The $\infty$-Elastica Problem on a Riemannian Manifold

Abstract

We consider the following problem: on any given complete Riemannian manifold (M, g), among all curves which have fixed length as well as fixed endpoints and tangents at the endpoints, minimise the $L^\infty$ norm of the curvature. We show that the solutions of this problem, as well as a wider class of curves, must satisfy a second-order ODE system. From this system,we obtain some geometric information about the behaviour of the curves.

1 Introduction

The study of so-called variational problems in $L^\infty$ originated in the 1960 s in a series of papers by Aronsson [1,2,3] with a focus on first-order functionals; these have now been studied extensively. In contrast, second-order functionals have a significantly shorter history, having started to be investigated only relatively recently. In the one-dimensional case, problems of second order and higher (with no immediate geometric context) are touched on in [4] and [5]. In [6], a variational problem depending on second derivatives only through the Laplacian is studied, and in [7], general second-order $L^\infty$ variational problems concerning the minimisation of an arbitrary $C^1$ function of the Hessian $D^2 u$ are examined.

Specific $L^\infty$ variational problems arising from geometric motivations have been studied for longer than the generic second-order problems, and some of these can be considered second order in the sense that (say) the quantities of interest require second derivatives when expressed in local co-ordinates. Some examples include [8], where surfaces minimising a weighted $L^\infty$ norm of the Gauss curvature are considered, and [9] where minimisers among a conformal class of the $L^\infty$ norm of the scalar curvature are studied for manifolds of dimension at least three.

In this paper, we consider the problem of minimising the $L^\infty$ norm of the curvature of a curve subject to boundary conditions and a side constraint. Specifically, we consider the problem with the following setup: let (M, g) be a complete Riemannian manifold (whose metric we shall also denote by $\langle \cdot , \cdot \rangle$ throughout this paper) and fix two points $x_1, x_2 \in M$ as well as two unit tangent vectors $v_1 \in T_{x_1}M$, $v_2 \in T_{x_2}M$ at these points. Finally, fix a length $L \ge d(x_1,x_2)$ where denotes the Riemannian distance function on M. We identify curves in M with their arclength parametrisation $\gamma : [0,L] \rightarrow M$ with $\vert \gamma ' \vert \, = \vert T \vert \, \equiv 1$, and the (unsigned) curvature is given by $\kappa = \vert \nabla _T T \vert$. Note that the arclength condition affects the problem only superficially since curvature is of course invariant under reparametrisation. Assuming that $\gamma$ belongs to open subset of immersions in the Sobolev space $W^{2,\infty }((0,L); M)$– the natural space to work in for this problem, which we will denote by the shorthand $W^{2,\infty }(0,L)$ or $W^{2,\infty }$ in places where the meaning is clear– we introduce the functional

$$\begin{aligned} {\mathcal {K}}_\infty \left[ \gamma \right] = \mathop {\mathrm {ess\,sup}}\limits _{s \in [0,L]} \, \vert \nabla _T T \vert . \end{aligned}$$

Denoting by ${\mathcal {G}}$,the set of all curves in $W^{2,\infty }(0,L)$ which satisfy the boundary conditions

$$\begin{aligned} \gamma (0) = x_1, \quad \gamma (L) = x_2, \quad \gamma '(0) = v_1, \quad \gamma '(L) = v_2 \end{aligned}$$

(1)

and the length constraint

$$\begin{aligned} {\mathcal {L}} \left[ \gamma \right] = L, \end{aligned}$$

(2)

as well as the arclength condition $\vert \gamma ' \vert \, \equiv 1$ (that is, ${\mathcal {G}}$ is the set of admissible curves for the problem), we seek to understand the behaviour of minimisers of ${\mathcal {K}}_\infty$ in ${\mathcal {G}}$. As we will see later on, such minimisers always exist and so the problem is always meaningful.

Note that here we are defining the space $W^{2,\infty }((0,L); M)$ as the space of all functions from (0, L) to M whose expressions in suitable co-ordinate charts belong to the Euclidean Sobolev space $W^{2,\infty }$, with similar definitions for the spaces $W^{2,q}$ with $1 \le q < \infty$.

This problem has not been considered in full generality previously, though specific cases of it and problems of a similar nature have been investigated. The problem in $L^p$ has been considered before, e.g. in [10] which focuses on manifolds of constant sectional curvature, and it has been studied extensively in the case $p=2$ where the theory is originated with the Bernoulli brothers as well as Euler [11]. However, the change from the finite $p < \infty$ to the infinite $p = \infty$ gives rise to a fundamentally different problem for which none of the usual tools of the calculus of variations apply.

In Euclidean space ${\mathbb {R}}^n$, the $\infty$-elastica problem has already been studied by Moser [12]. The main novelty in the present paper, therefore, is the generalisation to arbitrary complete Riemannian manifolds. Somewhat surprisingly, given that the $L^\infty$ norm is not differentiable in any material sense of the word, [12] shows that solutions of the Euclidean problem (as well as a more general class of curves) are characterised by a system of first-order ODEs, derived from the limit as $p \rightarrow \infty$ of the Euler-Lagrange equations for the $L^p$ problem. Moreover, the solutions admit a classification based on their structure, being either a chain of two-dimensional curves or a single three-dimensional curve.

As well as minimisers of ${\mathcal {K}}_\infty$, the analysis in this paper also lets us draw conclusions about a weaker class of “pseudo-minimisers”, called $\infty$-elastica, which are a sort of analogue of critical points for the $L^\infty$ norm. Due to the definition of the $L^\infty$ norm, there is no meaningful way to define critical points in the traditional sense. The underlying idea is that as well as minimisers of ${\mathcal {K}}_\infty$, the definition of an $\infty$-elastica also captures ‘weaker’ curves which minimise a modified ${\mathcal {K}}_\infty$ functional:

Definition 1

A curve $\gamma \in {\mathcal {G}}$ is called an $\infty$-elastica if there exists a constant $A \in {\mathbb {R}}$ such that the inequality

$$\begin{aligned} {\mathcal {K}}_\infty \left[ \gamma \right] \le {\mathcal {K}}_\infty \left[ {\tilde{\gamma }}\right] + \frac{A}{2L} \int _0^{L} d\left( {\tilde{\gamma }}(s), \gamma ([0,L]) \right) ^2 \, \textrm{d}s \end{aligned}$$

(3)

is satisfied for all ${\tilde{\gamma }}\in {\mathcal {G}}$.

Here, the function d denotes the (geodesic) distance between the point ${\tilde{\gamma }}(s)$ and the set $\gamma ([0,L])$, i.e. the image of the curve $\gamma$.

Although we use the same language (“$\infty$-elastica”) as [12], our definition is in fact slightly different (cf. [12, Definition 1]), although both definitions capture the minimisers of ${\mathcal {K}}_\infty$. The reason for this difference comes from the penalisation term introduced in Sect. 2, which is easier to use on Riemannian manifolds than the penalisation term used in the Euclidean case in [12]; it is unknown whether the two definitions give rise to different non-minimising $\infty$-elastica.

The purpose of this paper is to prove the following theorem, which extends some of the theory from [12] from Euclidean space ${\mathbb {R}}^n$ to an arbitrary complete Riemannian manifold (M, g):

Theorem 2

Assume that $\gamma \in {\mathcal {G}}$ is an $\infty$-elastica and that $\gamma$ is not a geodesic. Let $K = {\mathcal {K}}_\infty \left[ \gamma \right]$. Then the following statements hold:

• (1) There exist a vector field $\varphi \in W^{2,\infty }_{\text {loc}}(0,L) \backslash \{ 0 \}$ defined along $\gamma$ and a number $\lambda \in {\mathbb {R}}$ such that the equations

  $$\begin{aligned} \nabla _{T}^2 \varphi + R\left( \varphi ,T\right) T&= L \lambda \nabla _{T} T - 2\nabla _T \left( \left\langle \varphi , \nabla _{T} T \right\rangle T \right) , \end{aligned}$$

  (4)
  $$\begin{aligned} \vert \varphi \vert \nabla _T T&= K \varphi , \end{aligned}$$

  (5)

  are satisfied a.e. in (0, L).

• (2) The curvature of $\gamma$ takes on only two values up to null sets: $\kappa = K$ a.e. on the set where $\varphi \ne 0$, and $\kappa = 0$ a.e. on the set where $\varphi = 0$.

In particular, (1) and (2) hold when $\gamma$ is a minimiser of ${\mathcal {K}}_\infty$.

Note that $\kappa$ denotes the unsigned curvature, so it is possible that the normal vector may suddenly reverse direction in places. We therefore cannot expect that an $\infty$-elastica will possess any higher regularity than $W^{2,\infty }$ in general. The classification results of [12] demonstrate this: for curves in the plane, we can take the boundary and length conditions to be such that any single circular arc is not admissible for the problem. Then a minimiser of ${\mathcal {K}}_\infty$—which is an $\infty$-elastica in both the sense of this paper and the sense of [12]—must have its normal vector suddenly reverse direction somewhere by [12, Theorem 4], i.e. its second derivative must be discontinuous.

Theorem 2 opens up the possibility of further analysis of $\infty$-elastica through the ODE system (4)–(5) in a manner similar to that of [12]; however, such an analysis would most likely require us to restrict our attention to a single manifold M at a time, and would potentially only work for more ‘well-behaved’ and well-understood manifolds (e.g. the sphere ${\mathbb {S}}^n$ or hyperbolic space ${\mathbb {H}}^n$). Since the focus of this paper is on the general case, we do not consider this analysis here.

In the context of Theorem 2, and also throughout this paper, the condition that $\varphi \in W^{2,\infty }_{\text {loc}}(0,L)$ means that $\varphi \in W^{2,\infty }([a,b])$ for any $[a,b] \subset (0,L)$, where the space $W^{2,\infty }([a,b])$ may be interpreted as the space of all vector fields whose local co-ordinate representations as functions to ${\mathbb {R}}^n$ are in $W^{2,\infty }$ in suitable co-ordinate charts. It will be clear from context whether we are talking about a function or a vector field being in their respective Sobolev spaces. The restriction that $\gamma$ is not a geodesic does not take anything away from the problem, since if the conditions (1–2) allow geodesics as admissible curves then it is immediate that the only minimisers of ${\mathcal {K}}_\infty$ will be such geodesics.

Intriguingly, the results of Theorem 2 are different in some regards to the results for the problem in Euclidean space from [12]. Most notably, the ODE system (4–5) is of second order, compared to the first-order system in [12, Theorem 2]. This difference appears since in Euclidean space it is possible to consider everything in terms of the unit tangent vector field rather than the curve itself—indeed, the curve can be recovered from only the initial data and tangent field via integration– while such an approach is complicated on an arbitrary manifold. Moreover, the connection between the ODE system and $\infty$-elastica in [12] is an equivalence, whereas here the ODE system (4–5) has only been shown to be a necessary condition for $\infty$-elastica; this is a consequence of the specific way we have defined $\infty$-elastica, as the proof equivalence in [12] leans heavily on the (different) definition of $\infty$-elastica there, although the question of whether we in fact have equivalence in Theorem 2 is unresolved.

The paper is structured as follows. In Sect. 2, we introduce the penalisation term to our analysis, we approximate the $L^\infty$ problem by an $L^p$ one for which the Euler-Lagrange equations make sense, and we compute the Euler-Lagrange equations. In Sect. 3, we allow p to go to infinity and show that the ‘limiting’ version (4–5) of the Euler-Lagrange equations is satisfied when we take the limit, beginning the proof of Theorem 2. In Sect. 4, we continue the proof of Theorem 2, showing that the vector field $\varphi$ satisfying equations (4–5) is not identically zero and investigating what equations (4–5) tell us about $\infty$-elastica. We show that minimisers of ${\mathcal {K}}_\infty$ exist.

