{\mathbb {S}}^2}}\cap T_{\omega _{\!R}}{\mathcal {S}}^\perp$ such that $J_0'(\omega _{\!R})\varphi =u$.

Proof

One can argue by adapting the computations in [18, Sect. 5]. We provide here a simpler argument.

Since $J'_0(\omega _{\!R})(R\varphi )=R\big (J'_0(\omega )\varphi \big )$ for any $\varphi \in T_{\omega }C^2_{{\mathbb {S}}^2}$, it is not restrictive to assume that R is the identity matrix. By direct computations based on (3.1), one can check that

$$\begin{aligned} J'_0(\omega )(\psi ~\!\omega ')=-\psi ''~\!\omega '~, \quad J'_0(\omega )(\psi ~\!\omega \wedge \omega ')= \big (-\psi '' -\psi \big )~\!\omega \wedge \omega ' \end{aligned}$$

for any $\psi \in C^2({\mathbb {S}}^1,{\mathbb {R}})$. Since by (3.4) any function $\varphi \in T_{\omega }C^2_{{\mathbb {S}}^2}$ can be written as

$$\begin{aligned} \varphi =(\varphi \cdot \omega ')\omega '+(\varphi \cdot \omega \wedge \omega ')~\!\omega \wedge \omega '~\!, \end{aligned}$$

we are led to introduce the differential operator $B:C^2({\mathbb {S}}^1,{\mathbb {R}}^2)\rightarrow C^0({\mathbb {S}}^1,{\mathbb {R}}^2)$,

$$\begin{aligned} B(g)= -g_1''~\!e_1+(-g_2''-g_2)e_2~,\qquad g=(g_1,g_2)\in C^2({\mathbb {S}}^1,{\mathbb {R}}^2)~\!. \end{aligned}$$

and the function transform

$$\begin{aligned} \Psi \varphi =(\varphi \cdot \omega ')~\!e_1+~\!(\varphi \cdot \omega \wedge \omega ')~\!e_2~\!, \quad \Psi : T_{\omega }C^2_{{\mathbb {S}}^2}\rightarrow C^2({\mathbb {S}}^1,{\mathbb {R}}^2)~\!, \end{aligned}$$

so that

$$\begin{aligned} J_0'(\omega )\varphi =\Psi ^{-1}B(\Psi \varphi )\quad \text {for any } \varphi \in T_{\omega }C^2_{{\mathbb {S}}^2},\quad \Psi (\ker \! J'_0(\omega ))=\ker B~\!. \end{aligned}$$

(3.6)

We proved that $\ker \! J'_0(\omega )$ and $T_{\omega }{\mathcal {S}}$ have both dimension 3, thus they must coincide because $T_{\omega }{\mathcal {S}}\subseteq \ker \! J'_0(\omega )$.

For future convenience we notice that $\Psi$ is an isometry with respect to the $L^2$ norms, and in particular

$$\begin{aligned} \fint \limits _{{\mathbb {S}}^1}\big (\Psi \varphi \big )\cdot \big (\Psi {\tilde{\varphi }}\big )~\!dx= \fint \limits _{{\mathbb {S}}^1}\varphi \cdot {\tilde{\varphi }}~\!dx\quad \text {for any } \varphi ,{\tilde{\varphi }}\in T_{\omega }C^2_{{\mathbb {S}}^2}. \end{aligned}$$

(3.7)

Now we prove ii). If $\tau :=J_0'(\omega )\varphi \in T_{\omega }{\mathcal {S}}$, then $J_0'(\omega )\tau =0$, as $\displaystyle {\ker \! J_0'(\omega )=T_{\omega }{\mathcal {S}}}$. But then, using (3.5) we get

$$\begin{aligned} \fint \limits _{{\mathbb {S}}^1}|J_0'(\omega )\varphi |^2~\!dx=\fint \limits _{{\mathbb {S}}^1}J_0'(\omega )\varphi \cdot \tau ~\!dx=\fint \limits _{{\mathbb {S}}^1}J_0'(\omega )\tau \cdot \varphi ~\!dx=0. \end{aligned}$$

Thus $J_0'(\omega )\varphi =0$, that means $\varphi \in T_{\omega }{\mathcal {S}}$.

It remains to prove iii). Since $\Psi (T_\omega {\mathcal {S}})=\ker B$, from (3.6) and (3.7) we have that $u \in T_{\omega }{\mathcal {S}}^\perp$ if and only if $\Psi u\in \ker B^\perp$. In particular, if $u\in T_{\omega }{\mathcal {S}}^\perp$, then one can compute the unique solution $g_u\in \ker B^\perp$ to the system $Bg_u=\Psi u$. The function $\varphi :=\Psi ^{-1}g_u$ belongs to $T_{\omega }{\mathcal {S}}^\perp$; thanks to (3.6) it solves $J_0'(\omega )\varphi =u$, and is uniquely determined by u. The lemma is completely proved. $\square$

Remark 3.2

For future convenience we compute

$$\begin{aligned} m_{hj}=\fint \limits _{{\mathbb {S}}^1}(Re_h\wedge \omega _{\!R})\cdot (Re_j\wedge \omega _{\!R})~\!dx= \fint \limits _{{\mathbb {S}}^1}(e_h\wedge \omega )\cdot (e_j\wedge \omega )~\!dx= \delta _{hj}-\fint \limits _{{\mathbb {S}}^1}\omega _h\omega _j~\!dx. \end{aligned}$$

We see that the functions ${R}e_j\wedge \omega _{\!R}=R(e_j\wedge \omega )$ provide an orthogonal basis for $T_{\omega _{\!R}}{\mathcal {S}}$ endowed with the $L^2$ scalar product. More precisely, the matrix M associated to this scalar product with respect to the basis $\{{R}e_j\wedge \omega _{\!R}\}$ is given by

$$\begin{aligned} M=\left( \begin{array}{ccc} \frac{1}{2}&{}0&{}0\\ 0&{}\frac{1}{2}&{}0\\ 0&{}0&{}1 \end{array}\right) ~\!. \end{aligned}$$

3.1 Finite dimensional reduction

By the remarks at the beginning of Sect. 2, we are led to study problem (2.1) for $\varepsilon =c^{-1}$ close to 0. Further, since any solution u to (2.1) satisfies $|u'|\equiv L(u)$, we can rewrite (2.1) in the following, equivalent way,

$$\begin{aligned} u''+|u'|^2u={L(u)}\varepsilon K(u)~\! u\wedge u'~,\qquad u\in C^2_{{\mathbb {S}}^2}\setminus {\mathbb {S}}^2~\!. \end{aligned}$$

