#  A note on the regularity and the existence of Riemannian splines

D. Corona, R. Giambò, P. Piccione

###### Abstract

In this paper, we present a comprehensive proof concerning the regularity of
critical points for the spline energy functional on Riemannian manifolds, even
for the general higher-order case. Although this result is widely acknowledged
in the literature, a detailed proof was previously absent. Our proof relies on
a generalization of the DuBois-Reymond Lemma. Furthermore, we establish the
existence of minimizers for the spline energy functional in cases where
multiple interpolation points are prescribed alongside just one velocity.

MSC(2020): 49J15.

Key words and phrases: Riemannian cubic; spline interpolation.

##  1 Introduction

This paper analyzes the regularity and existence of spline curves on a
Riemannian manifold $(M,g)$, defined variationally as the critical points of
the _spline energy functional_ $f\colon\Gamma\to M$ that reads as follows:

$f(\gamma)\coloneqq\frac{1}{2}\int_{0}^{1}g\big{(}\mathrm{D}_{t}\dot{\gamma}(t),\mathrm{D}_{t}\dot{\gamma}(t)\big{)}\mathrm{d}t,$
(1.1)

where $\Gamma$ is a suitable subspace of the Sobolev functions
$W^{2,2}([0,1],M)=H^{2}([0,1],M)$ and $\mathrm{D}_{t}$ denotes the usual
covariant derivative given by $g$ (more formally, by its Levi-Civita
connection). Usually, $\Gamma$ is the set of curves that pass through some
prescribed points of the manifolds at prescribed time instants (interpolation
conditions), with some fixed velocities on some of these points. When $M$ is
an euclidean space $\mathbb{R}^{n}$, the critical points of such a functional
are the cubic splines, namely piecewise third degree polynomials, from which
this functional takes its name.

This problem has been introduced in the Riemannian setting by L. Noakes, G.
Heinzinger and B. Paden in [13], where $\Gamma$ is the set of _smooth_ curves
with prescribed initial and final points and velocities. Under this regularity
assumption on the curves one considers, [13, Theorem 1] states that if
$\gamma$ is a critical point of $f$ then

$\mathrm{D}^{3}_{t}\dot{\gamma}(t)+R(\mathrm{D}\dot{\gamma}(t),\dot{\gamma}(t))\dot{\gamma}(t)=0,\quad\text{for
any }t\in[0,1],$ (1.2)

where $R$ denotes the Riemannian curvature tensor of the metric $g$ and
$\mathrm{D}^{3}_{t}$ denotes the third order covariant derivative. The fourth-
order differential equation (1.2) is derived from the critical point
condition. This is done by starting from the first variation formula of $f$,
performing explicit computations, using the integration by parts formula, and
then applying the fundamental lemma of the calculus of variations, which
allows to obtain a differential equation from an integral one.

However, if one assume a priori that the critical point $\gamma$ has only the
$H^{2}$–regularity, which is the “natural” regularity coming from the
definition of $f$, the integration by parts is not allowed, and the above
procedure prematurely stops. Indeed, by standard computations, the first
variation of $f$ at $\gamma$ in the direction $\xi$ is as follows:

$\mathrm{d}f(\gamma)[\xi]=\int_{0}^{1}\Big{(}g\big{(}\mathrm{D}_{t}\dot{\gamma},\mathrm{D}^{2}_{t}\,\xi\big{)}-g\big{(}R\big{(}\dot{\gamma},\mathrm{D}_{t}\dot{\gamma}\big{)}\dot{\gamma},\xi\big{)}\Big{)}\mathrm{d}t,$
(1.3)

where, at this point in the presentation, one can assume that
$\xi\colon[0,1]\to TM$ is a smooth vector field on $\gamma$ with compact
support in $(0,1)$. If $\gamma$ is only of class $H^{2}$, then
$\mathrm{D}_{t}\dot{\gamma}$ is a vector field on $\gamma$ with $L^{2}$
regularity. Since the minimal regularity assumption to perform the integration
by parts is the existence of a weak derivative, the following step,

$\int_{0}^{1}g\big{(}\mathrm{D}_{t}\dot{\gamma},\mathrm{D}^{2}_{t}\,\xi\big{)}\mathrm{d}t=\int_{0}^{1}g\big{(}\mathrm{D}_{t}^{3}\dot{\gamma},\xi\big{)}\mathrm{d}t,$

cannot be executed since $\mathrm{D}_{t}^{2}\dot{\gamma}$ does not exist for a
curve of class $H^{2}$.

While the literature on Riemannian splines is extensive due to their numerous
applications in control theory, robotics, computational graphics, geometric
mechanics, to name a few, it appears that the regularity problem is often
circumvented by imposing certain smoothness conditions on the curves under
consideration. For example, the pioneering work [13] considers only smooth
functions, while [4], where multiple interpolation points are considered,
assumes to work with $C^{1}$ piecewise smooth curves, as done also in [2].
Jumping to more recent works, [9] provides a regularity result for the
minimizers of a slightly different functional (the one that weights the spline
energy functional with the path energy functional). Starting from the
$H^{2}$–regularity, [9, Theorem 2.19] states that the minimizers are in
$C^{2,\frac{1}{2}}([\delta,1-\delta],M)$, for any $\delta\in(0,1/2)$. However,
this is still not sufficient to perform an integration by parts and,
consequently, to obtain (1.2). Indeed, if a minimizer $\gamma$ is of class
$C^{2,\frac{1}{2}}$, then its second order derivative is a vector field of
class $C^{0,\frac{1}{2}}$ which can not admit a weak derivative. In the one
dimensional Riemannian setting, a typical function with the above properties
is the well-known Weierstrass function, which is Hölder continuous but it is
not absolutely continuous. If one requires the $C^{0,\frac{1}{2}}$–regularity,
then the real parameters $a,b$ in the Weierstrass function

$W(t)=\sum_{k=0}^{\infty}a^{k}\cos(b^{k}\pi t),$

have to be chosen such that $b=1/a^{2}\in\mathbb{N}$, $a\in(0,1)$ and
$1/a>1+\frac{3}{2}\pi$. With these parameters, $W\in
C^{0,\frac{1}{2}}(\mathbb{R},\mathbb{R})$ but, as it doesn’t admit a
derivative at any point, it is not absolutely continuous.

The study of the splines on Riemannian manifolds has also application in the
collision avoidance of multiple agents, as investigated, for instance, in [1,
3, 8], where the spline energy functional is combined both with the path
energy one and with a distance function between multiple curves. Once again,
in [1, 8] the regularity issue is not directly tackled, since the functional
space is given by the $C^{1}$ piecewise smooth curves. In [3], the functional
space is given among the curves with $H^{2}$–regularity and it is stated that
the critical points are smooth; however, the proof is omitted, mentioning the
results in [1], where the piecewise smoothness is assumed.

In recent years, the spline problem has also garnered interest for the
challenging and intriguing study of its flow, known as _elastic flow_ , which
involves the evolution of curves following the negative gradient of the spline
energy functional, leading to the analysis of a fourth-order parabolic
quasilinear PDE. The study of this flow began in [14], but for a detailed
survey and references, we refer to [12] and the works cited therein. Even
within this framework, the admissible curves exhibit more than
$H^{2}$-regularity. For instance, both in [12] and in [15], they are of class
$H^{4}$. A similar analysis was recently carried out in [10], where the
elastic flow is studied in a broader context. Even in this scenario, when
constrained to the original spline energy functional, the considered curves
are of class $C^{2}$ and piecewise $C^{4,\alpha}$, with $\alpha\in(0,1)$.