2 Approximation of the Problem

In this section, we consider a version of our problem where the $L^\infty$ norm is replaced by an $L^p$ norm, with the view that we will later let $p \rightarrow \infty$ and recover some information about the $L^\infty$ problem. In general, as $p \rightarrow \infty$, we would expect to recover a solution of the problem but not necessarily all solutions of the problem: although our approximation of the variational problem may lead to an $\infty$-elastica in the limit, this does not guarantee that every $\infty$-elastica will arise in this manner. This is a genuine issue, as in general we cannot expect that minimisers of ${\mathcal {K}}_\infty$ are unique; indeed, we can generalise the Euclidean constructions from [12, Sect. 1] to create examples where non-uniqueness is guaranteed. For example, when the boundary conditions (1) are symmetric with respect to some isometry of M yet no admissible curves exist which are invariant under this isometry, there are necessarily multiple minimisers. To overcome this issue, we add a penalisation term to our analysis similar to the one in [12] (albeit with a slightly different form), so that we can guarantee convergence to any given solution of the problem. We compute the Euler-Lagrange equations for the $L^p$ problem with this penalisation term added.

Recall that we are considering arclength-parametrised curves $\gamma : [0,L] \rightarrow M$ satisfying the boundary conditions (1) and length constraint (2), along with the curvature functional ${\mathcal {K}}_\infty$.

In practice, we will not work directly with Definition 1 when dealing with $\infty$-elastica; instead,we will make use of another inequality similar to (3). Whenever the inequality in Definition 1 is satisfied, so too will this other inequality hold. In this sense, Definition 1 is not the strongest possible definition we could use. However, it has the advantage of being easy to understand and the distinction between the two definitions is extremely technical.

Before introducing the inequality, we first need to introduce some notation. Fix a compact subset U of M which contains all admissible curves for the problem. For example, the geodesic ball of radius 2L centred at the start-point $x_1$ works as such a set. For a curve $\gamma \in {\mathcal {G}}$ with the $L^\infty$ norm of the curvature given by $\Vert \kappa _\gamma \Vert _{L^\infty } =: K$, we introduce the function

$$\begin{aligned} c(K) = \inf _{{\hat{\gamma }} \in X} \{ {\mathcal {L}} \left[ {\hat{\gamma }} \right] \} \end{aligned}$$

where the infimum is taken over the set X of all closed $W^{2,\infty }$ curves ${\hat{\gamma }} \subset U$ (closed in the sense that the start point coincides with the endpoint, although the tangents at the start and end need not be the same) satisfying the curvature bound $\Vert \kappa _{{\hat{\gamma }}}\Vert _{L^\infty } \le K$. Roughly speaking, c(K) is “the length of the shortest possible loop we can make under the restriction $\Vert \kappa _{{\hat{\gamma }}}\Vert _{L^\infty } \le K$.” Note that K is itself a function of $\gamma$ (and so c is too). For example, in Euclidean space, we can take $c(K) = 2 \pi / K$ by Fenchel’s theorem.

For a curve $\gamma \in W^{2,\infty }(0,L)$ with $\left\Vert \kappa _\gamma \right\Vert _{L^\infty (0,L)} = K$, we define the segment of $\gamma$ centred at s by

$$\begin{aligned} \text {seg}_{\gamma }(s) = \gamma ((s - c(K)/4, s + c(K)/4 ) \cap [0,L]). \end{aligned}$$

That is, $\text {seg}_{\gamma }(s)$ is the image of the restriction of $\gamma$ to the interval $(s - c(K)/4, s + c(K)/4 ) \cap [0,L]$.

The boundary of the segment $\text {seg}_{\gamma }(s)$ is given by the union of the two points

$$\begin{aligned} \gamma (s - c(K)/4) \cup \gamma (s + c(K)/4), \end{aligned}$$

where if one of the arguments is outside of the domain [0, L] then the corresponding term is replaced by the empty set. This definition coincides with the conventional definition of the endpoints of a curve, except when the centre of the segment lies close to the endpoints.

One of the key ingredients in this paper is the approximation of the ${\mathcal {K}}_\infty$ functional by the functionals

$$\begin{aligned} {\mathcal {K}}_p \left[ \gamma \right] = \left( \frac{1}{L} \int _0^L \vert \nabla _T T \vert ^p \, \textrm{d}s \right) ^{\frac{1}{p}} \end{aligned}$$

where $2 \le p < \infty$. Although the factor of 1/L disappears in the limit, and thus may seem superfluous, it plays a useful role in manipulating some inequalities later on and it makes the ${\mathcal {K}}_p$ functionals closer to “averages” in the traditional sense of the word.

Furthermore, given an admissible curve ${\tilde{\gamma }}\in {\mathcal {G}}$ and a real number $\sigma > 0$, we consider the “penalised ${\mathcal {K}}_p$ functional”

$$\begin{aligned} {\mathcal {J}}_p^\sigma \left[ \gamma ; {\tilde{\gamma }}\right]&= {\mathcal {K}}_p \left[ \gamma \right] + \frac{\sigma }{2L} \int _0^L d \left( \gamma (s), \text {seg}_{{\tilde{\gamma }}} (s) \right) ^2 \, \textrm{d}s \\&= {\mathcal {K}}_p \left[ \gamma \right] + {\mathcal {P}}_\sigma \left[ \gamma ; {\tilde{\gamma }}\right] . \end{aligned}$$

Using the direct method, it can be shown that minimisers of subject to the constraints (1–2) exist. Furthermore, we have

Lemma 3

Suppose ${\tilde{\gamma }}$ is an $\infty$-elastica and let $(\gamma _p)$ be a sequence of admissible curves which are minimisers of .

• (1) If $\sigma$ is sufficiently large then the sequence $(\gamma _p)$ converges weakly in $W^{2,q}(0,L)$ to ${\tilde{\gamma }}$ as $p \rightarrow \infty$ for every $1< q < \infty$.

• (2) If $\sigma$ is large enough that part 1) applies, then for p large enough and every $s \in (0,L)$,the minimal geodesic between $\gamma _p(s)$ and $\text {seg}_{\tilde{\gamma }}(s)$ is unique, and the endpoint of this geodesic does not lie on the boundary of the segment $\text {seg}_{\tilde{\gamma }}(s)$.

Proof

For $q \ge p$, observe the chain of inequalities

$$\begin{aligned}{} & {} {\mathcal {K}}_p\left[ \gamma _p \right] \le {\mathcal {J}}_p^\sigma \left[ \gamma _p; {\tilde{\gamma }}\right] \le {\mathcal {J}}_p^\sigma \left[ \gamma _q; {\tilde{\gamma }}\right] \nonumber \\{} & {} \qquad \le {\mathcal {J}}_q^\sigma \left[ \gamma _q; {\tilde{\gamma }}\right] \le {\mathcal {J}}_q^\sigma \left[ {\tilde{\gamma }}; {\tilde{\gamma }}\right] = {\mathcal {K}}_q \left[ {\tilde{\gamma }}\right] \le {\mathcal {K}}_\infty \left[ {\tilde{\gamma }}\right] , \end{aligned}$$

(6)

due to Hölder’s inequality and the choice of the curves $\gamma _p$. It follows that the sequence $(\gamma _p)_{q \le p < \infty }$ is bounded in $W^{2,q}(0,L)$ for any $2 \le q < \infty$, so there exists a subsequence $(p_i)$ such that $\gamma _{p_i}$ converges weakly in $W^{2,q}(0,L)$ (and therefore strongly in $C^0$) for every $q < \infty$ to a limit

$$\begin{aligned} \gamma _\infty \in \bigcap _{q < \infty } W^{2,q}(0,L). \end{aligned}$$

The well-known property of the $L^\infty$ norm as the limit of $L^p$ norms then implies that

$$\begin{aligned} {\mathcal {J}}_\infty ^\sigma \left[ \gamma _\infty ; {\tilde{\gamma }}\right]&= \lim _{q \rightarrow \infty } {\mathcal {J}}_q^\sigma \left[ \gamma _\infty ; {\tilde{\gamma }}\right] \\&= \lim _{q \rightarrow \infty } {\mathcal {K}}_q \left[ \gamma _\infty \right] + {\mathcal {P}}_\sigma \left[ \gamma _\infty ; {\tilde{\gamma }}\right] . \end{aligned}$$

By the strong $C^0$ convergence,we have that both the $\liminf$ and the limit as $i \rightarrow \infty$ of ${\mathcal {P}}_\sigma \left[ \gamma _{p_i}; {\tilde{\gamma }}\right]$ coincide and equal ${\mathcal {P}}_\sigma \left[ \gamma _\infty ; {\tilde{\gamma }}\right]$. Therefore, using the lower semicontinuity of the $L^q$ norm with respect to weak convergence, we find that

$$\begin{aligned} {\mathcal {J}}_\infty ^\sigma \left[ \gamma _\infty ; {\tilde{\gamma }}\right]&\le \lim _{q \rightarrow \infty } \liminf _{i \rightarrow \infty } {\mathcal {K}}_q \left[ \gamma _{p_i} \right] + \lim _{q \rightarrow \infty } \liminf _{i \rightarrow \infty } {\mathcal {P}}_\sigma \left[ \gamma _\infty ; {\tilde{\gamma }}\right] \nonumber \\&= \lim _{q \rightarrow \infty } \liminf _{i \rightarrow \infty } {\mathcal {J}}_q^\sigma \left[ \gamma _{p_i} ; {\tilde{\gamma }}\right] \nonumber \\&\le {\mathcal {K}}_\infty \left[ {\tilde{\gamma }}\right] , \end{aligned}$$

(7)

where the final inequality comes from (6). By assumption, however, there exists $A \in {\mathbb {R}}$ such that

$$\begin{aligned} {\mathcal {K}}_\infty \left[ {\tilde{\gamma }}\right] \le {\mathcal {K}}_\infty \left[ \gamma _\infty \right] + \frac{A}{2L}\int _0^L d(\gamma _\infty (s), \text {seg}_{\tilde{\gamma }}(s))^2 \, \textrm{d}s, \end{aligned}$$

and so

$$\begin{aligned} {\mathcal {J}}_\infty ^\sigma \left[ \gamma _\infty ; {\tilde{\gamma }}\right] \le {\mathcal {J}}_\infty ^A \left[ \gamma _\infty ; {\tilde{\gamma }}\right] . \end{aligned}$$

Choosing $\sigma > A$ then implies that

$$\begin{aligned} \int _0^L d(\gamma _\infty (s), \text {seg}_{\tilde{\gamma }}(s))^2 \, \textrm{d}s = 0. \end{aligned}$$

We claim that this equality can only be satisfied when $\gamma _\infty = {\tilde{\gamma }}$. Indeed, if there is any point on $\gamma _\infty$ which does not lie on ${\tilde{\gamma }}$ then by continuity, there must be some $\varepsilon > 0$ and some interval in which $d(\gamma _\infty (s), \text {seg}_{\tilde{\gamma }}(s))^2 > \varepsilon$, which contradicts the fact that the above integral evaluates to zero. It follows that every point on $\gamma _\infty$ is also a point on ${\tilde{\gamma }}$.