(3.8)

We will look for solutions to (3.8) by solving $J_\varepsilon (u)=0$, where $J_\varepsilon : C^2_{{\mathbb {S}}^2}\setminus {\mathbb {S}}^2\rightarrow C^0$,

$$\begin{aligned} J_\varepsilon (u)=J_0(u)+\varepsilon L(u)K(u)~\!u\wedge u' =-u''-|u'|^2u+L(u)\varepsilon K(u)~\! u\wedge u'.\nonumber \\ \end{aligned}$$

(3.9)

Thanks to (2.7), we can write

$$\begin{aligned} L(u)E'_{\varepsilon K}(p;u)\varphi =\fint \limits _{{\mathbb {S}}^1}J_\varepsilon (u)\cdot \varphi ~\!dx~,\quad \text {for } u\in C^2_{{\mathbb {S}}^2}\setminus {\mathbb {S}}^2,\, p\notin u({\mathbb {S}}^1),\, \varphi \in T_uC^2_{{\mathbb {S}}^2}.\nonumber \\ \end{aligned}$$

(3.10)

The regularity assumption on K implies that $J_\varepsilon$ is of class $C^1$ on its domain. In addition, $J_\varepsilon (u\circ \xi )=J_\varepsilon (u)$ for any $\xi \in {\mathbb {S}}^1$, and integration by parts gives

$$\begin{aligned} \fint \limits _{{\mathbb {S}}^1}J_\varepsilon (u)\cdot u'~\!dx=0 \qquad \text {for any } u\in C^2_{{\mathbb {S}}^2}\setminus {\mathbb {S}}^2. \end{aligned}$$

In general, the identities in (3.3) are not satisfied if $\varepsilon \ne 0$, because the perturbation term breaks the invariances of the operator $J_0$.

In the next lemma we provide the main step to obtain our multiplicity results.

Lemma 3.3

There exist ${\overline{\varepsilon }}>0$ and a $C^1$ function

$$\begin{aligned} {[}-{\overline{\varepsilon }},{\overline{\varepsilon }}]\times SO(3)\rightarrow C^2_{{\mathbb {S}}^2}\setminus {\mathbb {S}}^2~\quad (\varepsilon ,{R})\mapsto u^\varepsilon _{R} \end{aligned}$$

such that $u^\varepsilon _{R}$ is an embedded loop, and moreover

(i):

$u^0_{R}=\omega _{\!R}$;

(ii):

$u^\varepsilon _{R}\in T_{\omega _{\!R}}{\mathcal {S}}^\perp$;

(iii):

$J_\varepsilon (u^\varepsilon _{R}) \in T_{\omega _{\!R}}{\mathcal {S}}$;

(iv):

The function $[-{\overline{\varepsilon }},{\overline{\varepsilon }}]\times SO(3)\rightarrow {\mathbb {R}}$,

$$\begin{aligned} (\varepsilon ,{R})\mapsto {\mathcal {E}}^\varepsilon ({R}):=E_{\varepsilon K}(-Re_3;u^\varepsilon _R)= L(u^\varepsilon _R)+\varepsilon A_{K}(-Re_3;u^\varepsilon _R) \end{aligned}$$

is well defined, of class $C^1$ on its domain, and $d_R{\mathcal {E}}^\varepsilon ({R})(RT_3)=0$.

(v):

$R\in SO(3)$ is critical for ${\mathcal {E}}^\varepsilon : SO(3)\rightarrow {\mathbb {R}}$ if and only if $J_\varepsilon (u^\varepsilon _{R})=0$.

(vi):

Put ${\mathcal {E}}^\varepsilon _0({R})=E_{\varepsilon K}(-Re_3;\omega _{\!R})=1+\varepsilon A_{K}(-Re_3,\omega _{\!R})$. As $\varepsilon \rightarrow 0$, we have

$$\begin{aligned} {\mathcal {E}}^\varepsilon ({R})- {\mathcal {E}}^\varepsilon _0({R})=o(\varepsilon ) \end{aligned}$$

(3.11)

uniformly on SO(3), together with the derivatives with respect to $R\in SO(3)$.

Proof

Consider the differentiable functions

$$\begin{aligned} \displaystyle {{\mathcal {F}}_1:{\mathbb {R}}\!\times \! SO(3)\!\times \! (C^2_{{\mathbb {S}}^2}\!\!\setminus \!{\mathbb {S}}^2)\!\!\times \!\!{\mathbb {R}}^3\rightarrow C^0}~,~ \displaystyle {{\mathcal {F}}_1(\varepsilon ,{R},u;\zeta )= J_\varepsilon (u)- \sum _{j=1}^3 \zeta _j~\!(Re_j\wedge \omega _{\!R})} \\ \displaystyle { {\mathcal {F}}_2:{\mathbb {R}}\!\times \! SO(3)\!\times \! (C^2_{{\mathbb {S}}^2}\!\!\setminus \!{\mathbb {S}}^2)\!\!\times \!\!{\mathbb {R}}^3\rightarrow {\mathbb {R}}^3}~,~ {{\mathcal {F}}_2(\varepsilon ,{R},u;\zeta )= \sum _{j=1}^3\!\big (\displaystyle {\fint \limits _{{\mathbb {S}}^1}u\cdot Re_j\wedge \omega _{\!R}~\!dx\big )e_j}} \end{aligned}$$

where $\zeta =(\zeta _1,\zeta _2,\zeta _3)\in {\mathbb {R}}^3$, and then let

$$\begin{aligned} {\mathcal {F}}:{\mathbb {R}}\times SO(3)\times (C^2_{{\mathbb {S}}^2}\!\!\setminus \!{\mathbb {S}}^2)\!\times \!{\mathbb {R}}^3\rightarrow C^0\!\times \! {\mathbb {R}}^3~,\quad {\mathcal {F}}=\big ({\mathcal {F}}_1,{\mathcal {F}}_2). \end{aligned}$$

Fix ${R}\in SO(3)$. Since $J_0(\omega _{\!R})=0$ by (3.1), then ${\mathcal {F}}(0,{R},\omega _{\!R};0)=0$. Our first goal is to solve the equation ${\mathcal {F}}(\varepsilon ,{R},u;\zeta )=(0;0)$ in a neighborhood of $(0,{R},\omega _{\!R};0)$, via the implicit function theorem.