Finally, let us observe that in [6] and [7], the regularity of the critical
curves is not assumed a priori; however, its proof is given, without further
details, invoking a “standard” bootstrap method. The bootstrap method, usually
used to prove the regularity of critical points of a Lagrangian action
functional, requires integration by parts in the “wrong” direction, namely the
only one which is possible under the regularity assumed at the beginning. For
example, starting from (1.3), one should obtain the following:

$\mathrm{d}f(\gamma)([\xi])=\int_{0}^{1}\Big{(}g\big{(}\mathrm{D}_{t}\dot{\gamma},\mathrm{D}^{2}_{t}\,\xi\big{)}-g\big{(}\mu(t),\mathrm{D}^{2}_{t}\xi\big{)}\Big{)}\mathrm{d}t,$
(1.4)

where $\mu(t)$ is a vector field on $\gamma$ whose second order covariant
derivative is $R(\dot{\gamma},\mathrm{D}_{t}\dot{\gamma})\dot{\gamma}$, hence
it is of class $H^{2}$, and $\mu(0)=0$, $\mathrm{D}_{t}\mu(0)=0$. It then
applies the DuBois-Reymond Lemma, which, for the reader’s convenience, is
reported here in the simpler setting of real functions (see, e.g., [5,
Corollary 1.25]).

######  DuBois-Reymond Lemma.

Let $u\in L^{1}_{loc}((a,b),\mathbb{R})$ such that

$\int_{a}^{b}u(t)\eta(t)\mathrm{d}t=0,\qquad\forall\eta\in
C^{\infty}_{c}([a,b],\mathbb{R})\quad\text{s.t.
}\int_{a}^{b}\eta(t)\mathrm{d}t=0.$

Then, there exists $c\in\mathbb{R}$ such that $u=c$ a.e. in $(a,b)$.

The main problem to apply the “standard” bootstrap method is as follows. From
the critical point condition $\mathrm{d}f(\gamma)=0$, hence from the null
condition on (1.4), one can’t infer that
$\mathrm{D}_{t}\dot{\gamma}(t)=v+\mu(t)$ a.e. on $[0,1]$ for some _parallel_
vector $v$ field on $\gamma$ (i.e., $\mathrm{D}_{t}v\equiv 0$), obtaining that
$\mathrm{D}_{t}\dot{\gamma}$ is of class $H^{2}$ (so that one can infer the
spline equation (1.2) as previously described). This is not possible because
the set of possible variations one can apply to $\gamma$ it is strictly
contained within the set of the variations required to apply the DuBois-
Reymond Lemma. Indeed, if $\xi$ has compact support in $(0,1)$, not only
$\int_{0}^{1}\mathrm{D}^{2}_{t}\xi(t)\mathrm{d}t=0$, which is obtained from
the condition that the covariant derivative of $\xi$ is $0$ at the extreme
points, but we have the additional constraints that $\xi$ has to be null at
the extremes. As a consequence, the “standard” bootstrap method mentioned in
[6, 7] does not work, or rather, it requires some further clarification.

In this paper, we demonstrate that a bootstrap method can be indeed
successfully applied by utilizing a generalization of the DuBois-Reymond
Lemma, namely Lemma 2.3, thereby proving the regularity of critical points.
More formally, we have the following result.

######  Theorem 1.1.

Let $(M^{n},g)$ be an $n$–dimensional smooth and complete Riemannian manifold.
Let $\\{p_{0},\ldots,p_{N}\\}\subset M$ be a set of $N+1$ fixed points,
$0=t_{0}<t_{1}<\dots<t_{N}=1$ a partition of the unit interval, and let us fix
an index $j\in\\{0,\ldots,N\\}$, together with a vector $v\in T_{p_{j}}M$. Let

$\Gamma=\left\\{\gamma\in H^{2}([0,1],M):\gamma(t_{i})=p_{i},\,\forall
i=0,\ldots,N,\,\dot{\gamma}(t_{j})=v\right\\},$

and $f\colon\Gamma\to\mathbb{R}$ be defined by (1.1). Then, if
$\gamma\in\Gamma$ is a critical point of $f$, it is smooth on any interval
$[t_{i-1},t_{i}]$. As a consequence, for any $i=1,\ldots,N$ we have

$\mathrm{D}^{3}_{t}\dot{\gamma}(t)+R\big{(}\mathrm{D}_{t}\dot{\gamma}(t),\dot{\gamma}(t)\big{)}\dot{\gamma}(t)=0,\quad\text{for
any }t\in[t_{i-1},t_{i}].$ (1.5)

Moreover, both $\gamma|_{[0,t_{j}]}$ and $\gamma|_{[t_{j},1]}$ are of class
$C^{2}$; if $t_{j}\neq 0$, then $\mathrm{D}_{t}\gamma(0)=0$, and if $t_{j}\neq
1$, then $\mathrm{D}_{t}\gamma(1)=0$.

######  Remark 1.2.

The regularity results remain analogous even when velocities are prescribed at
multiple points. In such cases, a critical point $\gamma$ will be of class
$C^{2}$ within any interval delimited by two instants where velocity is
prescribed. The proof for this follows the same argument as that used in
Theorem 1.1, with the crucial observation that each point where velocity is
prescribed must be treated similarly to the only one in Theorem 1.1. To
maintain clarity and avoid heavy notation that might obscure the main proof
steps, we focus on using only one point with prescribed velocity. If no
velocity is prescribed, and a critical point exists, then it is of class
$C^{2}$ on the whole unit interval, and the covariant derivative of its
velocity vanishes at the extrema.

######  Remark 1.3.

The above regularity theorem holds even if $M$ is a differentiable manifold of
class $C^{4}$ and the metric tensor $g$ is of class $C^{3}$, as these are the
minimal (reasonable) assumptions so that (1.5) can be explicitly written in
local charts.

The regularity ensured by Theorem 1.1 can be generalized for higher-order
Riemannian splines, the so-called _$k$ -splines_, which are, for any integer
$k\geq 2$, the critical points of the energy functional
$f_{k}\colon\Gamma\subset H^{k}([0,1],M)\to\mathbb{R}$ given by

$f_{k}(\gamma)\coloneqq\int_{0}^{1}g\big{(}\mathrm{D}_{t}^{k-1}\dot{\gamma},\mathrm{D}_{t}^{k-1}\dot{\gamma}\big{)}\mathrm{d}t.$
(1.6)

Once again, if the regularity result is ensured, then applying the integration
by parts formula the necessary number of times and the fundamental lemma of
the calculus of variations leads to an ODE of order $2k$ on any interval
$I\subset[0,1]$ delimited by some interpolation conditions. However, even in
this case, we couldn’t find satisfactory regularity results for the critical
points of the functional (1.6), as usually one works in the set of piecewise
smooth curves or, at least, assuming the $C^{2k-2}$ regularity on each
interval of the admissible curves (see, e.g., [11, 10]). Therefore, we provide
the following result, which is based on the further generalization of the
DuBois-Reymond Lemma presented in Lemma 4.3.

######  Theorem 1.4.

Let $(M^{n},g)$ be an $n$–dimensional smooth and complete Riemannian manifold.
Let $\\{p_{0},\ldots,p_{N}\\}\subset M$ be a set of $N+1$ fixed points,
$0=t_{0}<t_{1}<\dots<t_{N}=1$ a partition of the unit interval, and let us fix
an index $j\in\\{0,\ldots,N\\}$, together with $k-1$ vectors $v_{\ell}\in
T_{p_{j}}M$, for $\ell=1,\dots,k-1$. Let

$\Gamma_{k}=\Big{\\{}\gamma\in H^{k}([0,1],M):\gamma(t_{i})=p_{i},\,\forall
i=0,\ldots,N,\,\mathrm{D}_{t}^{\ell-1}\dot{\gamma}(t_{j})=v_{\ell},\,\forall\ell=1,\dots,k-1\Big{\\}},$

and $f_{k}\colon\Gamma_{k}\to\mathbb{R}$ be defined by (1.6). Then, if
$\gamma\in\Gamma$ is a critical point of $f_{k}$, it is smooth on any interval
$[t_{i-1},t_{i}]$. As a consequence, for any $i=1,\ldots,N$ we have

$\mathrm{D}_{t}^{2k-1}\dot{\gamma}(t)+\sum_{\ell=0}^{k-2}(-1)^{\ell}R\big{(}\mathrm{D}_{t}^{2k-\ell-3}\dot{\gamma}(t),\mathrm{D}_{t}^{\ell}\dot{\gamma}(t)\big{)}\dot{\gamma}(t)=0,\quad\text{for
any }t\in(t_{i-1},t_{i}).$ (1.7)

Moreover, both $\gamma|_{[0,t_{j}]}$ and $\gamma|_{[t_{j},1]}$ are of class
$C^{2k-2}$; if $t_{j}\neq 0$, then
$\mathrm{D}_{t}^{k+\ell-2}\dot{\gamma}(0)=0$, for $\ell=1,\dots,k-1$, and
similarly for $t_{j}\neq 1$.