Assume for a contradiction that the two curves are not the same. Let $s_0 \in [0,L]$ be the unique number such that ${\tilde{\gamma }}(s) = \gamma _\infty (s)$ for all $s \in [0,s_0]$ and such that for all $\varepsilon > 0$ the interval $[s_0,s_0 + \varepsilon ]$ contains an element s with $\gamma _\infty (s) \ne {\tilde{\gamma }}(s)$. Then the following statement holds: there exist arbitrarily small $\varepsilon > 0$ for which we can say ${\tilde{\gamma }}(s_0 + \varepsilon ) \not \in \gamma _\infty ([s_0,s_0 + \varepsilon ])$. Otherwise, there would be $\delta > 0$ such that ${\tilde{\gamma }}(s) \in \gamma _\infty ([s_0,s]) \subseteq \gamma _\infty ([s_0, s_0 + \delta ])$ for all $s \in [s_0, s_0 + \delta ]$, hence ${\tilde{\gamma }}([s_0, s_0 + \delta ]) \subseteq \gamma _\infty ([s_0, s_0 + \delta ])$, which combined with the arclength parametrisations of both curves and the fact that ${\tilde{\gamma }}$ and hence $\gamma _\infty$ are injective on $[s_0, s_0 + \delta ]$ (because the interval is too short for any loops to form) implies that $\gamma _\infty \equiv {\tilde{\gamma }}$ on $[s_0, s_0 + \delta ]$ in contradiction to the definition of $s_0$.

Pick some small $0< \varepsilon < c/4$, with c the constant depending on the curvature of ${\tilde{\gamma }}$ as defined earlier, and set $s_1 = s_0 + \varepsilon$. Assume without loss of generality that ${\tilde{\gamma }}(s_1) \not \in \gamma _\infty ([s_0,s_1])$. Observe that for all $s \in (s_0,s_1)$, it must be the case that $\gamma _\infty (s) \not \in {\tilde{\gamma }}([ s_0 - c/4, s_0 ]) = \gamma _\infty ([ s_0 - c/4, s_0 ])$ (or if $s_0 < c/4$ the intervals should be replaced by $[0,s_0]$); if this inclusion did hold, $\gamma _\infty$ would contain a loop of length less than c/2. Thus, since $\gamma _\infty$ satisfies the same curvature bound as ${\tilde{\gamma }}$, the existence of such a loop contradicts the definition of c.

Define the set

$$\begin{aligned} \Omega = \{ s \in [s_0,s_1]:\, \gamma _\infty (s) \not \in {\tilde{\gamma }}([s_0,s_1]) \}. \end{aligned}$$

By length comparison, $\Omega$ must be non-empty, else we would conclude with the same arguments as mentioned previously that $\gamma _\infty \equiv {\tilde{\gamma }}$ on $[s_0,s_1]$ in contradiction of the definition of $s_0$. Moreover, $\Omega$ is open because $\gamma _\infty$ is continuous and the image of ${\tilde{\gamma }}$ is closed. $\Omega$ is therefore non-null, i.e. its measure is positive.

If $\Omega$ contains points arbitrarily close to $s_0$, we can consider a sequence $(s_n) \rightarrow s_0$ contained in $\Omega$. Because $\gamma _\infty (s_n) \in {\tilde{\gamma }}((s_n - c/4, s_n + c/4))$ by the conditions $\gamma _\infty (s) \in \text {seg}_{{\tilde{\gamma }}}(s)$ and $\gamma _\infty (s_n) \not \in {\tilde{\gamma }}([ s_0 - c/4, s_0 ])$, we can take a sequence of points $(t_n) \subset [s_1, s_1 + c/4]$ such that $\gamma _\infty (s_n) = {\tilde{\gamma }}(t_n)$. After taking a subsequence, we find upon taking the limit in n that there exists $t \in [s_1, s_1 + c/4]$ such that ${\tilde{\gamma }}(t) = \gamma _\infty (s_0) = {\tilde{\gamma }}(s_0)$, so ${\tilde{\gamma }}$ forms an impossibly short loop, a contradiction.

We can assume, then, that $s_0< {\bar{s}}:= \inf _{s \in \Omega } s < s_1$ (the second inequality coming from the fact that $\Omega$ is non-null). Because $\Omega$ is open, ${\bar{s}} \not \in \Omega$, i.e. $\gamma _\infty ({\bar{s}}) \in {\tilde{\gamma }}([s_0,s_1])$. As argued previously, we may take two sequences

$$\begin{aligned} ({\hat{s}}_n) \subset \Omega , {\hat{s}}_n \rightarrow {\bar{s}}, \text { and } ({\hat{t}}_n) \subset (s_1,s_1 + c/4) \end{aligned}$$

with $\gamma _\infty ({\hat{s}}_n) = {\tilde{\gamma }}({\hat{t}}_n)$. Again taking a subsequence then the limit in n, we find a number ${\hat{t}} \in [s_1, s_1 + c/4]$ such that $\gamma _\infty ({\bar{s}}) = {\tilde{\gamma }}({\hat{t}})$.

If ${\hat{t}} > s_1$, ${\tilde{\gamma }}$ forms an impossibly short loop because ${\tilde{\gamma }}({\hat{t}}) = \gamma _\infty ({\bar{s}}) \in {\tilde{\gamma }}([s_0,s_1])$.

Otherwise, we find that ${\tilde{\gamma }}(s_1) = {\tilde{\gamma }}({\hat{t}}) = \gamma _\infty ({\bar{s}}) \in \gamma _\infty ([s_0,s_1])$, giving the final contradiction we need to prove that the assumption $\gamma _\infty \ne {\tilde{\gamma }}$ cannot hold, as required.

For any $1< q < \infty$, the above argument shows that every subsequence of $(\gamma _p)$ has a further subsequence which converges weakly in $W^{2,q}(0,L)$ to ${\tilde{\gamma }}$. Thus the original sequence $(\gamma _p)$ also converges weakly in $W^{2,q}(0,L)$ to ${\tilde{\gamma }}$. This proves part 1).

To show part 2), we first note that since $\gamma _p \rightharpoonup {\tilde{\gamma }}$ weakly in $W^{2,q}$ we also have the uniform ($C^0$) convergence $\gamma _p \rightarrow {\tilde{\gamma }}$.

By construction, for all $s \in [0,L]$,the segment $\text {seg}_{\tilde{\gamma }}(s)$ has no self-intersections. Therefore for each $s \in [0,L]$, there exists $\varepsilon > 0$ such that the open ball $B_{3\varepsilon } ({\tilde{\gamma }}(s))$ centred at ${\tilde{\gamma }}(s)$ contains only one component of $\text {seg}_{\tilde{\gamma }}(s)$. By the compactness of [0, L], there must then exist an $\varepsilon > 0$ such that for every $s \in [0,L]$, the intersection of the ball $B_{3\varepsilon } ({\tilde{\gamma }}(s))$ and the segment $\text {seg}_{\tilde{\gamma }}(s)$ consists of only one component. At the same time, assume that $\varepsilon$ is small enough that the tails of $\text {seg}_{\tilde{\gamma }}(s)$ given by

$$\begin{aligned} {\tilde{\gamma }}\bigr |_{(s - c({\tilde{K}})/4, s - c({\tilde{K}})/5) \cap (0,L)} \quad \text { and } \quad {\tilde{\gamma }}\bigr |_{(s + c({\tilde{K}})/5, s + c({\tilde{K}})/4) \cap (0,L)} \end{aligned}$$

lie outside of $B_{3\varepsilon } ({\tilde{\gamma }}(s))$ (if s is near either end of (0, L) then one tail may be short or even non-existent). Also assume that $3\varepsilon < r$ where r is the infimum of the injectivity radius over all points x contained in the geodesic ball $B_L(x_0)$. Now, by the uniform convergence,we can take p so large that $d(\gamma _p(s), {\tilde{\gamma }}(s)) \le \varepsilon$ for all $s \in (0,L)$. Then the infimum distance between $\gamma _p(s)$ and $\text {seg}_{\tilde{\gamma }}(s)$ is equal to the infimum distance $d_0 \le \varepsilon$ between the fixed point $\gamma _p(s)$ and the compact set ${\tilde{\gamma }}([s - c({\tilde{K}})/5), s + c({\tilde{K}})/5) ])$, so it must be attained somewhere. Assuming for a contradiction that the minimal distance is attained at more than one point, we obtain two points on the geodesic circle $\partial B = \partial B_{d_0}(\gamma _p(s))$ of radius $d_0 \le \varepsilon$ at which ${\tilde{\gamma }}$ is tangent to B, connected by a segment of ${\tilde{\gamma }}$, contained entirely inside $B_{3 \varepsilon }(\gamma _p(s))$, of length less than or equal to $2c({\tilde{K}})/5$. Picking $\varepsilon$ arbitrarily small, the curvature of ${\tilde{\gamma }}$ is then forced to be arbitrarily large, but this is not possible due to the curvature bound $\kappa \le {\tilde{K}}$ on ${\tilde{\gamma }}$. Hence the minimiser must be unique. $\square$

When ${\tilde{\gamma }}$ is an $\infty$-elastica, Lemma 3 ensures that for sufficiently large p it is possible to compute the Euler-Lagrange equations for the minimiser $\gamma _p$ of the functional . Indeed, we find that

Proposition 4

Let $\sigma > 0$ and an $\infty$-elastica ${\tilde{\gamma }}\in {\mathcal {G}}$ be given. Assume $\sigma$ is large enough that part 1) of Lemma 3 applies. For every p large enough that part 2) of Lemma 3 applies, suppose that $\gamma _p \in W^{2,p}(0,L)$ minimises subject to the constraints (1–2). Let $K_p = {\mathcal {K}}_p \left[ \gamma _p \right]$ and let ${\hat{\varphi }}_p = \vert \nabla _{T_p} T_p \vert ^{p-2} \nabla _{T_p} T_p$. Then ${\hat{\varphi }}_p \in W^{2,1}(0,L)$ and there are Lagrange multipliers $\lambda _p \in {\mathbb {R}}$ such that the Euler-Lagrange equation

$$\begin{aligned} \begin{aligned} 0 =&\, \frac{1}{L}K_{p}^{1-p}R({\hat{\varphi }}_{p}, T_{p}) T_{p} + \frac{\sigma }{L} d_{p} \nu _{p} - \lambda _{p} \nabla _{T_{p}} T_{p} + \frac{1}{L}\frac{2p-1}{p}K_{p}^{1-p}\vert {\hat{\varphi }}_{p} \vert \kappa _{p} \nabla _{T_{p}} T_{p} \\&+ \frac{1}{L}\frac{2p-1}{p-1}K_{p}^{1-p} \kappa _{p} \vert {\hat{\varphi }}_{p} \vert ' T_{p} - \frac{\sigma }{2L}(d_{p}^2)'T_{p} - \frac{\sigma }{2L}d_{p}^2 \nabla _{T_{p}} T_{p} + \frac{1}{L}K_{p}^{1-p} \nabla _{T_{p}}^2 {\hat{\varphi }}_{p} \end{aligned} \end{aligned}$$

(8)

holds a.e. in (0, L), where $d_{p}^2(s) = d ( \gamma _{p} (s), \text {seg}_{{\tilde{\gamma }}} (s) )^2$ and $\nu _{p}(s)$ is the unit tangent vector at $\gamma _{p}(s)$ associated with the minimal geodesic from $\text {seg}_{{\tilde{\gamma }}} (s)$ to $\gamma _{p}(s)$.

To prove this proposition, we will need a standard Lemma concerning the derivative of the square of the distance function to a closed set:

Lemma 5

Let $\Omega \subset M$ be a closed set, and $x_0 \in M$ a point such that there is a unique minimal geodesic connecting $x_0$ and $\Omega$ (i.e. among all geodesics from $x_0$ to $\Omega$ there is exactly one with length equal to $d(x, \Omega )$). Then the function $f(x) = d(x,\Omega )^2$ is differentiable at $x_0$. At $x_0$, the directional derivative in the direction of $v \in T_{x_0}M$ is given by $-2d(x_0,\Omega )\langle v, \nu \rangle$ where $\nu$ is the unit tangent vector in $T_{x_0} M$ to the distance-minimising geodesic from $x_0$ to $\Omega$.