Consider the differentiable function

$$\begin{aligned} {\mathcal {F}}(0,{R},~\!\cdot ~\!;~\!\cdot ~\!) ~\!:~\!(u;\zeta )\mapsto {\mathcal {F}}(0,{R},u;\zeta )~\!, \quad (C^2_{{\mathbb {S}}^2}\!\!\setminus \!{\mathbb {S}}^2)\!\times \!{\mathbb {R}}^3\rightarrow C^0\!\times \! {\mathbb {R}}^3 \end{aligned}$$

and let

$$\begin{aligned} {\mathcal {L}}=({\mathcal {L}}_1,{\mathcal {L}}_2): (T_{\omega _{\!R}}C^2_{{\mathbb {S}}^2})\!\times \!{\mathbb {R}}^3\rightarrow C^0\!\times \!{\mathbb {R}}^3 \end{aligned}$$

be its differential evaluated at $(u;\zeta )=(\omega _{\!R};0)$. We need to prove that ${\mathcal {L}}$ is invertible.

Take $\varphi \in T_{\omega _{\!R}}C^2_{{\mathbb {S}}^2}$ and $p=(p_1,p_2,p_3)\in {\mathbb {R}}^3$. It is easy to compute

$$\begin{aligned} {\mathcal {L}}_1(\varphi ;{p})= J_0'(\omega _{\!R}) \varphi -\sum _{j=1}^3 ~\!p_j~\!(Re_j\wedge \omega _{\!R}), \quad {\mathcal {L}}_2(\varphi ;{p})=\sum _{j=1}^3 \big ( \fint \limits _{{\mathbb {S}}^1}\varphi \cdot {R}e_j\wedge \omega _{\!R}~\!dx\big )e_j~\!. \end{aligned}$$

Next, recall that $T_{\omega _{\!R}}{\mathcal {S}}$ is spanned by the functions $Re_j\wedge \omega _{\!R}$. If ${\mathcal {L}}_1(\varphi ;{p})=0$ then $J_0'(\omega _{\!R}) \varphi \in T_{\omega _{\!R}}{\mathcal {S}}$, hence $\varphi \in T_{\omega _{\!R}}{\mathcal {S}}$ by ii) in Lemma 3.1; if ${\mathcal {L}}_2(\varphi ;{p})=0$ then $\varphi \in T_{\omega _{\!R}}{\mathcal {S}}^\perp$. Therefore, the operator ${\mathcal {L}}$ is injective.

Before proving surjectivity we notice that

$$\begin{aligned} J'_0(\omega _{\!R})\varphi \in T_{\omega _{\!R}}{\mathcal {S}}^\perp \quad \hbox { for any}\ \varphi \in T_{\omega _{\!R}}C^2_{{\mathbb {S}}^2} \end{aligned}$$

(3.12)

because of (3.5) and since $T_{\omega _{\!R}}{\mathcal {S}}=\ker \! J'_0(\omega _{\!R})$.

Now take arbitrary $\psi \in C^0$ and ${q}=(q_1,q_2,q_3)\in {\mathbb {R}}^3$. We have to find functions $\varphi ^\top \in T_{\omega _{\!R}}{\mathcal {S}}, \varphi ^\perp \in T_{\omega _{\!R}}{\mathcal {S}}^\perp$ and ${p}=(p_1,p_2,p_3)\in {\mathbb {R}}^3$ such that ${\mathcal {L}}(\varphi ^\top +\varphi ^\perp ,p)=(\psi ,q)$. Since $T_{\omega _{\!R}}{\mathcal {S}}=\ker \! J'_0(\omega _{\!R})$ is spanned by the functions $Re_j\wedge \omega _{\!R}$, we only need to solve

$$\begin{aligned} \left\{ \begin{array}{ll} J_0'(\omega _{\!R})\varphi ^\perp =\psi +\sum _{j}~\!p_j ({R}e_j\wedge \omega _{\!R}),&{} \varphi ^\perp \in T_{\omega _{\!R}}\mathcal S,~p\in {\mathbb {R}}^3\\ \displaystyle \fint \limits _{{\mathbb {S}}^1}\varphi ^\top \cdot Re_j\wedge \omega _{\!R}~\!dx=q_j,&{} \varphi ^\top \in T_{\omega _{\!R}}\mathcal S^\perp . \end{array}\right. \end{aligned}$$

The tangential component $\varphi ^\top \in T_{\omega _{\!R}}{\mathcal {S}}$ is uniquely determined. Thanks to (3.12), we see that the function $\sum _{j}~\!p_j ({R}e_j\wedge \omega _{\!R})$ must coincide with the projection of $-\psi$ on $T_{\omega _{\!R}}{\mathcal {S}}$. This gives the unknown p. More explicitly, we have

$$\begin{aligned} e_h\cdot Mp = \sum _{j=1}^3 p_j\fint \limits _{{\mathbb {S}}^1}(Re_h\wedge \omega _{\!R})\cdot (Re_j\wedge \omega _{\!R})~\!dx = -\fint \limits _{{\mathbb {S}}^1}\psi \cdot Re_h\wedge \omega _{\!R}~\!dx~\!, \end{aligned}$$

where M is the invertible matrix in Remark 3.2. Once one knows p, the existence of $\varphi ^\perp$ follows from iii) in Lemma 3.1, and surjectivity is proved.

We are in position to apply the implicit function theorem for any fixed ${R}\in SO(3)$. Actually, by a compactness argument, we have that there exist $\varepsilon '>0$ and uniquely determined differentiable functions

$$\begin{aligned} \begin{array}{ll} u:(-\varepsilon ',\varepsilon ')\times SO(3)\rightarrow C^2_{{\mathbb {S}}^2}\!\!\setminus \!{\mathbb {S}}^2~,\quad &{}u:(\varepsilon ,{R})\mapsto u^\varepsilon _{R}\\ \zeta :(-\varepsilon ',\varepsilon ')\times SO(3)\rightarrow {\mathbb {R}}^3~,\quad &{}\zeta :(\varepsilon ,{R})\mapsto \zeta ^\varepsilon ({R})= (\zeta ^\varepsilon _1({R}),\zeta ^\varepsilon _2({R}), \zeta ^\varepsilon _3({R})) \end{array} \end{aligned}$$

such that

$$\begin{aligned} {\mathcal {F}}(\varepsilon ,{R},u^\varepsilon _{R};\zeta ^\varepsilon ({R}))=0~, \quad u^0_{R}=\omega _{\!R}~,\quad \quad \zeta ^0({R})=0. \end{aligned}$$

Clearly the function $(\varepsilon ,{R})\mapsto u^\varepsilon _{R}$ is differentiable. Since $\omega _{\!R}$ is embedded, then $u^\varepsilon _{R}$ is embedded as well, provided that $\varepsilon '$ is small enough.