######  Remark 1.5.

Theorem 1.4 can also be stated and proved in a slightly more general setting,
by fixing the covariant derivatives of the curves at different points and by
using different orders, where the latter are obviously always between $1$ and
$k-1$. More formally, for each $\ell=1,\dots,k-1$, one can choose an index
$j_{\ell}\in\\{0,\dots,N\\}$ and an order of derivative
$w_{\ell}\in\\{0,\dots,k-2\\}$, and define

$\Gamma_{k}=\Big{\\{}\gamma\in H^{k}([0,1],M):\gamma(t_{i})=p_{i},\,\forall
i=0,\ldots,N,\mathrm{D}_{t}^{w_{\ell}}\dot{\gamma}(t_{j_{\ell}})=v_{\ell},\,\forall\ell=1,\dots,k-1\Big{\\}},$

where each $v_{\ell}$ belongs to $T_{p_{j_{\ell}}}M$. In this setting, (1.7)
still holds, but the last regularity results stated in the theorem should be
modified accordingly. However, for the sake of clarity and simplicity, we
confine ourselves to the setting given in the theorem.

Beside a comprehensive proof of the regularity of critical points for the
spline energy functional, this paper contains an existence result for the
minimizers of the same functional. The existence result we give stems from the
analysis of [9, Lemma 2.15], where it is proven that, in general, a minimizer
of the spline energy functional can not exist if only the interpolation points
are prescribed. For the reader’s convenience, the counterexample provided as a
proof of [9, Lemma 2.15] is reported and discussed in details in Example 5.1.
Based on that remarkable example, [9] justifies the introduction of additional
constraints to obtain the existence of minimizers, which are as follows:

  * •

_natural_ boundary conditions, i.e.,
$\mathrm{D}_{t}\dot{\gamma}(0)=\mathrm{D}_{t}\dot{\gamma}(1)=0$;

  * •

_periodic_ boundary conditions, i.e., $p_{0}=p_{N}$ and
$\dot{\gamma}(0)=\dot{\gamma}(1)$;

  * •

_Hermite_ boundary conditions, i.e., the prescription of _the initial and
final velocities_ of the curves.

Additionally, [9] weights the spline energy functional with the path energy,
namely looking for the minimizers of

$F^{\sigma}(\gamma)\coloneqq
f(\gamma)+\sigma\int_{0}^{1}g\left(\dot{\gamma},\dot{\gamma}\right)\mathrm{d}t,$

for some $\sigma>0$. Indeed, in [9, Theorem 2.19], the existence of a
minimizer for $F^{\sigma}$ is guaranteed if $\sigma>0$ and if one of the above
constraints is assumed.

A direct inspection of the construction provided to show the non-existence of
a minimizer in [9, Lemma 2.15] (see also Example 5.1) suggests that, if _just
one_ velocity is prescribed, then a minimizer should exists, even when
$\sigma=0$. More formally, we have the following theorem, whose setting is
depicted in Figure 1.

######  Theorem 1.6.

Let $(M^{n},g)$ be an $n$–dimensional complete Riemannian manifold. Let
$\\{p_{0},\ldots,p_{N}\\}\subset M$ be a set of $N+1$ fixed points and
$0=t_{0}<t_{1}<\dots<t_{N}=1$ a partition of the unit interval. Let us fix a
vector $v\in T_{p_{0}}M$ and let

$\Gamma=\Big{\\{}\gamma\in H^{2}([0,1],M):\gamma(t_{i})=p_{i},\,\forall
i=0,\dots,N,\,\dot{\gamma}(0)=v\Big{\\}}.$

Then, the functional $f\colon\Gamma\to\mathbb{R}$ defined by (1.1) admits a
minimizer.

$M$$\gamma(0)=p_{0}$$\gamma(t_{1})=p_{1}$$\gamma(t_{2})=p_{2}$$\gamma(t_{N-1})=p_{N-1}$$\gamma_{N}(1)=p_{N}$$\dot{\gamma}(0)=v\in
T_{p_{0}}M$$\gamma$ Figure 1: The setting of Theorem 1.6: on a Riemannian
manifold $M$, we prescribe $N+1$ points, denoted by $p_{0},\dots,p_{N}$,
together with the initial velocity $v\in T_{p_{0}}M$.

######  Remark 1.7.

Prescribing the velocity at the initial time doesn’t affect the generality of
the theorem. Indeed, if one prescribes the velocity at an internal time
instant $t_{j}\neq\\{0,1\\}$, then Theorem 1.6 ensures the existence of two
minimizers of $f$ in the two time intervals $[0,t_{j}]$ and $[t_{j},1]$, say
$\gamma_{1}\colon[0,t_{j}]\to M$ and $\gamma_{2}\colon[t_{j},1]\to M$. For the
sake of precision, this follows also from the invariance of the spline energy
functional with respect to backward reparametrization of the curves. By
Theorem 1.1, the curve $\gamma\colon[0,1]\to M$ obtained by gluing together
$\gamma_{1}$ and $\gamma_{2}$ is of class $H^{2}$, as its second order
derivative could have just one step discontinuity at time $t_{j}$. Therefore,
$\gamma$ is a minimizer of $f$.

The rest of the paper is organized as follows. Section 2 presents a proof of
Theorem 1.1 in the one-dimensional case, allowing the key steps of the proof
to be appreciated without excessive notation. Section 3 then provides a
detailed proof of Theorem 1.1, encompassing all necessary details for the
Riemannian setting. Section 4 is dedicated to the proof of Theorem 1.4,
beginning again with the analogous result in the one dimensional setting.
Finally, Section 5 focuses on the proof of Theorem 1.6.

##  2 One dimensional setting

In this section, we provide the regularity result for the critical points of
the spline energy functional for one dimensional curves, namely maps defined
on $[0,1]$ with values in $\mathbb{R}$. Let $N+1\geq 2$ be the number of fixed
points, which we denote by $p_{0},p_{1},\ldots,p_{N}\in\mathbb{R}$, and let us
choose a partition of the unit interval $0=t_{0}<t_{1}<\ldots<t_{N}=1$. We
impose that every curve we consider should pass through the point $p_{i}$ at
the time $t_{i}$, for every $i=0,\ldots,N$, and we prescribe the velocity of
the curves at just _one_ point, i.e., there exists $j\in\\{0,\ldots,N\\}$ and
$v\in\mathbb{R}$ such that every curve we consider has a derivative equal to
$v$ at the time $t_{j}$. More formally, our functional space is defined as
follows:

$\Gamma\coloneqq\Big{\\{}\gamma\in
H^{2}([0,1],\mathbb{R}):\gamma(t_{i})=p_{i},\,\forall
i=0,\ldots,N,\,\dot{\gamma}(t_{j})=v\Big{\\}},$ (2.1)

where $H^{2}([0,1],\mathbb{R})$ denotes the space of Sobolev real functions
defined on $[0,1]$ that admits a second order weak derivative which belongs to
the Lebesgue space $L^{2}([0,1],\mathbb{R})$. The main objective of this
section is studying the regularity of the critical points of the spline energy
functional $f\colon\Gamma\to\mathbb{R}$, which in this case reads simply as
follows:

$f(\gamma)=\frac{1}{2}\int_{0}^{1}\ddot{\gamma}^{2}(t)\,\mathrm{d}t.$ (2.2)

By some standard computations, the set of admissible variations is given by

$V\coloneqq\left\\{\xi\in H^{2}([0,1],\mathbb{R}):\xi(t_{i})=0,\,\forall
i=0,\ldots,N,\,\dot{\xi}(t_{j})=0\right\\},$ (2.3)

so that, for every $\gamma\in\Gamma$, the differential of $f$ at $\gamma$,
that we denote by $\mathrm{d}f(\gamma)\colon V\to\mathbb{R}$, is