A reference for this lemma can be found, e.g. in Exercise 4.5.11 and Remark 4.5.12 of [13].

Proof of Proposition 4

Fix $p \in {\mathbb {N}}$ sufficiently large and let the minimiser of be denoted by $\gamma$ (for now, we drop the subscript p from our notation for simplicity).

Before computing the first variations, it will be useful to establish some notation. Let t be a regular parameter ranging over the interval [0, 1] (in particular t needs not be the arclength parameter). The speed v(t) of $\gamma$ at time t is given by $\left|\gamma '(t) \right|$. Take V to be the velocity vector field of $\gamma$, i.e. $V = vT = \frac{\partial \gamma }{\partial t}$. Consider a smooth variation $\gamma (t,w)$ of $\gamma$, such that $\gamma (t,0) =: \gamma _0(t) = \gamma (t)$, and assume it preserves the boundary conditions of the problem in the sense that for every fixed value of the variational parameter w, the curve given as a function of t by $\gamma _w(t):= \gamma (t,w)$ satisfies (1) (but not necessarily (2)). Note that we are slightly abusing notation here as we use $\gamma$ to refer to both the original curve and its variation, although from context the meaning will be clear.

Explicitly, in local co-ordinates,we may write $\gamma (t,w) = \gamma _0(t) + w\psi (t)$ for some smooth function $\psi$. To denote the variational vector field along $\gamma$, write

$$\begin{aligned} W(t) = \frac{\partial \gamma }{\partial w}(t,0). \end{aligned}$$

Note that since t and w are independent co-ordinates, their Lie bracket [V, W] vanishes, i.e. $\nabla _V W = \nabla _W V$.

We first compute the first variation of ${\mathcal {K}}_p^p$. Throughout this calculation, we follow the work of [14]. The first variation is given by

$$\begin{aligned} \frac{\partial }{\partial w}\Bigr |_{w=0} {\mathcal {K}}_p^p\left[ \gamma _w \right]&= \frac{\partial }{\partial w}\Bigr |_{w=0} \frac{1}{L} \int _0^{{\mathcal {L}}[\gamma ]} \kappa (s)^p \, \textrm{d}s \\&= \frac{\partial }{\partial w}\Bigr |_{w=0} \frac{1}{L} \int _0^1 \kappa (t)^p v(t) \, \textrm{d}t \\&= \frac{1}{L} \int _0^1 W(\kappa ^p) v + \kappa ^p W(v) \, \textrm{d}t \end{aligned}$$

The first term in the integrand becomes

$$\begin{aligned} W(\kappa ^p)v&= W\left( \left\langle \nabla _T T, \nabla _T T \right\rangle ^{p/2} \right) v \\&= p \kappa ^{p-2} \left\langle \nabla _W \nabla _T T, \nabla _T T \right\rangle v \\&= p \kappa ^{p-2} \left( \left\langle \nabla _T T, \nabla _T^2 W \right\rangle + \left\langle R(\nabla _T T,T)T , W \right\rangle - 2\kappa ^2\left\langle T, \nabla _T W \right\rangle \right) v \end{aligned}$$

where the third line follows from calculations similar to those in [14, Eq. 9]. Here, denotes the Riemann curvature tensor.

To compute the second term in the integrand, note that $W(v^2) = 2vW(v)$, and also

$$\begin{aligned} W(v^2)&= W \left( \left\langle V, V \right\rangle \right) \\&= 2 \left\langle \nabla _W V, V \right\rangle \\&= 2 \left\langle \nabla _V W, V \right\rangle \\&= 2 v^2 \left\langle \nabla _T W, T \right\rangle , \end{aligned}$$

where the vanishing Lie bracket condition $\nabla _W V = \nabla _V W$ and some linearities of the covariant derivative have been used. Thus $W(v) = \left\langle \nabla _T W, T \right\rangle v$.

The first variation of ${\mathcal {K}}_p^p$ now becomes

$$\begin{aligned} \frac{1}{L}\int _0^1 \left( p \kappa ^{p-2} \left( \left\langle \nabla _T T, \nabla _T^2 W \right\rangle + \left\langle R(\nabla _T T,T)T, W \right\rangle - 2\kappa ^2\left\langle T, \nabla _T W \right\rangle \right) + \kappa ^p \left\langle \nabla _T W, T \right\rangle \right) v \, \textrm{d}t, \end{aligned}$$

and reparametrising by arclength turns this into

$$\begin{aligned} \frac{1}{L}\int _0^{{\mathcal {L}}[\gamma ]} p \kappa ^{p-2} \left( \left\langle \nabla _T T, \nabla _T^2 W \right\rangle + \left\langle R(\nabla _T T,T)T, W \right\rangle \right) + (1-2p) \kappa ^p \left\langle T, \nabla _T W \right\rangle \, \textrm{d}s. \end{aligned}$$

We now compute the first variation of the penalisation term . Note that for $\gamma _w \not \in {\mathcal {G}}$, the distance term is given by the distance between the point $\gamma _w(s)$ at length s along $\gamma _w$ and the segment $\text {seg}_{\tilde{\gamma }}({\tilde{s}} \bigr |_{{\tilde{s}} = s(t,w) L / L_{\gamma _w}} )$ centred at the point ${\tilde{\gamma }}({\tilde{s}})$ at distance ${\tilde{s}} = s {\mathcal {L}}[{\tilde{\gamma }}]/{\mathcal {L}}[\gamma _w]$ (so,e.g. the point halfway along $\gamma _w$ is paired with the segment of ${\tilde{\gamma }}$ centred at the point halfway along ${\tilde{\gamma }}$).

The instantaneous direction of movement of the point $\gamma (t)$ is given by

$$\begin{aligned} \partial \gamma / \partial w\bigr |_{w=0} \gamma (t,w) = W(t). \end{aligned}$$

Now, by taking w sufficiently small,the difference between the arclength along $\gamma _w$ up to the point $\gamma _w(t)$ and the arclength along $\gamma$ up to the point $\gamma (t)$ can be made arbitrarily small, such that the endpoints of $\text {seg}_{\tilde{\gamma }}({\tilde{s}} \bigr |_{{\tilde{s}} = s(t,w) L / L_{\gamma _w}} )$ move so little as to leave unchanged the portion of $\text {seg}_{\tilde{\gamma }}({\tilde{s}} \bigr |_{{\tilde{s}} = s(t,w) L / L_{\gamma _w}} )$ contained within $B_{3\varepsilon } ({\tilde{\gamma }}(s))$ (with $\varepsilon$ as taken in the proof of the second part of Lemma 3). In effect, for small w and fixed t, the distance term is equal to the distance between $\gamma (t,w)$ and a fixed set of points. This lets us compute

$$\begin{aligned} \frac{\partial }{\partial w}\Bigr |_{w=0} {\mathcal {P}}_\sigma \left[ \gamma _w ; {\tilde{\gamma }}\right]&= \frac{\partial }{\partial w}\Bigr |_{w=0} \frac{\sigma }{2L} \int _0^{{\mathcal {L}}[\gamma ]} d(s)^2 \, \textrm{d}s \\&= \frac{\partial }{\partial w}\Bigr |_{w=0} \frac{\sigma }{2L} \int _0^1 d(t)^2 v(t) \, \textrm{d}t \\&= \frac{\sigma }{2L} \int _0^1 W(d^2) v + d^2 W(v) \, \textrm{d}t, \end{aligned}$$

where the shorthand d(t) has been used to denote the distance term for brevity. Moreover, using Lemma 5 and part 2) of Lemma 3 we obtain the expression $W(d^2) = -2d\langle \nu , W \rangle$, where $\nu$ is the tangent vector at $\gamma (t)$ associated with the distance-minimising curve starting at $\gamma (t)$ and ending on $\text {seg}_{\tilde{\gamma }}({\tilde{s}} \bigr |_{{\tilde{s}} = s(t) L / L_{\gamma }} )$.

The W(v) term was already computed when considering the first variation of ${\mathcal {K}}_p^p$, and so we can substitute in the expressions for $W(d^2)$ and W(v) to turn the first variation of into

$$\begin{aligned} \frac{\sigma }{2L} \int _0^1 \left( 2d \left\langle \nu , W \right\rangle + d^2 \left\langle \nabla _T W, T \right\rangle \right) v \, \textrm{d}t. \end{aligned}$$

Again, the presence of v makes it possible to reparametrise this by arclength, yielding

$$\begin{aligned} \frac{\sigma }{2L} \int _0^{{\mathcal {L}}[\gamma ]} \left( 2d \left\langle \nu , W \right\rangle + d^2 \left\langle \nabla _T W, T \right\rangle \right) \, \textrm{d}s. \end{aligned}$$

Finally, to account for the length constraint (2) we need to apply the Lagrange Multiplier Principle. We compute the first variation of the length functional

$$\begin{aligned} {\mathcal {L}}\left[ \gamma \right] = \int _0^1 \vert \gamma '(t) \vert \, \textrm{d}t = \int _0^1 v(t) \, \textrm{d}t. \end{aligned}$$

as

$$\begin{aligned} \frac{\partial }{\partial w}\Bigr |_{w=0} {\mathcal {L}}\left[ \gamma _w \right]&= \frac{\partial }{\partial w}\Bigr |_{w=0} \int _0^1 v(t) \, \textrm{d}t \\&= \int _0^1 W(v) \, \textrm{d}t \\&= \int _0^1 \left\langle \nabla _T W, T \right\rangle v \, \textrm{d}t \\&= \int _0^{{\mathcal {L}}[\gamma ]} \left\langle \nabla _T W, T \right\rangle \, \textrm{d}s \end{aligned}$$

so the Euler-Lagrange equation will have a term of the form

$$\begin{aligned} \lambda \int _0^{{\mathcal {L}}[\gamma ]} \left\langle \nabla _T W, T \right\rangle \, \textrm{d}s \end{aligned}$$

added to it, for some constant $\lambda \in {\mathbb {R}}$ (the Lagrange multiplier).

Combining the three calculations of the first variations, reintroducing the subscript p to our notation, and rewriting ${\mathcal {L}}[\gamma ] = L$, we obtain the following Euler-Lagrange equation in the weak sense:

$$\begin{aligned} \int _0^L \left\langle E_1, W \right\rangle + \left\langle E_2, \nabla _{T_p} W \right\rangle + \langle E_3, \nabla _{T_p}^2 W \rangle \, \textrm{d}s = 0, \end{aligned}$$

satisfied for every test variation W (smooth and compactly supported), where

$$\begin{aligned} E_1&= \frac{1}{L}K_p^{1-p}R({\hat{\varphi }}_p, T_p) T_p + \frac{\sigma }{L} d_p \nu _p ,\\ E_2&= \frac{1}{L}\frac{1-2p}{p} K_p^{1-p} \vert {\hat{\varphi }}_p \vert ^{p/(p-1)} T_p + \lambda _p T_p + \frac{\sigma }{2L}d_p^2 T_p , \end{aligned}$$

and

$$\begin{aligned} E_3&= \frac{1}{L}K_p^{1-p}{\hat{\varphi }}_p. \end{aligned}$$

After writing out the above Euler-Lagrange equation in local co-ordinates, we can make use of standard regularity results to deduce that ${\hat{\varphi }}_p \in W^{2,1}$, and this regularity allows us to integrate the Euler-Lagrange equation by parts:

$$\begin{aligned} \int _0^L \left\langle E_1, W \right\rangle - \langle \nabla _{T_p} E_2, W \rangle + \langle \nabla _{T_p}^2 E_3, W \rangle \, \textrm{d}s = 0, \end{aligned}$$

having used the fact that $d_p^2$ is Lipschitz continuous with Lipschitz constant 1 and thus is in $W^{1,\infty }$. The Fundamental Theorem of Calculus of Variations then implies that the Euler-Lagrange equation (8) holds a.e. in (0, L), finishing the proof. $\square$

Proposition 4 highlights a fundamental difference between p-elastica and $\infty$-elastica: when p is finite the fact that ${\hat{\varphi }}_p$ lies in $W^{2,1}(0,L)$ means that the curvature vector is continuous, yet the curvature vector of an $\infty$-elastica needs not be continuous and indeed in some cases is guaranteed by the constraints of the problem and the results of Theorem 2 to be discontinuous.