Condition i) in the Lemma is fulfilled; ii) follows from ${\mathcal {F}}_2(\varepsilon ,{R},u^\varepsilon _{R};\zeta ^\varepsilon ({R}))=0$ while ${\mathcal {F}}_1(\varepsilon ,{R},u^\varepsilon _{R};\zeta ^\varepsilon ({R}))=0$ gives iii).

Now we prove that iv) holds for any ${\overline{\varepsilon }}\in (0,\varepsilon ')$, provided that $\varepsilon '$ is small enough. Since $|\omega +e_3|\ge 1$ and $u^\varepsilon _R\rightarrow \omega _{\!R}$ uniformly on ${\mathbb {S}}^1$ as $\varepsilon \rightarrow 0$, we can assume that

$$\begin{aligned} |u^\varepsilon _R(x)+Re_3|\ge \frac{1}{2}\quad \text {for any } x\in {\mathbb {S}}^1, (\varepsilon ,R)\in (-\varepsilon ',\varepsilon ')\times SO(3). \end{aligned}$$

In particular, Lemma 2.3 guarantees that the function ${\mathcal {E}}^\varepsilon ({R})=E_{\varepsilon K}(-Re_3;u^\varepsilon _R)$ is well defined and of class $C^1$ on SO(3), for any $\varepsilon \in (-\varepsilon ',\varepsilon ')$. By iii) in Lemma 2.3 we have that the derivative of $p\mapsto E_{\varepsilon K}(p;u^\varepsilon _R)$ vanishes for $p\in {\mathbb {S}}^2\setminus u^\varepsilon _R({\mathbb {S}}^1)$, and we can compute

$$\begin{aligned} d_R {{\mathcal {E}}}^\varepsilon ({R})(RT_h)=E'_{\varepsilon K}(-Re_3;u^\varepsilon _R)(d_Ru^\varepsilon _R(RT_h)) \quad \text {for } h\in \{1,2,3\}, \end{aligned}$$

(3.13)

where $E'_{\varepsilon K}(-Re_3;~\!\cdot ~\!)$ is the differential of the energy with respect to curves running in $C^2_{{\mathbb {S}}^2\setminus \{-Re_3\}}$. The $C^1$ dependence of ${\mathcal {E}}^\varepsilon ({R})$ on $\varepsilon$ and thus on the pair $(\varepsilon , R)$ is evident.

Next, notice that $R^\xi _3\omega =\omega \circ \xi$ for any rotation $\xi \in {\mathbb {S}}^1$ (recall that $R^\xi _3$ rotates ${\mathbb {S}}^2$ about the $\langle e_3\rangle$ axis). Hence $RR^\xi _3\omega =\omega _{\!R}\circ \xi$ and $T_{RR^\xi _3\omega }{\mathcal {S}}=\big \{\tau \circ \xi ~|~\tau \in T_{\omega _{\!R}}{\mathcal {S}}~\big \}$ for any $R\in SO(3)$. Taking also ii), iii) into account, we have that

$$\begin{aligned} u^\varepsilon _R\circ \xi \in (T_{RR^\xi _3\omega }{\mathcal {S}})^\perp ~,\quad J_\varepsilon (u^\varepsilon _R\circ \xi )= J_\varepsilon (u^\varepsilon _R)\circ \xi \in T_{RR^\xi _3\omega }{\mathcal {S}}~\!. \end{aligned}$$

Since in addition $u^\varepsilon _R\circ \xi$ is close to $\omega _{\!R}\circ \xi =RR^\xi _3\omega$ in the $C^2$-norm by i), we see that

$$\begin{aligned} u^\varepsilon _{RR^\xi _3}=u^\varepsilon _{R}\circ \xi \end{aligned}$$

(3.14)

by the uniqueness of the function $\varepsilon \mapsto u^\varepsilon _R$ given by the implicit function theorem. By differentiating (3.14) with respect to $\xi$ at $\xi =1$ we obtain $d_Ru^\varepsilon _R(RT_3)=(u^\varepsilon _R)'$, that compared with (2.8) gives $E'_{\varepsilon K}(-Re_3;u^\varepsilon _R)(d_Ru^\varepsilon _R(RT_3))=E'_{\varepsilon K}(-Re_3;u^\varepsilon _R)(u^\varepsilon _R)'=0$. Thus $d_R{\mathcal {E}}^\varepsilon ({R})(RT_3)=0$ by (3.13), and iv) is proved.

To prove that v) holds for ${\overline{\varepsilon }}$ small enough, first take $R\in SO(3)$, $h\in \{1,2,3\}$ and notice that the condition $u^\varepsilon _R\in T_{\omega _{\!R}}{\mathcal {S}}^\perp$ trivially gives

$$\begin{aligned} d_R\Big (\fint \limits _{{\mathbb {S}}^1}u^\varepsilon _R\cdot ~\!R(e_j\wedge \omega )~\!dx\Big )(RT_h)=0~\!. \end{aligned}$$

We compute $d_R R(e_j\wedge \omega )(RT_h)=Re_h\wedge (R(e_j\wedge \omega )) =R\big (e_h\wedge (e_j\wedge \omega )\big )$. Since in addition $u^\varepsilon _R\cdot R(e_h\wedge (e_j\wedge \omega ))= -(Re_h\wedge u^\varepsilon _R)\cdot (Re_j\wedge \omega _{\!R})$ we obtain

$$\begin{aligned} m^\varepsilon _{hj}({R}):=\fint \limits _{{\mathbb {S}}^1}d_Ru^\varepsilon _R(RT_h)\cdot Re_j\wedge \omega _{\!R}~\!dx= \fint \limits _{{\mathbb {S}}^1}(Re_h\wedge u^\varepsilon _R)\cdot (Re_j\wedge \omega _{\!R})~\!dx.\nonumber \\ \end{aligned}$$

(3.15)

Since $u^\varepsilon _R\rightarrow \omega _{\!R}$ uniformly for $R\in SO(3)$, from (3.15) we obtain

$$\begin{aligned} m^\varepsilon _{hj}({R})=\fint \limits _{{\mathbb {S}}^1}(Re_h\wedge \omega _{\!R})\cdot (Re_j\wedge \omega _{\!R})~\!dx+o(1)= m_{hj}+o(1), \end{aligned}$$

where $m_{hj}$ are the entries of the invertible matrix M in Remark 3.2. It follows that the $3\times 3$ matrix $M^\varepsilon _R=(m^\varepsilon _{hj}({R}))_{j,h=1,2,3}$ is invertible for any $R\in SO(3)$, if $\varepsilon$ is small enough.