$\mathrm{d}f(\gamma)[\xi]=\int_{0}^{1}\ddot{\gamma}(t)\,\ddot{\xi}(t)\,\mathrm{d}t.$
(2.4)

######  Remark 2.1.

It is important to notice that if $\gamma$ is a critical point of $f$, we
cannot apply the DuBois-Raymond lemma to deduce its regularity. Let us proceed
with the computation to better highlight this statement, which is a key
observation for all the subsequent discussion about the regularity of the
critical points. Assume that $\gamma$ is a critical point, so that
$\mathrm{d}f(\gamma)[\xi]=0$ for every $\xi\in V$. Let us fix $i=1,\dots,N$
and restrict our analysis on the variations with compact support in
$(t_{i-1},t_{i})$, so we have

$\mathrm{d}f(\gamma)[\xi]=\int_{t_{i-1}}^{t_{i}}\ddot{\gamma}(t)\,\ddot{\xi}(t)\,\mathrm{d}t=0,\quad\forall\xi\in
V,\,\operatorname{supp}\xi\subset(t_{i-1},t_{i}).$

Notice that, if $\operatorname{supp}\xi\subset(t_{i-1},t_{i})$, then
$\dot{\xi}(t_{i-1})=\dot{\xi}\,(t_{i})=0$, so that
$\int_{0}^{1}\ddot{\xi}\,\mathrm{d}t=0$. By the DuBois-Raymond lemma, if we
can consider _all_ the variations $\xi$ such that
$\int_{0}^{1}\ddot{\xi}\,\mathrm{d}t=0$, then we can conclude that
$\ddot{\gamma}=c$ a.e. on $(t_{i-1},t_{i})$, hence we obtain a regularity
result. However, we have to take into account also the condition
$\xi(t_{i-1})=\xi(t_{i})=0$, so _not all_ the variations required for the
application of the DuBois-Raymond lemma are available.

Building on the previous remark, we need a characterization result about the
admissible variations.

######  Lemma 2.2.

For every $\xi\in V$ and $i=1,\dots,N$, we have

$\int_{t_{i-1}}^{t_{i}}\dot{\xi}\,\mathrm{d}t=0,$ (2.5)

and

$t_{i}\dot{\xi}(t_{i})-t_{i-1}\dot{\xi}(t_{i-1})=\int_{t_{i-1}}^{t_{i}}t\,\ddot{\xi}\mathrm{d}t.$
(2.6)

###### Proof.

The condition (2.5) naturally arises from the hypotheses that $\xi(t_{i})=0$
for each $i=0,\dots,N$. Indeed, since $\xi(t_{0})=0$, we have

$\xi(t_{1})=\xi(t_{0})+\int_{t_{0}}^{t_{1}}\dot{\xi}\,\mathrm{d}t=\int_{t_{0}}^{t_{1}}\dot{\xi}\,\mathrm{d}t=0,$

and similarly for all subsequent indices. Then, (2.6) follows directly from
(2.5) and the integration by parts formula as follows:

$\int_{t_{i-1}}^{t_{i}}t\,\ddot{\xi}(t)\mathrm{d}t=t_{i}\dot{\xi}(t_{i})-t_{i-1}\dot{\xi}(t_{i-1})-\int_{t_{i-1}}^{t_{i}}\dot{\xi}\,\mathrm{d}t=t_{i}\dot{\xi}(t_{i})-t_{i-1}\dot{\xi}(t_{i-1}).$

∎

To state the next result, we need some further notation. Let us define

$V_{0}\coloneqq\left\\{\xi\in V:\dot{\xi}(t_{i})=0,\,\forall
i=0,\dots,N\right\\}\subset V.$

By Lemma 2.2, if $\xi\in V_{0}$ then for every $i=0,\dots,N-1$ we have

$\int_{t_{i-1}}^{t_{i}}\ddot{\xi}(t)\,\mathrm{d}t=0\quad\text{and}\quad\int_{t_{i-1}}^{t_{i}}t\,\ddot{\xi}(t)\,\mathrm{d}t=0,$

and viceversa, since $\dot{\xi}(t_{j})=0$. In other words, setting

$W_{0}\big{(}[a,b],\mathbb{R}\big{)}\coloneqq\left\\{\eta\in
L^{2}([a,b],\mathbb{R}):\int_{a}^{b}\eta(t)\,\mathrm{d}t=0\text{ and
}\int_{a}^{b}t\,\eta(t)\,\mathrm{d}t=0\right\\},$

for every interval $[a,b]\subset\mathbb{R}$, we have

$V_{0}=\left\\{\xi\in V:\ddot{\xi}\big{|}_{[t_{i-1},t_{i}]}\in
W_{0}([t_{i-1},t_{i}],\mathbb{R})\text{ for every }i=1,\dots,N\right\\}.$
(2.7)

Now, we are ready to state the following result, which generalizes the DuBois-
Reymond lemma.

######  Lemma 2.3.

If $u\in L^{2}([a,b],\mathbb{R})$ is such that

$\int_{a}^{b}u(t)\eta(t)\mathrm{d}t=0,\qquad\forall\eta\in
W_{0}([a,b],\mathbb{R}),$

then there exist two constants $c_{0},c_{1}\in\mathbb{R}$ such that
$u(t)=c_{1}t+c_{0}$ a.e. in $[a,b]$.

###### Proof.

Setting

$A([a,b],\mathbb{R})=\big{\\{}u\in
L^{2}([a,b],\mathbb{R}):u(t)=c_{1}s+c_{0},\,\text{for some
}c_{0},c_{1}\in\mathbb{R}\big{\\}},$

by definition of $W_{0}([a,b],\mathbb{R})$ we have that $A([a,b],\mathbb{R})$
and $W_{0}([a,b],\mathbb{R})$ are orthogonal in $L^{2}([a,b],\mathbb{R})$. To
conclude the proof, let us show that
$L^{2}([a,b],\mathbb{R})=A([a,b],\mathbb{R})\oplus W_{0}([a,b],\mathbb{R})$.
This can be achieved by proving that for every $u\in L^{2}([a,b],\mathbb{R})$,
there exist exactly two real numbers $c_{0},c_{1}$ such that

$u-c_{1}t-c_{0}\in W_{0}([a,b],\mathbb{R}).$

A straightforward computation shows that $c_{0},c_{1}$ must solve the
following system:

$\begin{dcases}c_{1}\frac{(b-a)^{2}}{2}+c_{0}(b-a)=\int_{a}^{b}u(t)\,\mathrm{d}t,\\\
c_{1}\frac{(b-a)^{3}}{3}+c_{0}\frac{(b-a)^{2}}{2}=\int_{a}^{b}t\,u(t)\,\mathrm{d}t,\end{dcases}$

whose determinant is different from $0$ if $a\neq b$, and so $c_{0}$ and
$c_{1}$ are unique. ∎

By directly applying Lemma 2.3, we have the following regularity result in the
one dimensional setting.

######  Proposition 2.4.

Let $\gamma\in\Gamma$ be a critical point for $f$, where $\Gamma$ is defined
by (2.1). Then, for every $i=1,\dots,N$ the restriction
$\gamma\big{|}_{[t_{i-1},t_{i}]}$ is smooth and

$\frac{\mathrm{d}^{4}}{\mathrm{d}t^{4}}\gamma(t)=0,\qquad\forall
t\in(t_{i-1},t_{i}).$ (2.8)

Moreover, both $\gamma\big{|}_{[0,t_{j}]}$ and $\gamma\big{|}_{[t_{j},1]}$ are
of class $C^{2}$; if $t_{j}\neq 0$ then $\ddot{\gamma}(0)=0$ and if $t_{j}\neq
1$ then $\ddot{\gamma}(1)=0$.