3 Limiting Equations for $\infty$-Elastica

In this section, we show that along the limiting $\infty$-elastica which the p-minimisers converge to, a limiting version of a normalisation of the Euler-Lagrange equation (8) is satisfied. In addition,we derive another equation from our definition of the rescaled curvature vector; these two equations together act as a system of equations that can be considered in a sense to be Euler-Lagrange equations for the $L^\infty$ problem. This proves most of statement 1) of Theorem 2, except that the limiting $\varphi$ is not identically zero.

Before we can obtain the limiting Euler-Lagrange equations (4–5), we must know that the terms in (8) do not become too large when $p \rightarrow \infty$. To this end, we prove two lemmata. The first lemma concerns the size of the ${\mathcal {K}}_p$ functionals when evaluated at minimisers of .

Lemma 6

Suppose ${\tilde{\gamma }}$ is an $\infty$-elastica and let $(\gamma _p)$ be a sequence of admissible curves which are minimisers of . Assume that $\sigma$ is large enough that both parts of Lemma 3 apply. Then the sequence $(K_p)$ converges to $K:= {\mathcal {K}}_\infty \left[ {\tilde{\gamma }}\right]$.

Proof

Take $\sigma$ large enough so the situation in the proof of Lemma 3 applies. Consider the inequalities in (6). They show that the sequence of real numbers given by $({\mathcal {J}}_p^\sigma \left[ \gamma _p; {\tilde{\gamma }}\right] )$ is increasing and bounded above by ${\mathcal {K}}_\infty \left[ {\tilde{\gamma }}\right]$, hence convergent with

$$\begin{aligned} \lim _{p \rightarrow \infty } {\mathcal {J}}_p^\sigma \left[ \gamma _p; {\tilde{\gamma }}\right] \le {\mathcal {K}}_\infty \left[ {\tilde{\gamma }}\right] . \end{aligned}$$

However, the inequalities in (7) combined with the fact that ${\tilde{\gamma }}= \gamma _\infty$ mean that

$$\begin{aligned}{} & {} {\mathcal {K}}_\infty \left[ {\tilde{\gamma }}\right] = {\mathcal {J}}_\infty ^\sigma \left[ {\tilde{\gamma }}; {\tilde{\gamma }}\right] \le \lim _{q \rightarrow \infty } \liminf _{i \rightarrow \infty } {\mathcal {J}}_q^\sigma \left[ \gamma _{p_i}; {\tilde{\gamma }}\right] \\ {}{} & {} \le \liminf _{i \rightarrow \infty } {\mathcal {J}}_{p_i}^\sigma \left[ \gamma _{p_i}; {\tilde{\gamma }}\right] = \lim _{p \rightarrow \infty } {\mathcal {J}}_p^\sigma \left[ \gamma _p; {\tilde{\gamma }}\right] . \end{aligned}$$

It follows that $\lim _{p \rightarrow \infty } {\mathcal {J}}_p^\sigma \left[ \gamma _p; {\tilde{\gamma }}\right] = {\mathcal {K}}_\infty \left[ {\tilde{\gamma }}\right]$. Now,

$$\begin{aligned} {\mathcal {K}}_\infty \left[ {\tilde{\gamma }}\right] \,{=}\, \lim _{p \rightarrow \infty } {\mathcal {J}}_p^\sigma \left[ \gamma _p; {\tilde{\gamma }}\right] = \lim _{p \rightarrow \infty } {\mathcal {K}}_p \left[ \gamma _p \right] + \lim _{p \rightarrow \infty } \frac{\sigma }{2L} \int _0^L d(\gamma _p(s),\text {seg}_{\tilde{\gamma }}(s))^2 \, \textrm{d}s, \end{aligned}$$

and by the uniform convergence of $\gamma _p$ to ${\tilde{\gamma }}$,the second term on the right-hand side vanishes, finishing the proof. $\square$

The second lemma concerns the size of the Lagrange multipliers $\lambda _p$.

Lemma 7

Suppose ${\tilde{\gamma }}$ is an $\infty$-elastica, and that $\sigma$ is as large as required by Lemma 3. Then the sequence of Lagrange multipliers $(\lambda _p)$ in the Euler-Lagrange equation (8) is bounded.

Before proving this lemma, we introduce some common but not ubiquitous notation: given sequences $(\alpha _p)$, $(\beta _p)$ we say that $\alpha _p \lesssim \beta _p$ if there exists a positive constant c independent of p such that $\alpha _p \le c \beta _p$ for every $p \in {\mathbb {N}}$.

Proof of Lemma 7

Rearranging (8) gives us the equation

$$\begin{aligned} {\tilde{E}}_p = L \lambda _p \nabla _{T_p} T_p, \end{aligned}$$

(9)

where for $K_p = {\mathcal {K}}_p \left[ \gamma _p \right]$ we have

$$\begin{aligned} {\tilde{E}}_p := \,&K_p^{1-p}\nabla _{T_p}^2 \left( \kappa _p^{p-2} \nabla _{T_p} T_p \right) + K_p^{1-p}\kappa _p^{p-2} R(\nabla _{T_p} T_p, T_p ) T_p + \sigma d_p \nu _p \\&+ K_p^{1-p}\frac{2p-1}{p} \nabla _{T_p} \left( \kappa _p^p T_p \right) - \frac{\sigma }{2} (d_p^2)' T - \frac{\sigma }{2} d_p^2 \nabla _{T_p} T_p. \end{aligned}$$

Note here that we have combined two of the terms from (8) into a single term expressed as a derivative:

$$\begin{aligned}{} & {} \frac{1}{L}\frac{2p-1}{p}K_{p}^{1-p}\vert {\hat{\varphi }}_{p} \vert \kappa _{p} \nabla _{T_{p}} T_{p} + \frac{1}{L}\frac{2p-1}{p-1}K_{p}^{1-p} \kappa _{p} \vert {\hat{\varphi }}_{p} \vert ' T_{p}\\{} & {} \qquad = \frac{1}{L} \frac{2p-1}{p} \nabla _{T_{p}} \left( \langle {\hat{\varphi }}_p, \nabla _{T_p} T_p \rangle T_p \right) . \end{aligned}$$

If eventually the $\lambda _p$s all become 0, the lemma is immediate, so assume this is not the case and discard from the sequence any $\lambda _p$s which are zero. Then

$$\begin{aligned} \frac{1}{L \lambda _p} {\tilde{E}}_p = \nabla _{T_p} T_p \rightharpoonup \nabla _T T \ne 0 \text { in }L^q(0,L)\text { for any }q < \infty , \end{aligned}$$

recalling that $\nabla _T T$ is not identically zero because it is assumed a priori that no geodesic solutions exist. Since $\nabla _T T$ does not vanish, there exists a test vector field $\psi$ such that $\int _0^L \left\langle \nabla _T T, \psi \right\rangle \, \textrm{d}s = 1$. Without loss of generality, we can assume that the support of $\psi$ is contained in a single co-ordinate chart and thus we can write $\psi (s) = \psi ^i \frac{\partial }{\partial x^{i}} |_{\gamma (s)}$. In these co-ordinates, for sufficiently large $p \in {\mathbb {N}}$, consider the test vector field $\psi _p$ defined along $\gamma _p$ by $\psi ^i \frac{\partial }{\partial x^{i}} |_{\gamma _p(s)}$ (this is valid due to the uniform convergence of $\gamma _p$ to $\gamma$), so the coefficients $\psi ^i$ are smooth and do not depend on p. Taking the inner product of ${\tilde{E}}_p$ with $\psi _p$, integrating from 0 to L and integrating by parts tells us that

$$\begin{aligned} \int _0^L \left\langle {\tilde{E}}_p, \psi _p \right\rangle \, \textrm{d}s&\le I_1 + I_2 + I_3 + I_4 + I_5 + I_6, \end{aligned}$$

where

$$\begin{aligned} I_1&= K_p^{1-p}\int _0^L \kappa _p^{p-2} \vert \nabla _{T_p} T_p \vert \vert \nabla _{T_p}^2 \psi _p \vert \, \textrm{d}s, \\ I_2&= K_p^{1-p}\int _0^L \kappa _p^{p-2} \vert R\left( \nabla _{T_p} T_p, T_p \right) T_p \vert \vert \psi _p \vert \, \textrm{d}s, \\ I_3&= \sigma \int _0^L d_p \vert \nu _p \vert \vert \psi _p \vert \, \textrm{d}s, \\ I_4&= 2K_p^{1-p}\int _0^L \vert \kappa _p^{p-2} \nabla _{T_p} T_p \vert \vert \nabla _{T_p} \psi _p \vert \, \textrm{d}s, \\ I_5&= \frac{\sigma }{2} \int _0^L \vert (d_p^2)' \vert \vert \psi _p \vert \, \textrm{d}s, \end{aligned}$$

and

$$\begin{aligned} I_6&= \frac{\sigma }{2} \int _0^L d_p^2 \vert \nabla _{T_p} T_p \vert \vert \psi _p \vert \, \textrm{d}s. \end{aligned}$$

Using the definition of $\psi _p$, Hölder’s inequality, the definitions of $\kappa _p$ and $K_p$, Lemma 6, the boundedness of R on compact subsets of M, and the boundedness of $d_p^2$ and $(d_p^2)'$, it follows that each of the six integrals $I_n$ is bounded above by some constant $C(\psi )$ depending on the components $\psi ^i$ of $\psi$ but independent of p, and hence so too is $\int _0^L \langle {\tilde{E}}_p, \psi \rangle \, \textrm{d}s$.

For example, we have that

$$\begin{aligned} I_2&\lesssim \left\Vert \psi _p\right\Vert _{L^\infty (0,L)} K_p^{1-p} \int _0^L \kappa _p^{p-1} \, \textrm{d}s \\&\lesssim C(\psi ) K_p^{1-p} \left\Vert \kappa _p\right\Vert _{L^p}^{p-1} \left\Vert 1\right\Vert _{L^p} \\&\lesssim C(\psi ), \end{aligned}$$

where we have made the dependence on $\psi$ explicit. Here we have used the boundedness of R and $\psi$ and the definition of $\kappa _p$ in the first line, Hölder’s inequality in the second line, and the definition of $K_p$ in the third line. The other five integrals are treated similarly.

By the weak $L^q(0,L)$ convergence of $\nabla _{T_p} T_p$ and the choice of $\psi$,

$$\begin{aligned} \int _0^L \left\langle \frac{1}{L \lambda _p} {\tilde{E}}_p, \psi _p \right\rangle \, \textrm{d}s \ge \frac{1}{2} \end{aligned}$$

for sufficiently large p, i.e.

$$\begin{aligned} L \lambda _p \le 2\int _0^L \left\langle {\tilde{E}}_p, \psi _p \right\rangle \, \textrm{d}s \le 2 C(\psi ). \end{aligned}$$

This gives the desired bound. $\square$

With these two lemmata established we are well positioned to begin to prove Theorem 2.