We are in position to conclude the proof of v). We know that there exists a differentiable function $(\varepsilon ,R)\mapsto \zeta ^\varepsilon ({R})\in {\mathbb {R}}^3$ such that

$$\begin{aligned} J_\varepsilon (u^\varepsilon _{R})=\sum _{j=1}^3 ~\!\zeta ^\varepsilon _j({R})~\!(Re_j\wedge \omega _{\!R}). \end{aligned}$$

(3.16)

On the other hand, (3.13) and (3.10) give

$$\begin{aligned} L(u^\varepsilon _R)d_R {{\mathcal {E}}}^\varepsilon ({R})(RT_h)= \fint \limits _{{\mathbb {S}}^1}J_\varepsilon (u^\varepsilon _R)\cdot d_Ru^\varepsilon _R(RT_h)~\!dx, \end{aligned}$$

(3.17)

by (3.16) and recalling (3.15) we obtain

$$\begin{aligned} L(u^\varepsilon _R)d_R {{\mathcal {E}}}^\varepsilon ({R})(RT_h)= \sum _{j=1}^3m^\varepsilon _{hj}({R})\zeta ^\varepsilon _j({R}) =~\!e_h\cdot M^\varepsilon _R(\zeta ^\varepsilon ({R}))~\!. \end{aligned}$$

If $\varepsilon \approx 0$ so that the matrix $M^\varepsilon _R$ is invertible, then R is a critical matrix for ${{\mathcal {E}}}^\varepsilon$ if and only if $\zeta ^\varepsilon ({R})=0$, which is equivalent to say that $J_\varepsilon (u^\varepsilon _R)=0$.

To prove the last claim of the lemma we take $R\in SO(3)$ and compute the Taylor expansion formula of the function

$$\begin{aligned} \begin{aligned} f_R(\varepsilon )={\mathcal {E}}^\varepsilon ({R})- {\mathcal {E}}^\varepsilon _0({R})=L(u^\varepsilon _R)-1+ \varepsilon \big (A_{K}(-Re_3;u^\varepsilon _R)- A_{K}(-Re_3;\omega _{\!R})\big ) \end{aligned} \end{aligned}$$

at $\varepsilon =0$. Clearly $f_R(0)=0$. Now we recall that $L'(\omega _{\!R})=0$ because $\omega _{\!R}$ is a geodesic, and we write

$$\begin{aligned} f'_R(\varepsilon )= & {} \big (L'(u^\varepsilon _R)-L'(\omega _{\!R})\big )(\partial _\varepsilon u^\varepsilon _R)+ \varepsilon ~\!A'_{K}(-Re_3;u^\varepsilon _R)(\partial _\varepsilon u^\varepsilon _R)\\&+ \big (A_{K}(-Re_3;u^\varepsilon _R)- A_{K}(-Re_3;\omega _{\!R})\big ). \end{aligned}$$

To take the limit as $\varepsilon \rightarrow 0$, we notice that $\partial _\varepsilon u^\varepsilon _R$ is uniformly bounded in $C^2_{{\mathbb {S}}^2}$ because the function $(\varepsilon ,R) \mapsto u^\varepsilon _R$ is of class $C^1$. Further, $L'(u^\varepsilon _R)\rightarrow L'(\omega _{\!R})$ in the norm operator, $A'_K(-Re_3;u^\varepsilon _R)(\partial _\varepsilon u^\varepsilon _R)$ remains bounded and $A_{K}(-Re_3;u^\varepsilon _R)\rightarrow A_{K}(-Re_3;\omega _{\!R})$. In conclusion, we have that $f'_R(0)=0$, and therefore $f_R(\varepsilon )=o(\varepsilon )$ as $\varepsilon \rightarrow 0$, uniformly on SO(3). That is, (3.11) holds true ”at the zero order”.

To conclude the proof we have to handle the derivatives of ${\mathcal {E}}^\varepsilon ({R})- {\mathcal {E}}^\varepsilon _0({R})$ with respect to R, along any direction $RT_h\in T_{R}SO(3)$. We use (3.16), the second equality in (3.15) and then (3.16) again to obtain

$$\begin{aligned} \fint \limits _{{\mathbb {S}}^1}J_\varepsilon (u^\varepsilon _R)\cdot (d_Ru^\varepsilon _R(RT_h))~\!dx= & {} \sum _{j=1}^3 \zeta ^\varepsilon _j({R})\fint \limits _{{\mathbb {S}}^1}(d_Ru^\varepsilon _R(RT_h))\cdot (Re_j\wedge \omega _{\!R})~\!dx\\= & {} \sum _{j=1}^3 \zeta ^\varepsilon _j({R})\fint \limits _{{\mathbb {S}}^1}(Re_h\wedge u^\varepsilon _R)\cdot (Re_j\wedge \omega _{\!R})~\!dx\\= & {} \fint \limits _{{\mathbb {S}}^1}J_\varepsilon (u^\varepsilon _R)\cdot (Re_h\wedge u^\varepsilon _R) ~\!dx. \end{aligned}$$

By (3.9), the last integral can be written as

$$\begin{aligned} \begin{aligned} \fint \limits _{{\mathbb {S}}^1}J_0(u^\varepsilon _R)\cdot (Re_h\wedge u^\varepsilon _R)~\!dx ~\!+~\!&\varepsilon L(u^\varepsilon _R)A'_K(-Re_3;u^\varepsilon _R)(Re_h\wedge u^\varepsilon _R)\\ =&~\!\varepsilon ~\!L(u^\varepsilon _R) A'_K(-Re_3;u^\varepsilon _R)(Re_h\wedge u^\varepsilon _R) \end{aligned} \end{aligned}$$

because of (3.3). Thus (3.17) leads to the new formula

$$\begin{aligned} d_R {{\mathcal {E}}}^\varepsilon ({R})(RT_h) =\varepsilon A'_K(-Re_3;u^\varepsilon _R)(Re_h\wedge u^\varepsilon _R)~\!. \end{aligned}$$