###### Proof.

As an initial step, let us select an index $i\in\\{1,\dots,N\\}$ and
demonstrate that the restriction of $\gamma$ to $[t_{i-1},t_{i}]$ is smooth.
We consider the subset of $V_{0}$ consisting of variations $\xi$ with compact
support within the interval $[t_{i-1},t_{i}]$. Given that $\gamma$ is a
critical point for the functional $f$, it follows that

$\mathrm{d}f(\gamma)[\xi]=\int_{t_{i-1}}^{t_{i}}\ddot{\gamma}(t)\,\ddot{\xi}(t)\,\mathrm{d}t=0,$

for every $\xi$ within this particular subset of $V_{0}$. As $\xi$ is an
element of $V_{0}$, according to (2.7) $\ddot{\xi}$ belongs to
$W_{0}([t_{i-1},t_{i}],\mathbb{R})$. Conversely, for any $\eta\in
W_{0}([t_{i-1},t_{i}],\mathbb{R})$, there exists a $\xi$ in this subset of
$V_{0}$ such that $\ddot{\xi}(t)=\eta(t)$ for all $t\in[t_{i-1},t_{i}]$. This
equivalence is achieved by integrating $\eta$ twice and adopting zero as the
integration constants. By Lemma 2.3 we obtain two constants
$c_{0}^{i},c_{1}^{i}\in\mathbb{R}$ such that

$\ddot{\gamma}(t)=c_{1}^{i}t+c_{0}^{i},\qquad\text{a.e. in }(t_{i-1},t_{i}).$

Therefore, $\gamma\big{|}_{[t_{i-1},t_{i}]}$ is smooth and (2.8) holds for
every $i=0,\dots,N-1$.

To demonstrate the final part of the proposition, consider any $\xi\in V$.
Given that $\gamma\in\Gamma$ is a critical point and exhibits smoothness over
each interval $[t_{i-1},t_{i}]$, integrating by parts within each interval
yields to the following equation:

$\mathrm{d}f(\gamma)[\xi]=\sum_{i=1}^{N}\int_{t_{i-1}}^{t_{i}}\ddot{\gamma}(t)\ddot{\xi}(t)\,\mathrm{d}t=\sum_{i=1}^{N}\left(\ddot{\gamma}(t_{i}^{-})\dot{\xi}(t_{i})-\ddot{\gamma}(t_{i-1}^{+})\dot{\xi}(t_{i-1})\right)=0,\qquad\forall\xi\in
V,$

where $\ddot{\gamma}(t^{-})$ and $\ddot{\gamma}(t^{+})$ denote the second-
order left and right derivatives of $\gamma$ at time $t$, respectively. The
integral terms vanish because $\mathrm{d}^{3}\gamma/\mathrm{d}t^{3}$ remains
constant and $\dot{\xi}$ has a zero mean value over each interval.

Given that for any $i\neq j$, there exists $\xi\in V$ such that
$\dot{\xi}(t_{i})\neq 0$ and $\dot{\xi}(t_{k})=0$ for all $k\neq i$, it
follows that

$\ddot{\gamma}(t_{i}^{-})=\ddot{\gamma}(t_{i}^{+}),\quad\text{if }i\neq j,$

which indicates that both sections of $\gamma$, specifically
$\gamma\big{|}_{[0,t_{j}]}$ and $\gamma\big{|}_{[t_{j},1]}$, are of class
$C^{2}$. When $t_{j}$ does not coincide with $0$ and $1$, we deduce that
$\ddot{\gamma}(0)=0$ and $\ddot{\gamma}(1)=0$, respectively. ∎

##  3 The regularity result in the Riemannian setting

In this section, we prove Theorem 1.1, thus generalizing the results achieved
by Proposition 2.4 in the one-dimensional setting to the more general context
of Riemannian manifolds.

From now on, let $(M^{n},g)$ is a smooth $n$–dimensional complete Riemannian
manifold, endowed with a smooth metric tensor $g$. For any absolute continuous
curve $\gamma\colon[a,b]\to M$ and any vector field $\eta$ along $\gamma$ of
class $L^{1}$, we denote by

$\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}\eta(t)\mathrm{d}t$

the _covariant integral of $\eta$ along $\gamma$_, that is the unique vector
field $\mu$ along $\gamma$ such that $\mu(a)=0$ and $D_{t}\mu(t)=\eta(t)$ for
every $t\in[a,b]$. Hence, for any vector field $\xi$ of class $H^{1}$ along
$\gamma$, we have

$\xi(t)=\xi_{t}(a)+\int\\!\\!\\!\\!\\!\\!\int_{a}^{t}\mathrm{D}_{t}\xi(\tau)\mathrm{d}\tau,$
(3.1)

where we denote by $\xi_{t}(a)\in T_{\gamma(t)}M$ the parallel transport of
the vector $\xi(a)\in T_{\gamma(a)}M$ along the curve $\gamma$ at time
$t\in[a,b]$.

Let $\Gamma\subset H^{2}([0,1],M)$ and $f\colon\Gamma\to\mathbb{R}$ be defined
as in Theorem 1.1. Due to the constraints of the curves we are considering,
the set of admissible variations depends on each $\gamma\in\Gamma$ and is the
following:

$\displaystyle V_{\gamma}\coloneqq\Big{\\{}\xi\in H^{2}([0,1],TM):$
$\displaystyle\xi(t)\in T_{\gamma(t)}M,\,\forall t\in[0,1],$ (3.2)
$\displaystyle\xi(t_{i})=0,\,\forall i=0,\dots,N,\,$
$\displaystyle\mathrm{D}_{t}\xi(t_{j})=0\Big{\\}}.$

In other words, $V_{\gamma}$ is the set of vector fields along $\gamma$ of
Sobolev class $H^{2}$ vanishing at each time $t_{i}$, for $i=0,\dots,N$, and
with first covariant derivative equal to $0$ at time $t_{j}$.

We are concerning about the regularity and existence of the critical points of
$f\colon\Gamma\to\mathbb{R}$, hence of those curves such that

$\mathrm{d}f(\gamma)[\xi]=0,\qquad\forall\xi\in V_{\gamma}.$

By standard arguments (see, e.g., [6]), it can be proved that the differential
$\mathrm{d}f(\gamma)\colon V_{\gamma}\to\mathbb{R}$ reads as follows:

$\mathrm{d}f(\gamma)[\xi]=\int_{0}^{1}\Big{(}g\big{(}\mathrm{D}_{t}\dot{\gamma},\mathrm{D}^{2}_{t}\,\xi\big{)}-g\big{(}R\big{(}\dot{\gamma},\mathrm{D}_{t}\dot{\gamma}\big{)}\dot{\gamma},\xi\big{)}\Big{)}\mathrm{d}t,$
(3.3)

where $R\colon TM\times TM\times TM\to TM$ denotes the Riemannian curvature
tensor of the manifold. Even in this case, due to the constraints on the
admissible variations, we can’t employ the DuBois Raymond’s lemma to prove the
regularity of the critical points. Similarly to the one dimensional case, we
proceed as follows. For every $\gamma\in\Gamma$, let us consider

$V_{\gamma,0}=\left\\{\xi\in V_{\gamma}:\mathrm{D}_{t}\,\xi(t_{i})=0,\,\forall
i=0,\dots,N\right\\},$

and let us define the following vector field along $\gamma$,

$\eta_{\gamma}(t)\coloneqq\int\\!\\!\\!\\!\\!\\!\int_{0}^{t}\left(\int\\!\\!\\!\\!\\!\\!\int_{0}^{\tau}R\big{(}\dot{\gamma},\mathrm{D}_{s}\dot{\gamma}\big{)}\dot{\gamma}\,\mathrm{d}s\right)\mathrm{d}\tau,$
(3.4)

so that $\eta_{\gamma}(t)\in T_{\gamma(t)}M$ for every $t\in[0,1]$ and the
second order covariant derivative of $\eta_{\gamma}$ is
$R\big{(}\dot{\gamma},\mathrm{D}_{t}\dot{\gamma}\big{)}\dot{\gamma}$. With
this notation, by integrating by parts (in the admissible direction) we obtain

$\mathrm{d}f(\gamma)[\xi]=\int_{0}^{1}g\left(\mathrm{D}_{t}\dot{\gamma}-\eta_{\gamma}(t),\mathrm{D}^{2}_{t}\,\xi\right)\mathrm{d}t,\qquad\forall\xi\in
V_{\gamma,0}.$ (3.5)

We are going to prove that, if $\mathrm{d}f(\gamma)[\xi]=0$ for every $\xi\in
V_{\gamma,0}\subset V_{\gamma}$, in particular if $\gamma$ is a critical
point, then $\gamma$ is smooth on every time interval $[t_{i},t_{i+1}]$. As a
first step, we give the following characterization result.