Beginning of proof of Theorem2 We begin by normalising equation (8), setting $\varphi _p = K_p^{1-p} {\hat{\varphi }}_p$. Thus we obtain the equation

$$\begin{aligned} \begin{aligned} 0 =&\, R(\varphi _p, T_p) T_p + \sigma d_p \nu _p - L \lambda _p \nabla _{T_p} T_p + \frac{2p-1}{p}\vert \varphi _p \vert \kappa _p \nabla _{T_p} T_p \\&+ \frac{2p-1}{p-1} \kappa _p \vert \varphi _p \vert ' T_p - \frac{\sigma }{2} (d_p^2)' T_p - \frac{\sigma d_p^2}{2} \nabla _{T_p} T_p + \nabla _{T_p}^2 \varphi _p. \end{aligned}\nonumber \\ \end{aligned}$$

(10)

Our goal now is to obtain higher regularity of the $\varphi _p$ terms along with bounds on them which are uniform in p.

Choose $\psi \in C^\infty _c(0,L)$ to be a test function such that

$$\begin{aligned} \left\Vert \frac{\psi '(s)^2}{\psi (s)} \right\Vert _{L^\infty (0,L)} < \infty . \end{aligned}$$

For example, the square of any test function suffices here, as does a rescaling of the standard “bump” function given by $\exp (-1/(1-x^2))$. Now consider the function $f(s) = \frac{1}{2}\vert \varphi _p(s)\vert ^2$. Observing that $\int _0^L (f' \psi ^2)' \, \textrm{d}s = 0$ then integrating by parts shows that

$$\begin{aligned} 0&= \int _0^L \psi ^2 f'' - 2 \psi \psi '' f - 2 (\psi ')^2 f \, \textrm{d}s. \end{aligned}$$

Substituting in the definition of f gives

$$\begin{aligned} 0 = \int _0^L \psi ^2\langle \varphi _p , \nabla _{T_p}^2 \varphi _p \rangle + \vert \psi \nabla _{T_p} \varphi _p \vert ^2 -\psi \psi '' \vert \varphi _p \vert ^2 - (\psi ')^2 \vert \varphi _p \vert ^2 \, \textrm{d}s. \end{aligned}$$

Again substituting, this time using the Euler-Lagrange equation (10) to replace the $\nabla _{T_p}^2 \varphi _p$ term, we find that

$$\begin{aligned} \int _0^L \vert \psi \nabla _{T_p} \varphi _p \vert ^2 \, \textrm{d}s \le I_1 + I_2 + I_3 + I_4 + I_5 + I_6 + I_7, \end{aligned}$$

(11)

where

$$\begin{aligned} I_1&= \int _0^L \psi ^2 \vert R(\varphi _p,T_p)T_p \vert \vert \varphi _p \vert \, \textrm{d}s ,\\ I_2&= \sigma \int _0^L \psi ^2 d_p \vert \varphi _p \vert \, \textrm{d}s ,\\ I_3&= L \lambda _p \int _0^L \psi ^2 \vert \varphi _p \vert \vert \nabla _{T_p} T_p \vert \, \textrm{d}s ,\\ I_4&= \frac{2p - 1}{p} \int _0^L \psi ^2 \kappa _p \vert \varphi _p \vert ^2 \vert \nabla _{T_p} T_p \vert \, \textrm{d}s ,\\ I_5&= \frac{\sigma }{2} \int _0^L \psi ^2 d_p^2 \vert \varphi _p \vert \vert \nabla _{T_p} T_p \vert \, \textrm{d}s, \\ I_6&= \int _0^L \vert \psi \vert \vert \psi '' \vert \vert \varphi _p \vert ^2 \, \textrm{d}s, \end{aligned}$$

and

$$\begin{aligned} I_7&= \int _0^L \vert \psi ' \vert ^2 \vert \varphi _p \vert ^2 \, \textrm{d}s. \end{aligned}$$

The contributions from all the terms in (10) parallel to $T_p$ vanish since we take the inner product $\langle \varphi _p , \nabla _{T_p}^2 \varphi _p \rangle$ and $T_p$ is orthogonal to $\varphi _p$.

The first integral can be estimated by

$$\begin{aligned} I_1&\lesssim \int _0^L \psi ^2 \vert \varphi _p \vert ^2 \, \textrm{d}s \\&\lesssim K_p^{2-2p} \int _0^L \psi ^2 \kappa _p^{2p-2} \, \textrm{d}s \\&\lesssim K_p^{2-2p} \left\Vert \psi ^{\frac{p}{p-1}}\right\Vert _{L^\infty } \left\Vert \psi ^{\frac{p-2}{p-1}} \kappa _p^{p-2}\right\Vert _{L^\infty } \int _0^L \kappa _p^p \, \textrm{d}s \\&\lesssim K_p^{2-p} \left\Vert \psi \right\Vert _{L^\infty }^{\frac{p}{p-1}} \left\Vert \psi ^{\frac{p-2}{p-1}} \kappa _p^{p-2}\right\Vert _{L^\infty } \\&\lesssim \left\Vert \psi \right\Vert _{L^\infty }^{\frac{p}{p-1}} \left\Vert \psi \varphi _p\right\Vert _{L^\infty }^{\frac{p-2}{p-1}} \end{aligned}$$

having used the definitions of $\varphi _p = K_p^{1-p} \vert \nabla _{T_p} T_p \vert ^{p-2} \nabla _{T_p} T_p$ and $K_p$ as well as the boundedness of the Riemann curvature tensor. Note that $\left\Vert \psi \kappa _p^{p-2}\right\Vert _{L^\infty }$ is finite because $\psi$ is smooth and the fact from Proposition 4 that ${\hat{\varphi }}_p \in W^{2,1}$ implies that ${\hat{\varphi }}_p$ and in particular $\kappa _p$ is bounded (though this does not give a uniform bound). Now, since $\left\Vert f\right\Vert _{L^\infty } \le \left\Vert f\right\Vert _{L^1}/L + \left\Vert f'\right\Vert _{L^1}$ (this can be seen, e.g. by the Sobolev embedding of $W^{1,\infty }(0,L)$ into $L^\infty (0,L)$) we get that

$$\begin{aligned} I_1&\lesssim \left\Vert \psi \right\Vert _{L^\infty }^{\frac{p}{p-1}} \left( \sqrt{L} \left\Vert \psi \nabla _{T_p} \varphi _p\right\Vert _{L^2} + \left\Vert \varphi _p\right\Vert _{L^1} \Vert \psi '\Vert _{L^\infty } + \frac{1}{L} \left\Vert \psi \varphi _p\right\Vert _{L^1} \right) ^{\frac{p-2}{p-1}} \\&\lesssim \left\Vert \psi \right\Vert _{L^\infty }^{\frac{p}{p-1}} \left( \left\Vert \psi \nabla _{T_p} \varphi _p\right\Vert _{L^2} + \left\Vert \varphi _p\right\Vert _{L^1} \Vert \psi '\Vert _{L^\infty } + \left\Vert \psi \varphi _p\right\Vert _{L^1} \right) ^{\frac{p-2}{p-1}} \\&\lesssim \left\Vert \psi \right\Vert _{L^\infty }^{\frac{p}{p-1}}\left( \left\Vert \psi \nabla _{T_p} \varphi _p\right\Vert _{L^2}^{\frac{p-2}{p-1}} + \left( \left\Vert \varphi _p\right\Vert _{L^1} \Vert \psi '\Vert _{L^\infty } + \left\Vert \varphi _p\right\Vert _{L^1}\left\Vert \psi \right\Vert _{L^\infty } \right) ^{\frac{p-2}{p-1}}\right) \\&\lesssim \left\Vert \psi \right\Vert _{L^\infty }^{\frac{p}{p-1}}\left\Vert \psi \nabla _{T_p} \varphi _p\right\Vert _{L^2}^{\frac{p-2}{p-1}} + \Vert \psi '\Vert _{L^\infty }^{\frac{p-2}{p-1}} \left\Vert \psi \right\Vert _{L^\infty }^{\frac{p}{p-1}} + \left\Vert \psi \right\Vert _{L^\infty }^2 \end{aligned}$$

because of the uniform $L^1$ bound on $\varphi _p$ which comes from its definition and an immediate application of Hölder’s inequality. Applying Young’s inequality in the form

$$\begin{aligned} a^{\frac{p-2}{p-1}} b^{\frac{p}{p-1}} = \left( a^{\frac{p-2}{p-1}} \varepsilon ^{\frac{p-2}{2p-2}} \right) \left( b^{\frac{p}{p-1}} \varepsilon ^{-\frac{p-2}{2p-2}} \right) \le \frac{p-2}{2p-2} \varepsilon a^2 + \frac{p}{2p-2} \varepsilon ^{\frac{2-p}{p}} b^2 \end{aligned}$$

to the first term of this expression, we find that

$$\begin{aligned} I_1&\le \varepsilon \frac{p-2}{2p-2} \left\Vert \psi \nabla _{T_p} \varphi _p\right\Vert _{L^2}^2 + C \left( \varepsilon ^{\frac{2-p}{p}} \frac{p}{2p-2} \left\Vert \psi \right\Vert _{L^\infty }^{2} + \Vert \psi '\Vert _{L^\infty }^{\frac{p-2}{p-1}} \left\Vert \psi \right\Vert _{L^\infty }^{\frac{p}{p-1}} + \left\Vert \psi \right\Vert _{L^\infty }^2 \right) \\&\le \varepsilon \left\Vert \psi \nabla _{T_p} \varphi _p\right\Vert _{L^2}^2 + C \left( \varepsilon ^{\frac{2-p}{p}}\left\Vert \psi \right\Vert _{L^\infty }^{2} + \Vert \psi '\Vert _{L^\infty }^{\frac{p-2}{p-1}} \left\Vert \psi \right\Vert _{L^\infty }^{\frac{p}{p-1}} + \left\Vert \psi \right\Vert _{L^\infty }^2 \right) \end{aligned}$$

where $C \in {\mathbb {R}}$ is some constant independent of p. We have therefore shown that

$$\begin{aligned} I_1&\le \varepsilon \left\Vert \psi \nabla _{T_p} \varphi _p\right\Vert _{L^2}^2 + C, \end{aligned}$$

where C is a constant which depends on $\varepsilon$ and $\psi$ but importantly not on p.