On the other hand, it is easy to see that

$$\begin{aligned} d_R {{\mathcal {E}}}^\varepsilon _0({R})(RT_h)=\varepsilon A'_K(-Re_3;\omega _{\!R})(d_R(\omega _{\!R})(RT_h))= \varepsilon A'_K(-Re_3;\omega _{\!R})(Re_h\wedge \omega _{\!R}), \end{aligned}$$

because $A_K(~\!\cdot ~\!;\omega _{\!R})$ is locally constant, and we can conclude that

$$\begin{aligned} d_R\big ( {{\mathcal {E}}}^\varepsilon ({R})-{{\mathcal {E}}}^\varepsilon _0({R})\big )(RT_h)\\ = \varepsilon \big (A'_K(-Re_3;u^\varepsilon _R)(Re_h\wedge u^\varepsilon _R)- A'_K(-Re_3;u^\varepsilon _R)(Re_h\wedge \omega _{\!R})\big )=o(\varepsilon ), \end{aligned}$$

because $u^\varepsilon _R\rightarrow \omega _{\!R}$. The lemma is completely proved. $\square$

4 Two solutions

In the present section we use Lemma 3.3 together with the local variational approach in Sect. 2 to provide a more direct, self-contained and analytical treatment to Viterbo’s and Bottkoll’s result which avoids the deep and general theories of characteristics and symplectic actions.

We stress the fact that, differently from [11, 18] and [16], in the next theorem we do not make any sign assumptions on K. For instance, K might vanish on some geodesic circle of radius ${{\pi }/{2}}$ about a point $z\in {\mathbb {S}}^2$ and thus $\partial {{\mathcal {D}}}_{\!\frac{\pi }{2}\!}(z)$ can be parameterized by two K-magnetic geodesics that coincide up to orientation². This is the reason why, in that case, we have to add an extra assumption to obtain two distinct solutions.

Theorem 4.1

Let $K\in C^1({\mathbb {S}}^2)$ be given. For every $c>0$ large enough, Problem (${\mathcal {P}}_{K,c}$) has at least a solution $\gamma$. If in addition K does not vanish on any closed geodesic, or

$$\begin{aligned} \int \limits _{{{\mathcal {D}}}_{\!\frac{\pi }{2}\!}(z)} K(q)~\!d\sigma _{\!q}=\int \limits _{{{\mathcal {D}}}_{\!\frac{\pi }{2}\!}(-z)} K(q)~\!d\sigma _{\!q} \quad \text {whenever } K\equiv 0 \text { on } ~\partial {{\mathcal {D}}}_{\!\frac{\pi }{2}\!}(z), \end{aligned}$$

(4.1)

then for every $c>0$ large enough, Problem (${\mathcal {P}}_{K,c}$) has at least two embedded, distinct solutions.

Proof

Let ${\overline{\varepsilon }}$ be given by Lemma 3.3. For any $c > {\overline{\varepsilon }}^{-1}$, let $\varepsilon := c^{-1} < {\overline{\varepsilon }}$ and $(\varepsilon ,{R})\mapsto u^\varepsilon _R$, $(\varepsilon ,{R})\mapsto {\mathcal {E}}^\varepsilon ({R})$ be the functions in Lemma 3.3. To every critical point $R^\varepsilon$ for ${\mathcal {E}}^\varepsilon$ corresponds a curve $u^\varepsilon _{R^\varepsilon }$ that solves $J_\varepsilon (u^\varepsilon _{R^\varepsilon })=0$. Hence $u^\varepsilon _{R^\varepsilon }$ solves (3.8) and, as explained at the beginning of Sect. 2, yields a solution to (${\mathcal {P}}_{K,\varepsilon ^{-1}}$) = (${\mathcal {P}}_{K,c}$).

Now, if ${\mathcal {E}}^\varepsilon$ is constant, then $u^\varepsilon _{R}$ solves (3.8) for every $R\in SO(3)$ and the conclusions in Theorem 4.1 hold. Otherwise, take ${\underline{R}}^\varepsilon , {\overline{R}}^\varepsilon \in SO(3)$ achieving the minimum and the maximum value of ${\mathcal {E}}^\varepsilon$, respectively. Then $\underline{u}^\varepsilon :=u^\varepsilon _{\underline{R}^\varepsilon }$ and $\overline{u}^\varepsilon :=u^\varepsilon _{\overline{R}^\varepsilon }$ solve (3.8) and this concludes the proof of the existence part.

Next, assume that ${\mathcal {E}}^\varepsilon$ is not constant, and that $\underline{u}^\varepsilon =\overline{u}^\varepsilon \circ g$ for a diffeomorphism g of ${\mathbb {S}}^1$. To conclude the proof we have to show that (4.1) can not hold.

We have $E_{\varepsilon K}(\underline{z}^\varepsilon , \underline{u}^\varepsilon )< E_{\varepsilon K}(\overline{z}^\varepsilon , \overline{u}^\varepsilon )$, that is,

$$\begin{aligned} L(\underline{u}^\varepsilon )+\varepsilon A_{K}(\underline{z}^\varepsilon , \underline{u}^\varepsilon )< L(\overline{u}^\varepsilon )+\varepsilon A_{K}(\overline{z}^\varepsilon , \overline{u}^\varepsilon ) \end{aligned}$$

(4.2)

where $\underline{z}^\varepsilon =-\underline{R}^\varepsilon e_3, \overline{z}^\varepsilon =-\overline{R}^\varepsilon e_3$. Since $|(\underline{u}^\varepsilon )'|, |(\overline{u}^\varepsilon )'|$ are constant, then $|g'|$ is constant as well. Thus $|g'|=1$ and $L(\underline{u}^\varepsilon )=L(\overline{u}^\varepsilon )$. Therefore, (4.2) implies

$$\begin{aligned} A_{K}(\underline{z}^\varepsilon , \underline{u}^\varepsilon )\ne A_{ K}(\overline{z}^\varepsilon , \overline{u}^\varepsilon ) \end{aligned}$$