######  Lemma 3.1.

Let $\gamma\in H^{2}([a,b],M)$ and let $\xi\in
V_{\gamma,0}([a,b],\mathbb{R})$. Then, for every $i=1,\dots,N$ we have

$\int\\!\\!\\!\\!\\!\\!\int_{t_{i-1}}^{t_{i}}\mathrm{D}^{2}_{t}\xi\,\mathrm{d}t=0,$
(3.6)

and

$\int\\!\\!\\!\\!\\!\\!\int_{t_{i-1}}^{t_{i}}t\,\mathrm{D}^{2}_{t}\xi\,\mathrm{d}t=0.$
(3.7)

###### Proof.

Since $\mathrm{D}_{t}\xi(t_{i})=0$ for any $i=0,\dots,N$, (3.6) directly
follows from (3.1). Now, let us fix $i=1,\dots,N$. Since
$\xi(t_{i-1})=\xi(t_{i})=0$, and $\mathrm{D}_{t}\xi(t_{i-1})=0$, we have

$0=\int\\!\\!\\!\\!\\!\\!\int_{t_{i-1}}^{t_{i}}\mathrm{D}_{t}\xi(t)\mathrm{d}t=\int\\!\\!\\!\\!\\!\\!\int_{t_{i-1}}^{t_{i}}\left(\int\\!\\!\\!\\!\\!\\!\int_{t_{i-1}}^{t}\mathrm{D}^{2}_{\tau}\xi(\tau)\mathrm{d}\tau\right)\mathrm{d}t.$

Then, by using an integration by parts, we obtain

$\int\\!\\!\\!\\!\\!\\!\int_{t_{i-1}}^{t_{i}}t\,\mathrm{D}_{\tau}^{2}\xi(t)\mathrm{d}t=-\int\\!\\!\\!\\!\\!\\!\int_{t_{i-1}}^{t_{i}}\left(\int\\!\\!\\!\\!\\!\\!\int_{t_{i}}^{t}\mathrm{D}^{2}_{\tau}\xi(\tau)\mathrm{d}\tau\right)\mathrm{d}t=0.$

∎

As a consequence of the previous lemma, let us introduce the following
notation. Let $[a,b]\subset\mathbb{R}$ and let $\gamma\colon[a,b]\to M$ be a
curve of class $H^{2}$. Defining $W_{\gamma,0}([a,b],M)$ as follows,

$W_{\gamma,0}([a,b],M)\coloneqq\Big{\\{}\eta\in L^{2}([a,b],TM):\,\eta(t)\in
T_{\gamma(t)}M,\,\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}\eta(t)\,\mathrm{d}t=0,\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}t,\
\eta(t)\,\mathrm{d}t=0\Big{\\}},$

we are ready to state the generalization of the DuBois-Reymond lemma in the
Riemannian setting.

######  Lemma 3.2.

If $\mu\colon[a,b]\to TM$ is a $L^{1}$ vector field along a continuous curve
$\gamma\colon[a,b]\to M$ such that

$\int_{a}^{b}g\left(\mu,\eta\right)\mathrm{d}t=0,\qquad\forall\eta\in
W_{\gamma,0}([a,b],M),$

then there exist two parallel vector fields $\nu$ and $\zeta$ along $\gamma$
such that

$\mu(t)=\nu(t)+t\zeta(t),\qquad\text{a.e. in }(a,b).$ (3.8)

###### Proof.

By definition of $W_{\gamma,0}$, any $\mu$ given by (3.8), with $\nu$ and
$\zeta$ two arbitrary parallel vector fields along $\gamma$, is orthogonal to
$W_{\gamma,0}$. Indeed, for any $\eta\in W_{\gamma,0}$ and a vector field
$\nu$ along the curve $\gamma$ with zero covariant derivative we have

$\int_{a}^{b}g(\nu,\eta)\mathrm{d}t=g\Big{(}\nu(b),\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}\eta\,\mathrm{d}t\Big{)}-\int_{a}^{b}g\Big{(}\mathrm{D}_{t}\nu,\int\\!\\!\\!\\!\\!\\!\int_{a}^{t}\eta\,\mathrm{d}s\Big{)}\mathrm{d}t=0,$

since the both terms vanish. Similarly, for any vector field $\zeta$ along
$\gamma$ with zero covariant derivative we obtain

$\int_{a}^{b}g(t\zeta,\eta)\mathrm{d}t=\int_{a}^{b}g(\zeta,t\eta)\mathrm{d}t=g\Big{(}\zeta(b),\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}t\eta\,\mathrm{d}t\Big{)}-\int_{a}^{b}g\Big{(}\mathrm{D}_{t}\zeta,\int\\!\\!\\!\\!\\!\\!\int_{a}^{t}s\eta\,\mathrm{d}s\Big{)}\mathrm{d}t=0.$

Therefore, it remains to prove that any arbitrary vector field $\rho$ along
$\gamma$ can be obtained as a direct sum of an element in $W_{\gamma,0}$ and
one given by (3.8). In other words, we need to show the existence and
uniqueness of two parallel vector fields $\nu$ and $\zeta$ along $\gamma$ such
that $\rho-\nu-t\zeta$ belongs to $W_{\gamma,0}$, namely

$\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}(\nu+t\zeta)\mathrm{d}t=\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}\rho\,\mathrm{d}t\qquad\text{and}\qquad\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}(t\nu+t^{2}\zeta)\mathrm{d}t=\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}t\rho\,\mathrm{d}t.$

Denoting by $\nu_{b},\zeta_{b}\in T_{\gamma(b)}M$ the two vectors that
uniquely determined the parallel vector fields $\nu$ and $\zeta$,
respectively, we have that $\rho-\nu-t\zeta$ belongs to $W_{\gamma,0}$ if and
only if there exist $\nu_{b},\zeta_{b}\in T_{\gamma(b)}M$ such that

$\begin{dcases}(b-a)\nu_{b}+\frac{(b-a)^{2}}{2}\zeta_{b}=\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}\rho\,\mathrm{d}t,\\\
\frac{(b-a)^{2}}{2}\nu_{b}+\frac{(b-a)^{3}}{3}\zeta_{b}=\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}t\rho\,\mathrm{d}t.\end{dcases}$

Since the last system uniquely provides the two vectors $\nu_{b},\zeta_{b}$
(unless $b\neq a$), the thesis is obtained. ∎

###### Proof of Theorem 1.1.

The main steps of the proof are the same as those of Proposition 2.4; however,
in this setting, we can appreciate the iterative procedure of the bootstrap
method.

Let us fix a time interval $[t_{i-1},t_{i}]$, with $i=1,\dots,N$. Since
$\gamma\in\Gamma$ is a critical point, for any $\xi\in V_{\gamma,0}$ with
compact support in $(t_{i-1},t_{i})$, by (3.5) we have

$\mathrm{d}f(\gamma)[\xi]=\int_{t_{i-1}}^{t_{i}}g\left(\mathrm{D}_{t}\dot{\gamma}-\eta_{\gamma}(t),\mathrm{D}^{2}_{t}\,\xi\right)\mathrm{d}t=0,$

where we recall that $\eta_{\gamma}$ is the vector field along $\gamma$ given
by (3.4). Using Lemma 3.1, we have that $\mathrm{D}^{2}_{t}\xi\in
W_{\gamma,0}([t_{i-1},t_{i}],M)$. Therefore, by the generality of $\xi$ and
using Lemma 3.2, we have the existence of two parallel vector fields
$\nu^{i},\zeta^{i}$ along $\gamma|_{[t_{i-1},t_{i}]}$ such that

$\mathrm{D}_{t}\dot{\gamma}(t)-\eta_{\gamma}(t)=\nu^{i}(t)+t\zeta^{i}(t),\qquad\text{a.e.
in }(t_{i-1},t_{i}).$ (3.9)