Similarly, the fourth integral can be estimated by

$$\begin{aligned} I_4&\le C \left\Vert \psi ^{\frac{p-2}{p-1}}\right\Vert _{L^\infty } \Vert \kappa _p^p \psi ^{\frac{p}{p-1}}\Vert _{L^\infty } K_p^{2-2p} \int _0^L \kappa _p^p \, \textrm{d}s \\&\lesssim \left\Vert \psi \right\Vert ^{\frac{p-2}{p-1}}_{L^\infty } K_p^{2-p} \left\Vert \kappa _p^{p-1} \psi \right\Vert _{L^\infty }^{\frac{p}{p-1}} \\&\lesssim \left\Vert \psi \right\Vert ^{\frac{p-2}{p-1}}_{L^\infty } \left\Vert \psi \varphi _p \right\Vert _{L^\infty }^{\frac{p}{p-1}} \\&\lesssim \left\Vert \psi \right\Vert _{L^\infty }^{\frac{p-2}{p-1}} \left( \left\Vert \psi \nabla _{T_p} \varphi _p\right\Vert _{L^2} + \Vert \psi ' \varphi _p\Vert _{L^1} + \left\Vert \psi \varphi _p\right\Vert _{L^1} \right) ^{\frac{p}{p-1}} \\&\lesssim \left\Vert \psi \right\Vert _{L^\infty }^{\frac{p-2}{p-1}}\left\Vert \psi \nabla _{T_p} \varphi _p\right\Vert _{L^2}^{\frac{p}{p-1}} + \Vert \psi '\Vert _{L^\infty }^{\frac{p}{p-1}}\left\Vert \psi \right\Vert _{L^\infty }^{\frac{p-2}{p-1}} + \left\Vert \psi \right\Vert _{L^\infty }^2 \\&\lesssim \varepsilon \left\Vert \psi \nabla _{T_p} \varphi _p\right\Vert _{L^2}^{2} + \varepsilon ^{\frac{-p}{p-2}}\left\Vert \psi \right\Vert _{L^\infty }^{\frac{-2}{p-1}} + \Vert \psi '\Vert _{L^\infty }^{\frac{p}{p-1}}\left\Vert \psi \right\Vert _{L^\infty }^{\frac{p-2}{p-1}} + \left\Vert \psi \right\Vert _{L^\infty }^2 \end{aligned}$$

which again results in an inequality of the form

$$\begin{aligned} I_4&\le \varepsilon \left\Vert \psi \nabla _{T_p} \varphi _p\right\Vert _{L^2}^2 + C. \end{aligned}$$

The sixth and seventh integrals yield similar estimates, where in the calculations for $I_7$ we make use of the assumption on the form of $\psi$:

$$\begin{aligned} I_7&= \int _0^L \vert \psi ' \vert ^2 \vert \varphi _p \vert ^2 \, \textrm{d}s \\&= \int _0^L \frac{\vert \psi ' \vert ^2}{\vert \psi \vert } \vert \varphi _p \vert \vert \psi \varphi _p \vert \, \textrm{d}s \\&\le \left\Vert \frac{(\psi ')^2}{\psi }\right\Vert _{L^\infty } \left\Vert \psi \varphi _p\right\Vert _{L^\infty } \left\Vert \varphi _p\right\Vert _{L^1}, \end{aligned}$$

and from this point the calculations with the $\left\Vert \psi \varphi _p\right\Vert _{L^\infty }$ follow similarly to the calculations for $I_1$.

We also compute from their definitions that

$$\begin{aligned} I_2, I_3, I_5 \le C(\psi ) \end{aligned}$$

using Lemma 6 and the uniform $L^1$ bound on $\kappa _p \varphi _p$ coming from its definition.

Combining these seven estimates and picking $\varepsilon$ sufficiently small, we obtain a uniform bound for $\Vert \psi \nabla _{T_p} \varphi _p\Vert _{L^2}^2$. By picking $\psi$ such that $\psi \ge 1$ on $[a,b] \subset (0,L)$, we get a bound on $\Vert \nabla _{T_p} \varphi _p\Vert _{L^2([a,b])}^2$ which is uniform in p. From the Sobolev Embedding Theorem, we obtain a uniform bound on $\Vert \varphi _p\Vert _{L^\infty ([a,b])}$, and then going back to the Euler-Lagrange equation (10) we obtain a uniform $L^2([a,b])$ bound on $\nabla _{T_p}^2 \varphi _p$ by expressing it as the sum of terms each possessing such a uniform bound. Applying the Sobolev Embedding Theorem again, we get a uniform bound on $\Vert \nabla _{T_p} \varphi _p\Vert _{L^\infty ([a,b])}$, and again going back to (10),we get a uniform bound on $\Vert \nabla _{T_p}^2 \varphi _p\Vert _{L^\infty ([a,b])}$.

From this, we obtain a subsequence of $(\varphi _p)$ which converges weakly in $W^{2,q}([a,b])$ to a limit $\varphi \in W^{2,\infty }([a,b])$ for every $q < \infty$. Since [a, b] was arbitrary we have that in fact $\varphi \in W^{2,\infty }_{\text {loc}}(0,L)$. Now, the sequence $(\lambda _p)$ is bounded by Lemma 7 and so has a convergent subsequence, and by the uniform convergence of the curves $\gamma _p$ to $\gamma$ and the distance functions $d_p$ to 0 we see after expressing (10) in local co-ordinates and taking a suitable subsequence (not explicitly labelled) that we obtain the limiting equation (4), with the final term coming from the fact that

$$\begin{aligned}{} & {} \frac{2p-1}{p} \nabla _{T_p} \left( \langle \varphi _p, \nabla _{T_p} T_p \rangle T_p \right) \\{} & {} \qquad = \frac{2p-1}{p}K_{p}^{1-p}\vert {\hat{\varphi }}_{p} \vert \kappa _{p} \nabla _{T_{p}} T_{p} + \frac{2p-1}{p-1}K_{p}^{1-p} \kappa _{p} \vert {\hat{\varphi }}_{p} \vert ' T_{p}. \end{aligned}$$

To obtain equation (5), we consider

$$\begin{aligned} \nabla _{T_p} T_p = \vert \varphi _p \vert ^{1/(p-1)} K_p \frac{\varphi _p}{\vert \varphi _p \vert } \end{aligned}$$

and let $[a,b] \subset (0,L)$ be arbitrary. By Lemma 6, $K_p \rightarrow K$. As $\varphi$ is continuous, and the convergence of the $\varphi _p$s to $\varphi$ is locally uniform, for any fixed t with $\varphi (t) \ne 0$ there exist $\delta , \varepsilon > 0$ such that $\delta \le \vert \varphi _p \vert \le 1/\delta$ in $(t - \varepsilon , t + \varepsilon ) \cap [a,b]$ for p sufficiently large. Thus

$$\begin{aligned} \frac{\varphi _p}{\vert \varphi _p \vert } \rightarrow \frac{\varphi }{\vert \varphi \vert } \text { uniformly in }(t - \varepsilon , t + \varepsilon ) \cap (a,b). \end{aligned}$$

Also, since $\delta \le \vert \varphi _p \vert \, \le 1/\delta$ in $(t - \varepsilon , t + \varepsilon ) \cap (a,b)$, it holds in this interval that $\vert \varphi _p \vert ^{1/(p-1)} \rightarrow 1$ uniformly as $p \rightarrow \infty$.

Combining these convergence results, (5) is obtained at the points in (a, b) where $\vert \varphi \vert \, \ne 0$. Because $[a,b] \subset (0,L)$ was arbitrary and (5) holds wherever $\varphi = 0$, it must therefore hold a.e. in (0, L).

4 Properties of $\infty$-Elastica and Existence of ${\mathcal {K}}_\infty$ Minimisers

In this section,we examine the limiting system of Eqs. (4–5) and how they relate to solutions of our problem. We prove the remainder of the first statement of Theorem 2—that the limiting vector field $\varphi$ is not identically zero—and we finish the proof of Theorem 2 by proving the second statement: that if a curve admits a solution of (4–5), then its curvature may take on at most two values. We show the existence of minimisers of ${\mathcal {K}}_\infty$.

Proof of Theorem 2 continued

To finish proving statement 1) of Theorem 2, it remains only to show that $\varphi$ is not identically zero. To this end, choose a sequence of vector fields $(Z_p)$ with each vector field $Z_p$ defined along the curve $\gamma _p$ such that $Z_p(0) = Z_p(L) = 0$ with $\nabla _{T_p} Z_p = T_p$ on $[0,\delta ] \cup [L - \delta , L]$ for some small $\delta > 0$ (such a choice is made possible by the standard existence theorem for ODEs with initial data). Take a function $\eta \in C^\infty ([0,L])$ such that $\eta (0) = \eta (L) = 1$, $\vert \eta \vert \,\le 1$, and $\text {supp}(\eta ) \subseteq [0,\delta ] \cup [L - \delta , L]$. Moreover, take the vector fields $Z_p$ to have continuous covariant derivatives up to second order on (0, L). By the definitions of $\varphi _p$ and $Z_p$,

$$\begin{aligned} K_p^{1-p} \int _0^L \eta \kappa _p^p \, \textrm{d}s&= \int _0^L \eta \langle \varphi _p, \nabla _{T_p}^2 Z_p \rangle \, \textrm{d}s \\&= \left[ \eta \langle \varphi _p, \nabla _{T_p} Z_p \rangle \right] _0^L - \int _0^L \eta ' \langle \varphi _p, \nabla _{T_p} Z_p \rangle + \eta \langle \nabla _{T_p} \varphi _p, \nabla _{T_p} Z_p \rangle \, \textrm{d}s \\&= - \int _0^L \eta \langle \nabla _{T_p} \varphi _p, \nabla _{T_p} Z_p \rangle \, \textrm{d}s \\&= -\left[ \eta \langle \nabla _{T_p} \varphi _p, Z_p \rangle \right] _0^L + \int _0^L \eta ' \langle \nabla _{T_p} \varphi _p, Z_p \rangle + \eta \langle \nabla _{T_p}^2 \varphi _p, Z_p \rangle \, \textrm{d}s \\&= \int _0^L \eta ' \langle \nabla _{T_p} \varphi _p, Z_p \rangle \, \textrm{d}s + \int _0^L \eta \langle \nabla _{T_p}^2 \varphi _p, Z_p \rangle \, \textrm{d}s \\&= \left[ \eta ' \langle \varphi _p, Z_p \rangle \right] _0^L - \int _0^L \eta '' \langle \varphi _p, Z_p \rangle \\&\qquad + \eta ' \langle \varphi _p, \nabla _{T_p} Z_p \rangle \, \textrm{d}s + \int _0^L \eta \langle \nabla _{T_p}^2 \varphi _p, Z_p \rangle \, \textrm{d}s \\&= - \int _0^L \eta '' \langle \varphi _p, Z_p \rangle \, \textrm{d}s + \int _0^L \eta \langle \nabla _{T_p}^2 \varphi _p, Z_p \rangle \, \textrm{d}s, \end{aligned}$$

having used integration by parts as well as the facts that $Z_p$ vanishes at the boundary and $\nabla _{T_p} Z_p = T_p$ is orthogonal to $\varphi _p$.

Substituting in for $\nabla _{T_p}^2 \varphi _p$ using the Euler-Lagrange equation (10) then integrating by parts turns this expression into

$$\begin{aligned} K_p^{1-p} \int _0^L \eta \kappa _p^p \, \textrm{d}s =&- \int _0^L \eta '' \langle \varphi _p, Z_p \rangle \, \textrm{d}s \\&- \int _0^L \eta \langle R(\varphi _p,T_p)T_p, Z_p \rangle \, \textrm{d}s - \sigma \int _0^L \eta d_p \langle \nu _p, Z_p \rangle \, \textrm{d}s \\&+ L \lambda _p \int _0^L \eta \langle \nabla _{T_p} T_p, Z_p \rangle \, \textrm{d}s + \frac{2p-1}{p-1} \int _0^L \eta \langle \varphi _p, \nabla _{T_p} T_p \rangle \, \textrm{d}s \\&+ \frac{2p-1}{p-1} \int _0^L \eta ' \langle \varphi _p, \nabla _{T_p} T_p \rangle \langle T_p, Z_p \rangle \, \textrm{d}s + \frac{\sigma }{2} \int _0^L \eta (d_p^2)' \langle T_p, Z_p \rangle \, \textrm{d}s \\&+ \frac{\sigma }{2}\int _0^L \eta d_p^2 \langle \nabla _{T_p} T_p, Z_p \rangle \, \textrm{d}s, \end{aligned}$$

where the fourth and fifth integrals come from

$$\begin{aligned}&- \frac{2p-1}{p} \int _0^L K_p^{1-p} \langle \nabla _{T_p} ( \kappa _p^p T_p ) , \eta Z_p \rangle \, \textrm{d}s\\&\quad = \frac{2p-1}{p} \int _0^L K_p^{1-p} \langle \kappa _p^p T_p, \eta T_p + \eta ' Z_p \rangle \, \textrm{d}s \\&\quad = \frac{2p-1}{p} K_p^{1-p} \left( \int _0^L \kappa _p^p \vert T_p \vert ^2 \eta + \kappa _p^p \eta ' \langle T_p, Z_p \rangle \, \textrm{d}s \right) \\&\quad = \frac{2p-1}{p} \int _0^L \eta \langle \varphi _p, \nabla _{T_p} T_p \rangle + \eta ' \langle \varphi _p, \nabla _{T_p} T_p \rangle \langle T_p, Z_p \rangle \, \textrm{d}s. \end{aligned}$$