(4.3)

for any $\varepsilon \ne 0$. In particular, g can not be a positive rotation of the circle by the property A2) of the area functional. Thus g is a counterclockwise rotation of ${\mathbb {S}}^1$. Recall that $\underline{u}^\varepsilon$ has curvature $\varepsilon K(\underline{u}^\varepsilon )$ and $\overline{u}^\varepsilon$ has curvature $\varepsilon K(\overline{u}^\varepsilon )$. Since changing the orientation of a curve changes the sign of its curvature, we have that at any point $p\in \Gamma :={\underline{u}}^\varepsilon ({\mathbb {S}}^1)={\overline{u}}^\varepsilon ({\mathbb {S}}^1)$ we have $K(p)=-K(p)$. It follows that $K\equiv 0$ on $\Gamma$, and hence $\Gamma$ is the boundary of a half-sphere ${{\mathcal {D}}}_{\!\frac{\pi }{2}\!}(w^\varepsilon )$. We can assume that $\underline{u}^\varepsilon$ is a positive parameterization of $\partial {{\mathcal {D}}}_{\!\frac{\pi }{2}\!}(w^\varepsilon )$. Then $\underline{z}^\varepsilon \notin \overline{{{\mathcal {D}}}_{\!\frac{\pi }{2}\!}(w^\varepsilon )}$ because $\underline{u}^\varepsilon \approx \omega _{\underline{R}^\varepsilon }$, see i) in Lemma 3.3. Next, since $\overline{u}^\varepsilon$ parameterizes the same geodesic with opposite direction, then $\overline{u}^\varepsilon$ a positive parameterization of $\partial {{\mathcal {D}}}_{\!\frac{\pi }{2}\!}(-w^\varepsilon )$ and $\overline{z}^\varepsilon \notin \overline{{{\mathcal {D}}}_{\!\frac{\pi }{2}\!}(-w^\varepsilon )}$. In particular, from the properties A3) and A4) of the area functional we infer

$$\begin{aligned} \begin{aligned} A_{K}(\underline{z}^\varepsilon , \underline{u}^\varepsilon )=&~\!A_{K}(-w^\varepsilon , \underline{u}^\varepsilon )~\!=~\!-\frac{1}{2\pi }\int \limits _{{{\mathcal {D}}}_{\!\frac{\pi }{2}\!}(w^\varepsilon )}K(q)~\!d\sigma _{\!q}\\ A_{K}(\overline{z}^\varepsilon , \overline{u}^\varepsilon )=&A_{K}(w^\varepsilon , \overline{u}^\varepsilon )= -\frac{1}{2\pi }\int \limits _{{{\mathcal {D}}}_{\!\frac{\pi }{2}\!}(-w^\varepsilon )}K(q)~\!d\sigma _{\!q}~\!, \end{aligned} \end{aligned}$$

that compared with (4.3) shows that (4.1) is violated. The theorem is completely proved. $\square$

5 Many solutions

In this section we suggest a way to obtain more and more distinct K-magnetic geodesics. It involves the $C^1$ Mel’nikov-type function

$$\begin{aligned} F_{\!K}(z)=\int \limits _{{\mathcal {D}}_{\!\frac{\pi }{2}\!}(z)} K({p})~\!d\sigma _{\!p}~,\quad F_{\!K}:{\mathbb {S}}^2\rightarrow {\mathbb {R}}~\!~\!, \end{aligned}$$

(5.1)

where $K\in C^1({\mathbb {S}}^2)$ is given. We start by recalling the definition of stable critical point proposed in [3, Chapter 2], see also [14].

Definition 5.1

Let $\Omega \subset {\mathbb {S}}^2$ be open. We say that ${{F_K}}$ has a stable critical point in $\Omega$ if there exists $r>0$ such that any function ${G}\in C^1({\overline{\Omega }})$ satisfying $\displaystyle {\Vert {G}-{{F_K}}\Vert _{C^1({\overline{\Omega }})}<r}$ has a critical point in $\Omega$.

If $F_K$ is not constant, then it has at least two distinct stable critical points, namely, its minimum and its maximum. Different sufficient conditions to have the existence of (possible multiple) stable critical points $z\in \Omega$ for ${F_K}$ are easily given via elementary calculus. For instance, one can assume that one of the following conditions holds:

(i):

$\nabla {{F_K}}(z)\ne 0$ for any $z\in \partial \Omega$, and $\mathrm{deg}(\nabla {{F_K}},\Omega , 0)\ne 0$, where ”$\mathrm{deg}$” is Browder’s topological degree;

(ii):

$\displaystyle {{\min _{\partial \Omega } {{F_K}}>\min _{\Omega } {{F_K}}}}$  or  $\displaystyle {\max _{\partial \Omega } {{F_K}}<\max _{\Omega } {{F_K}}}$;

(iii):

${{F_K}}$ is of class $C^2$ on $\Omega$, it has a critical point $z_0\in \Omega$, and the Hessian matrix of ${{F_K}}$ at $z_0$ is invertible.

In the next result we show that any stable critical point $z_0$ for $F_K$ gives rise, for any $c>0$ large enough, to a solution $\gamma ^c$ to Problem (${\mathcal {P}}_{K,c}$) which is a perturbation of the closed geodesic about $z_0$. Taking advantage of the remarks at the beginning of Sect. 2, we only need to show that for any stable critical point $z_0$ for $F_K$ and for any $\varepsilon = c^{-1}\approx 0^+$, there exists a solution $u^\varepsilon$ to (3.8), such that $u^\varepsilon$ is close to the closed geodesic about $z_0$.

Theorem 5.2

Let $K\in C^1({\mathbb {S}}^2)$ be given. Assume that $F_{\!K}$ has a stable critical point in an open set $\Omega \subset {\mathbb {S}}^2$, such that ${\overline{\Omega }}\subsetneq {\mathbb {S}}^2$.

Then for every $\varepsilon \in {\mathbb {R}}$ close enough to 0, there exists a point $z_\varepsilon \in \Omega$, an embedding $\omega ^\varepsilon :{\mathbb {S}}^1\rightarrow {\mathbb {S}}^2$ parameterizing the boundary of a circle of geodesic radius $\pi /2$ about $z_\varepsilon$, and a solution $u^\varepsilon$ to (${\mathcal {P}}_{K,\varepsilon ^{-1}}$), such that $\Vert u^\varepsilon -\omega ^\varepsilon \Vert _{C^2}=O(\varepsilon )$.