Now, the iterative procedure of the bootstrap method can start. Since
$\eta_{\gamma}$ is of class $H^{2}$ and $\nu^{i}$ and $\zeta^{i}$ are parallel
vector fields, from the previous equation we have that
$\mathrm{D}_{t}\dot{\gamma}$ has $H^{2}$–regularity on $[t_{i-1},t_{i}]$,
hence $\gamma$ is of class $H^{4}$ on this interval. Since by definition of
$\eta_{\gamma}$ we have

$\mathrm{D}^{2}_{t}\eta_{\gamma}(t)=R\big{(}\dot{\gamma}(t),\mathrm{D}_{t}\dot{\gamma}(t)\big{)}\dot{\gamma}(t),\qquad\forall
t\in[t_{i-1},t_{i}],$ (3.10)

and $\gamma|_{[t_{i-1},t_{i}]}$ is of class $H^{4}$, we have that
$\eta_{\gamma}$ is actually of class $H^{4}$ and, by using again (3.9),
$\gamma|_{[t_{i-1},t_{i}]}$ is of class $H^{6}$. By applying alternatively
(3.9) and (3.10), we have that $\gamma|_{[t_{i-1},t_{i}]}$ belongs to
$H^{2k}([t_{i-1},t_{i}],M)\subset C^{2k-1}([t_{i-1},t_{i}],M)$ for every
$k\in\mathbb{N}$, hence it is smooth on $[t_{i-1},t_{i}]$. Since the above
procedure can be applied for every $i=1,\dots,N$, the desired regularity
result is achieved. As a consequence, by taking the second order covariant
derivative on both sides of (3.9), we obtain the spline equation (1.5).

To prove the last part of the theorem, it suffices to compute the first
variation of $\gamma$ for every vector field $\xi\in V_{\gamma}$. By the above
regularity result, we can integrate by parts the first variation formula and,
recalling that $\xi(t_{i})=0$ for every $i=0,\ldots,N$, and that (1.5) holds
on every interval $[t_{i-1},t_{i}]$, we obtain

$\mathrm{d}f(\gamma)[\xi]=\sum_{i=1}^{N}\int_{t_{i-1}}^{t_{i}}\Big{(}g\big{(}\mathrm{D}_{t}\dot{\gamma},\mathrm{D}^{2}_{t}\,\xi\big{)}-g\big{(}R\big{(}\dot{\gamma},\mathrm{D}_{t}\dot{\gamma}\big{)}\dot{\gamma},\xi\big{)}\Big{)}\mathrm{d}t\\\
=\sum_{i=1}^{N}g\big{(}\mathrm{D}_{t}\dot{\gamma},\mathrm{D}_{t}\xi\big{)}\big{|}_{t_{i-1}}^{t_{i}}-\int_{t_{i-1}}^{t_{i}}\Big{(}g\big{(}\mathrm{D}^{2}_{t}\dot{\gamma},\mathrm{D}_{t}\,\xi\big{)}+g\big{(}R\big{(}\dot{\gamma},\mathrm{D}_{t}\dot{\gamma}\big{)}\dot{\gamma},\xi\big{)}\Big{)}\mathrm{d}t\\\
=\sum_{i=1}^{N}g\big{(}\mathrm{D}_{t}\dot{\gamma},\mathrm{D}_{t}\xi\big{)}\big{|}_{t_{i-1}}^{t_{i}}+\int_{t_{i-1}}^{t_{i}}g\big{(}\mathrm{D}^{3}_{t}\dot{\gamma}-R\big{(}\dot{\gamma},\mathrm{D}_{t}\dot{\gamma}\big{)}\dot{\gamma},\xi\big{)}\mathrm{d}t\\\
=\sum_{i=1}^{N}\left(g\big{(}\mathrm{D}_{t}\dot{\gamma}(t_{i}^{-}),\mathrm{D}_{t}\xi(t_{i}^{-})\big{)}-g\big{(}\mathrm{D}_{t}\dot{\gamma}(t_{i-1}^{+}),\mathrm{D}_{t}\xi(t_{i-1}^{+})\big{)}\right)=0.$
(3.11)

Since for every index $i\neq j$, $i\neq 0,N$, we can choose $\xi\in
V_{\gamma}$ such that $\mathrm{D}_{t}\xi$ does not vanish only at that index,
for such a variation we obtain

$g\big{(}\mathrm{D}_{t}\dot{\gamma}(t_{i}^{-})-\mathrm{D}_{t}\dot{\gamma}(t_{i}^{+}),\mathrm{D}_{t}\xi(t_{i})\big{)}=0,$

and by the arbitrariness of $\xi$ we obtain the joining condition
$\mathrm{D}_{t}\dot{\gamma}(t_{i}^{-})=\mathrm{D}_{t}\dot{\gamma}(t_{i}^{+})$,
for every $i\neq j$. This implies that both $\gamma|_{[0,t_{j}]}$ and
$\gamma|_{[t_{j},1]}$ are of class $C^{2}$. Finally, if $j\neq 0$ or $j\neq
1$, then an analogous argument shows that $\mathrm{D}_{t}\dot{\gamma}(0)=0$ or
$\mathrm{D}_{t}\dot{\gamma}(1)=0$, respectively, and we are done. ∎

##  4 Regularity of $k$–splines

This section generalizes the previous regularity results for the $k$–splines,
for every integer $k\geq 2$. As we did for the splines, at first we show it in
the one dimensional setting, where the notation is not overwhelming, and then
we give it in the Riemannian setting, thus proving Theorem 1.4.

For the more general case of the $k$–splines, Lemma 3.2 is not sufficient to
start the bootstrap method if $k\geq 3$, hence a generalization of that result
is required to obtained the desired regularity, which in the one dimensional
is the following.

######  Lemma 4.1.

Let $k$ be a non-negative integer. Let $u\in L^{1}([a,b],\mathbb{R})$ such
that

$\int_{a}^{b}u(t)\eta(t)\mathrm{d}t=0,$

for any $\eta\in C^{\infty}_{c}((a,b),\mathbb{R})$ that satisfies

$\int_{a}^{b}t^{\ell}\eta(t)\mathrm{d}t=0,\qquad\forall\ell=0,\dots,k.$

Then, there exist $k+1$ constants $c_{0},\dots,c_{k}\in\mathbb{R}$ such that

$u(t)=c_{k}t^{k}+\dots+c_{1}t+c_{0}=\sum_{\ell=0}^{k}c_{\ell}t^{\ell},\qquad\text{a.e.
in }[a,b].$

###### Proof.

The proof relies on an induction argument. When $k=0$, the statement is
nothing but the DuBois-Reymond Lemma.

Assume that the thesis holds for $k$ and let us show it for $k+1$. Let $\xi\in
C^{\infty}_{c}((a,b),\mathbb{R})$ be such that

$\int_{a}^{b}t^{\ell}\xi(t)\mathrm{d}t=0,\qquad\forall\ell=0,\dots,k,$

and we fix a function $\phi\in C^{\infty}_{c}((a,b),\mathbb{R})$ such that
$\int_{a}^{b}\phi(t)\mathrm{d}t=1$. Moreover, let us define

$c=\frac{(-1)^{k}}{(k+1)!}\int_{a}^{b}t^{k+1}\xi(t)\mathrm{d}t\in\mathbb{R}.$

Then, we define $\eta\in C^{\infty}_{c}((a,b),\mathbb{R})$ as follows:

$\eta(t)=\xi(t)+c\,\phi^{(k+1)}(t),$

where

$\phi^{(k+1)}(t)=\frac{\mathrm{d}^{k+1}}{\mathrm{d}t^{k+1}}\phi(t).$

By using the integration by parts formula $\ell$ times, for every
$\ell=0,\dots,k$ we have

$\int_{a}^{b}t^{\ell}\eta(t)\mathrm{d}t=\int_{a}^{b}t^{\ell}\xi(t)\mathrm{d}t+c\int_{a}^{b}t^{\ell}\phi^{(k+1)}(t)\mathrm{d}t\\\
=(-1)^{\ell}c\,\ell!\int_{a}^{b}\phi^{(k+1-\ell)}(t)\mathrm{d}t=(-1)^{\ell}c\,\ell!\big{(}\phi^{(k-\ell)}(b)-\phi^{(k-\ell)}(a)\big{)}=0.$