After some manipulation, we obtain

$$\begin{aligned}&\frac{p}{p-1} K_p^{1-p} \int _0^L \eta \kappa _p^p \, \textrm{d}s\\&\quad = \, \int _0^L \eta '' \langle \varphi _p, Z_p \rangle \, \textrm{d}s + \int _0^L \eta \langle R(\varphi _p,T_p)T_p, Z_p \rangle \, \textrm{d}s + \sigma \int _0^L \eta d_p \langle \nu _p, Z_p \rangle \, \textrm{d}s \\&\qquad - L\lambda _p \int _0^L \eta \langle \nabla _{T_p} T_p, Z_p \rangle \, \textrm{d}s - \frac{2p-1}{p-1} \int _0^L \eta ' \langle \varphi _p, \nabla _{T_p} T_p \rangle \langle T_p, Z_p \rangle \, \textrm{d}s \\&\qquad - \frac{\sigma }{2}\int _0^L \eta (d_p^2)' \langle T_p, Z_p \rangle \, \textrm{d}s - \frac{\sigma }{2}\int _0^L \eta d_p^2 \langle \nabla _{T_p} T_p, Z_p \rangle \, \textrm{d}s \\&\quad = \, I_1 + I_2 + I_3 + I_4 + I_5 + I_6 +I_7. \end{aligned}$$

Now, suppose the limiting $\varphi$ is identically zero, so that $\varphi _p$ converges uniformly to 0 on compact subintervals of (0, L). Set $\eta$ so that the supports of $\eta '$ and of $\eta ''$ are contained in such a subinterval, denoted by [a, b]. Using the fact that $\vert Z_p \vert \, \le \delta$ on $\text {supp}\, \eta$ by construction ($Z_p$ vanishes at 0 and L and $\vert \nabla _{T_p} Z_p \vert \, = \vert T_p \vert \, = 1$, while $\text {supp}\, \eta \subseteq [0,\delta ] \cup [L-\delta , L]$ is contained in two intervals of width $\delta$), along with the same techniques we have used to bound integrals earlier in this paper, we obtain the inequalities

$$\begin{aligned} I_1, I_5&\lesssim C_\eta \delta \varepsilon _p , \\ I_2, I_3, I_4, I_6, I_7&\lesssim \delta , \end{aligned}$$

where $\varepsilon _p$ represents a term that decays to 0 as $p \rightarrow \infty$. The constant $C_\eta$ denotes the quantity $\max ( \left\Vert \eta '\right\Vert _{L^\infty }, \left\Vert \eta ''\right\Vert _{L^\infty })$.

The inequality

$$\begin{aligned} K_p^{1-p} \int _0^L \eta \kappa _p^p \, \textrm{d}s \le C \left( \delta + C_\eta \delta \varepsilon _p \right) \end{aligned}$$

(12)

follows, where C is some constant independent of both p and $\eta$.

On the other hand, note that from the definition of $\varphi _p$ it follows that

$$\begin{aligned} \left\Vert \varphi _p\right\Vert _{L^{p'}(0,L)} = L^{1/p'}, \end{aligned}$$

so the “$p'$ mass” of $\varphi _p$ is uniformly bounded from both below and above. The uniform convergence of $\varphi _p$ to 0 on [a, b] means that for large p, the inequality

$$\begin{aligned} \left\Vert \varphi _p\right\Vert _{L^{p'}([a,b])} \le \epsilon \le \frac{1}{2} \left\Vert \varphi _p\right\Vert _{L^{p'}(0,L)} \end{aligned}$$

holds. That is, at least half of the $p'$ mass of $\varphi _p$ is concentrated at the tails (0, a) and (b, L) of (0, L). It is then true that

$$\begin{aligned}{} & {} \left\Vert \eta ^{1/p'} \varphi _p\right\Vert _{L^{p'}(0,L)} \ge \left\Vert \eta ^{1/p'} \varphi _p\right\Vert _{L^{p'}((0,L) \backslash [a,b])}\\{} & {} \quad = \left\Vert \varphi _p\right\Vert _{L^{p'}((0,L) \backslash [a,b])} \ge \frac{1}{2} \left\Vert \varphi _p\right\Vert _{L^{p'}(0,L)} = \frac{L^{1/p'}}{2}, \end{aligned}$$

having used the definition of $\varphi _p$ to compute its $L^{p'}$ norm as well as the facts that $\eta (0) = \eta (L) = 1$ and that $\text {supp}\,\eta ' \subseteq [a,b]$ to infer that $\eta \equiv 1$ on $(0,L) \backslash [a,b]$. Therefore

$$\begin{aligned} K_p^{-p} \int _0^L \eta \kappa _p^p \, \textrm{d}s \ge \frac{L}{2^{p'}}, \end{aligned}$$

which means

$$\begin{aligned} K_p^{1-p} \int _0^L \eta \kappa _p^p \, \textrm{d}s \ge \frac{K_p L}{2^{p'}} \rightarrow \frac{KL}{2} > 0 \end{aligned}$$

as $p \rightarrow \infty$ by Lemma 6. Choosing $\delta$ sufficiently small and p large enough (to control the $C_\eta \delta \varepsilon _p$ term, since $C_\eta \delta$ may grow large when $\delta \rightarrow 0$) gives a contradiction to inequality (12).

The only step left is to prove statement 2) of Theorem 2.

Where $\varphi \ne 0$, it is immediate from equation (5) that $\vert \nabla _T T \vert = K$. It remains therefore to show that $\nabla _T T = 0$ almost everywhere on the set $\Sigma = \varphi ^{-1}(\{ 0 \} )$. Consider two different cases depending on the value of $\lambda$.

First, assume that $\lambda \ne 0$. By Rademacher’s theorem and the fact that $W^{1,\infty } = C^{0,1}$, we see that the second derivative $\nabla _T^2 \varphi$ exists in the classical sense almost everywhere in (0, L) and so it makes sense to interpret $\nabla _T^2 \varphi$ pointwise. Substituting (5) into (4), we see that at almost every given point $s_1 \in \Sigma$ the equation

$$\begin{aligned} \nabla _{T}^2 \varphi = L \lambda \nabla _{T} T - 2K \vert \varphi \vert ' T \end{aligned}$$

(13)

holds. If $\nabla _{T}^2 \varphi (s_1) = 0$, taking the inner product with $\nabla _T T$ in (13) shows that $\nabla _T T(s_1) = 0$. If instead $\nabla _{T}^2 \varphi (s_1) \ne 0$, it follows that $s_1$ is an isolated point of $\Sigma$ and can therefore be ignored for the purposes of this proof.

Now, assume instead that $\lambda = 0$. Substituting (5) into (4) as before gives us the equation

$$\begin{aligned} \nabla _{T}^2 \varphi = - R(\varphi , T)T - 2K \vert \varphi \vert ' T - 2K \vert \varphi \vert \nabla _T T, \end{aligned}$$

which implies the inequality

$$\begin{aligned} \vert \nabla _{T}^2 \varphi \vert \,\le C\left( \vert \varphi \vert + \vert \varphi \vert ' \right) \end{aligned}$$

after using the boundedness of K (by Lemma 6), $\nabla _T T$ (because $\gamma$ is an $\infty$-elastica and hence belongs to $W^{2,\infty }$), and the Riemann curvature tensor (because we are effectively working on a compact subset of the manifold M). Writing out the expression for $\nabla _T^2 \varphi$ in local co-ordinates

$$\begin{aligned} \nabla _{T}^2 \varphi = \varphi ^i{}'' \frac{\partial }{\partial x^{i}} + 2 \varphi ^i{}'T^j\Gamma _{ji}^k\frac{\partial }{\partial x^{k}} + \varphi ^i \left( T^j \Gamma _{ji}^k \right) '\frac{\partial }{\partial x^{k}} + \varphi ^i T^j \Gamma _{ji}^k T^m \Gamma _{mk}^s \frac{\partial }{\partial x^{s}}, \end{aligned}$$

rearranging to get an expression for $\varphi ''$ then using the triangle inequality and equivalence of norms implies the inequality

$$\begin{aligned} \vert \varphi '' \vert \, \le C\left( \vert \varphi \vert + \vert \varphi ' \vert \right) . \end{aligned}$$

where the norms here are taken to be the Euclidean norms of the vector $\varphi$ and its derivatives, viewed as vectors in ${\mathbb {R}}^n$ via local co-ordinates. From this, we may obtain for $\phi = (\varphi , \varphi ')$ the (Euclidean) inequality

$$\begin{aligned} \vert \phi \vert '&\le \vert \phi ' \vert \\&\le \vert \varphi ' \vert \,+\, \vert \varphi '' \vert \\&\le \vert \varphi ' \vert \,+\, C \left( \vert \varphi \vert \,+\, \vert \varphi ' \vert \right) \\&\le C \left( \vert \varphi \vert \,+\, \vert \varphi ' \vert \right) \\&\le C \vert ( \varphi , \varphi ' ) \vert \\&\le C \vert \phi \vert \end{aligned}$$

(note that here C denotes a constant whose value may change line-to-line). By Grönwall’s inequality, $\phi$ cannot have any zeroes, or else $\phi$ and hence $\varphi$ would be identically zero, yet we already know $\varphi$ does not vanish everywhere. Hence at any points where $\varphi = 0$ it must be that $\varphi ' \ne 0$, i.e. the zeroes of $\varphi$ must be isolated. $\square$

All the results we have established so far apply to $\infty$-elastica, and therefore to minimisers of ${\mathcal {K}}_\infty$. However, we have not yet shown that such minimisers actually exist; the following remark rectifies this and ensures that the analysis of this paper is meaningful.

Remark 8

Under the conditions of the problem described in Sect. 1, there exists at least one $\infty$-elastica. In particular, there exists a minimiser of ${\mathcal {K}}_\infty$.

Since the proof of this remark is so similar to the proofs already contained in this paper, we do not go into full detail here but rather give a sketch. Consider a sequence of curves $(\gamma _p)$ which are minimisers of the ${\mathcal {K}}_p$ functionals (that is, the ${\mathcal {J}}_p^\sigma$ functionals with $\sigma = 0$, i.e. no penalisation term). Using arguments similar to those contained in Sects. 2, 3, we obtain the (subsequential) convergence to a limiting curve $\gamma _\infty$ satisfying conditions (1–2) with ${\mathcal {K}}_\infty \left[ \gamma _\infty \right] = \lim _p {\mathcal {K}}_p \left[ \gamma _p \right]$. A straightforward manipulation of inequalities shows that there cannot exist any admissible curves such that the $L^\infty$ norm of their curvature is less than that of $\gamma _\infty$, or else there would exist p such that $\gamma _p$ does not minimise ${\mathcal {K}}_p$, a contradiction. Indeed, supposing there exists a curve $\gamma$ such that ${\mathcal {K}}_\infty [ \gamma ] = {\mathcal {K}}_\infty [ \gamma _\infty ] - \varepsilon$, we may take p sufficiently large so that

$$\begin{aligned} {\mathcal {K}}_p \left[ \gamma _p \right]> {\mathcal {K}}_\infty [ \gamma _\infty ] - \frac{\varepsilon }{2} > {\mathcal {K}}_\infty [ \gamma ] \ge {\mathcal {K}}_p [\gamma ], \end{aligned}$$

having used Hölder’s inequality in the final step. This contradicts the assumption that $\gamma _p$ is minimal for the functional ${\mathcal {K}}_p$, and so $\gamma _\infty$ must in fact be a minimiser of ${\mathcal {K}}_\infty$.