Proof

We can assume $-e_3\notin {\overline{\Omega }}$. Otherwise, take any rotation $R\in SO(3)$ such that $-e_3\notin R{\overline{\Omega }}$, and look for a solution ${\tilde{u}}^\varepsilon$ to

$$\begin{aligned} u''+|u'|^2u=L(u)~\!\varepsilon ~\!(K\circ ^{t}{}{\!R})(u)~\!u\wedge u'~\qquad \text {on } {\mathbb {S}}^1, \end{aligned}$$

in a $C^2$-neighborhood of a geodesic circle about some point ${\tilde{z}}^\varepsilon \in R\Omega$. Conclude by noticing that $u^\varepsilon :=^{t}{}{\!R}{\tilde{u}}^\varepsilon$ solves (3.8) and approaches the geodesic circle about $R^{\text {t}} {\tilde{z}}^\varepsilon \in \Omega$.

Next, for $z\in {\mathbb {S}}^2\setminus \{-e_3\}$ consider the rotation

$$\begin{aligned} N(z)=\left( \begin{array}{ccc} 1-\frac{z_1^2}{1+z_3}&{} -\frac{z_1z_2}{1+z_3}&{}z_1\\ &{}\\ -\frac{z_1z_2}{1+z_3}&{}1-\frac{z_2^2}{1+z_3}&{}z_2\\ &{}\\ -z_1&{}-z_2&{}z_3 \end{array}\right) ~\!, \end{aligned}$$

that maps $e_3$ to z. Clearly the function $N:{\mathbb {S}}^2\setminus \{-e_3\}\rightarrow SO(3)$ is differentiable; its differential dN(z) at any $z\in {\mathbb {S}}^2\setminus \{-e_3\}$ is a linear map $T_z{\mathbb {S}}^2\rightarrow T_{N(z)}SO(3)$. We have

$$\begin{aligned} T_z{\mathbb {S}}^2= & {} \langle N(z)e_1,N(z)e_2\rangle \end{aligned}$$

(5.2)

$$\begin{aligned} T_{N(z)}SO(3)= & {} \langle dN(z)\big ( N(z)e_1\big ),dN(z)\big ( N(z)e_2\big )\rangle \oplus \langle N(z)T_3\rangle ~\!. \end{aligned}$$

(5.3)

Equality (5.2) and the inclusion $\supseteq$ in (5.3) are trivial. To conclude the proof of (5.3) we need to show that the matrices

$$\begin{aligned} dN(z)\big ( N(z)e_1\big )~,\quad dN(z)\big ( N(z)e_2\big )~,\quad N(z)T_3 \end{aligned}$$

are linearly independent. By differentiating the identity $N(z)e_3=z$ one gets

$$\begin{aligned} dN(z)\tau \cdot e_3=\tau ~,\quad \tau \in T_z{\mathbb {S}}^2~\!. \end{aligned}$$

By choosing $\tau =N(z)e_h$, $h=1,2$ we infer that the third columns of the matrices $dN(z)\big ( N(z)e_1\big ), dN(z)\big ( N(z)e_2\big )$ are linearly independent. Thus the matrices $dN(z)\big ( N(z)e_1\big )~, dN(z)\big ( N(z)e_2\big )$ are linearly independent as well. On the other hand, the third column on $N(z)T_3$ is identically zero, that concludes the proof of (5.3).

Now, take the differentiable functions $(\varepsilon ,R)\mapsto u^\varepsilon _R\in C^2_{{\mathbb {S}}^2}$, $(\varepsilon ,R)\mapsto {\mathcal {E}}^\varepsilon ({R})\in {\mathbb {R}}$ given by Lemma 3.3. To simplify notations, for $z\in {\mathbb {S}}^2\setminus \{-e_3\}$ we write

$$\begin{aligned} \widetilde{{\mathcal {E}}^\varepsilon }(z)={\mathcal {E}}^\varepsilon (N(z))=E_{\varepsilon K}(-z;u^\varepsilon _{N(z)})~,\quad \widetilde{{\mathcal {E}}^\varepsilon _0}(z)={\mathcal {E}}^\varepsilon _0(N(z))=E_{\varepsilon K}(-z;N(z)\omega )~\!. \end{aligned}$$

Notice that $N(z)\omega$ parameterizes $\partial {{\mathcal {D}}}_{\!\pi /2}(z)$. Therefore, using ii) in Lemma 2.3, property A4) and elementary computations we get

$$\begin{aligned} \begin{aligned} \widetilde{{\mathcal {E}}^\varepsilon _0}(z)=&L(N(z)\omega ) +\varepsilon A_{K}(-z;N(z)\omega ) \\ =&L(\omega )\!-\frac{\varepsilon }{2\pi } \!\!\!\int \limits _{D_{\!\pi /2}(z)} \!\!\!K(q)~\!\! d\sigma _{\!q}= L(\omega )\!-\frac{\varepsilon }{2\pi } F_{\!K}(z). \end{aligned} \end{aligned}$$

(5.4)

Next, for any small $\varepsilon \ne 0$ consider the function

$$\begin{aligned} G^\varepsilon (z)= \frac{2\pi }{\varepsilon }(\widetilde{{\mathcal {E}}^\varepsilon }(z)-L(\omega )) \end{aligned}$$

and use (5.4) together with iv) in Lemma 3.3 to get

$$\begin{aligned} \Vert G^\varepsilon +F_{\!K}\Vert _{C^1({\overline{\Omega }})}= \frac{2\pi }{|\varepsilon |}\big \Vert E_{\varepsilon K}(-z;u^\varepsilon _{N(z)})-E_{\varepsilon K}(-z;N(z)\omega )\big \Vert _{C^1({\overline{\Omega }})}=o(1) \end{aligned}$$

as $\varepsilon \rightarrow 0$. We see that for $\varepsilon$ small enough the function $G^\varepsilon$ has a critical point $z^\varepsilon \in \Omega$. Thus, for any $\tau \in T_{z^\varepsilon } {\mathbb {S}}^2$ we have

$$\begin{aligned} 0=d_z\widetilde{{\mathcal {E}}^\varepsilon }(z^\varepsilon )\tau =d_R{\mathcal {E}}^\varepsilon (N(z^\varepsilon ))\big (d_zN(z^\varepsilon )\tau \big )~\!. \end{aligned}$$

Taking (5.3) and iv) in Lemma 3.3 into account, we infer that the matrix $N(z^\varepsilon )$ is critical for ${{\mathcal {E}}^\varepsilon }$. Thus, by arguing as for Theorem 4.1 we have that the curve $u^\varepsilon :=u^\varepsilon _{N(z_\varepsilon )}$ is a solution to (${\mathcal {P}}_{K, \varepsilon ^{-1}}$). $\square$