By the same procedure and by definition of $c$, we obtain

$\int_{a}^{b}t^{k+1}\eta(t)\mathrm{d}t=\int_{a}^{b}t^{k+1}\xi(t)\mathrm{d}t+c\int_{a}^{b}t^{k+1}\phi^{(k+1)}(t)\mathrm{d}t\\\
=\int_{a}^{b}t^{k+1}\xi(t)\mathrm{d}t+(-1)^{k+1}c\,(k+1)!\int_{a}^{b}\phi(t)\mathrm{d}t=\int_{a}^{b}t^{k+1}\xi(t)\mathrm{d}t+(-1)^{k+1}c\,(k+1)!=0.$

Hence, we have that $\int_{a}^{b}t^{\ell}\eta(t)\mathrm{d}t=0$ for any
$\ell=0,\dots,k+1$, and by hypothesis we have

$\int_{a}^{b}u(t)\eta(t)\mathrm{d}t=0.$

Setting,

$c_{k+1}=-\frac{(-1)^{k}}{(k+1)!}\int_{a}^{b}u(t)\phi^{(k+1)}(t)\mathrm{d}t$

we can expand the previous equality as follows:

$\int_{a}^{b}u(t)\eta(t)\mathrm{d}t=\int_{a}^{b}u(t)\xi(t)\mathrm{d}t+c\int_{a}^{b}u(t)\phi^{(k+1)}(t)\mathrm{d}t\\\
=\int_{a}^{b}u(t)\xi(t)\mathrm{d}t+\left(\frac{(-1)^{k}}{(k+1)!}\int_{a}^{b}t^{k+1}\xi(t)\mathrm{d}t\right)\int_{a}^{b}u(t)\phi^{(k+1)}(t)\mathrm{d}t\\\
=\int_{a}^{b}\big{(}u(t)-c_{k+1}t^{k+1}\big{)}\xi(t)\mathrm{d}t=0.$

By the arbitrariness of $\xi$ and by the induction hypothesis, we then obtain
the existence of $k+1$ constants $c_{0},\dots,c_{k}\in\mathbb{R}$ such that

$u(t)-c_{k+1}t^{k+1}=c_{k}t^{k}+\dots+c_{0},\qquad\text{a.e. in }(a,b).$

Therefore, the thesis holds also for $k+1$ and this ends the proof. ∎

Thanks to the previous generalization, we have the following regularity result
in the one dimensional setting.

######  Proposition 4.2.

Let $k$ be a positive integer and let $v_{1},\dots,v_{k-1}\in\mathbb{R}$. Let
us also fix $p_{0},\dots,p_{N}\in\mathbb{R}$ for some positive integer $N$ and
let us fix $j\in\\{0,\dots,N\\}$. Setting

$\Gamma_{k}=\left\\{\gamma\in
H^{k}([0,1],\mathbb{R}):\gamma(t_{i})=p_{i},\,\forall
i=0,\ldots,N,\gamma^{(\ell)}(t_{j})=v_{\ell},\,\forall\ell=1,\dots,k-1\right\\}$

and $f_{k}\colon\Gamma_{k}\to\mathbb{R}$ as follows:

$f_{k}(\gamma)=\int_{0}^{1}|\gamma^{(k)}(t)|^{2}\mathrm{d}t.$

If $\gamma$ is a critical point of $f_{k}$, then, for every $i=1,\dots,N$ the
restriction $\gamma\big{|}_{[t_{i-1},t_{i}]}$ is smooth and

$\frac{\mathrm{d}^{2k}}{\mathrm{d}t^{2k}}\gamma(t)=0,\qquad\forall
t\in[t_{i-1},t_{i}].$ (4.1)

Moreover, both $\gamma\big{|}_{[0,t_{j}]}$ and $\gamma\big{|}_{[t_{j},1]}$ are
of class $C^{2k-2}$; if $t_{j}\neq 0$ then $\gamma^{(k+\ell-1)}(0)=0$ for
every $\ell=1,\dots,k-1$, and similarly if $t_{j}\neq 1$.

###### Proof.

By standard computations, it can be shown that for every
$\gamma\in\Gamma_{k}$, the set of admissible variations is given by

$V_{k}=\left\\{\xi\in H^{k}([0,1],\mathbb{R}):\xi(t_{i})=0,\,\forall
i=0,\dots,N,\,\xi^{(\ell)}(t_{j})=0,\,\forall\ell=1,\dots,k-1\right\\}.$

Let us fix $i=1,\dots,N$ and a variation $\xi\in V$ such that
$\operatorname{supp}\xi\subset(t_{i-1},t_{i})$. Therefore,
$\xi^{(\ell)}(t_{i-1})=\xi^{(\ell)}(t_{i})=0$ for every $\ell=0,\dots,k-1$,
and this implies that

$\int_{t_{i-1}}^{t_{i}}t^{\ell}\xi^{(k)}(t)\mathrm{d}t=(-1)^{\ell}\,\ell!\int_{t_{i-1}}^{t_{i}}\xi^{(k-\ell)}(t)\mathrm{d}t\\\
=(-1)^{\ell}\,\ell!\big{(}\xi^{(k-\ell-1)}(t_{i})-\xi^{(k-\ell-1)}(t_{i-1})\big{)}=0,\qquad\forall\ell=1,\dots,k-1.$

By a standard computation, we have

$\mathrm{d}f_{k}(\gamma)[\xi]=\int_{t_{i-1}}^{t_{i}}\gamma^{(k)}(t)\xi^{(k)}(t)\mathrm{d}t=0,$

and therefore, by Lemma 4.1, we have that

$\gamma^{(k)}(t)=c_{k-1}^{i}t^{k-1}+\dots+c_{1}^{i}t+c_{0}^{i},\quad\text{a.e.
in }(t_{i-1},t_{i}),$

for some real constants $c_{k-1}^{i},\dots,c_{0}^{i}$. Hence,
$\gamma|_{[t_{i-1},t_{i}]}$ is smooth and (4.1) holds, for every
$i=1,\dots,N$.

Now, let $\xi$ be a general variation. A careful application of the
integration by parts formula leads to the following equality:

$\mathrm{d}f_{k}(\gamma)[\xi]=\sum_{i=1}^{N}\left(\sum_{\ell=1}^{k-1}(-1)^{\ell}\Big{(}\gamma^{(k+\ell-1)}(t_{i}^{-})\xi^{(k-\ell)}(t_{i})-\gamma^{(k+\ell-1)}(t_{i-1}^{+})\xi^{(k-\ell)}(t_{i-1})\Big{)}\right)=0.$
(4.2)

For every $i\neq j$, $i\neq 0,N$, and for every $\ell=1,\dots,k-1$, there
exists $\xi\in V_{k}$ such that $\xi^{(k-\ell)}(t_{i})\neq 0$ only for these
$i$ and $\ell$. In this case we have

$\mathrm{d}f_{k}(\gamma)[\xi]=(-1)^{\ell}\Big{(}\gamma^{(k+\ell-1)}(t_{i}^{-})-\gamma^{(k+\ell-1)}(t_{i}^{+})\Big{)}\xi^{(k-\ell)}(t_{i})=0,$

meaning that

$\gamma^{(k+\ell-1)}(t_{i}^{-})=\gamma^{(k+\ell-1)}(t_{i}^{+}),\qquad\forall\ell=1,\dots,k-1,\,\forall
i\neq j,i\neq 0,N,$

from which we infer that both $\gamma|_{[0,t_{j}]}$ and $\gamma|_{[t_{j},1]}$
are of class $C^{2k-2}$. If $t_{j}\neq 0$, by choosing a variation $\xi\in
V_{k}$ such that $\operatorname{supp}\xi\subset[0,t_{1})$ we obtain

$\mathrm{d}f_{k}(\gamma)[\xi]=\sum_{\ell=1}^{k-1}\gamma^{(k+\ell-1)}(0)\xi^{(k-\ell)}(0)=0,$

from which we infer that $\gamma^{(k+\ell-1)}(0)=0$ for every
$\ell=1,\dots,k-1$, and an analogous result can be obtained if $t_{j}\neq 1$.
∎

###  4.1 Regularity of $k$–splines in the Riemannian setting

This section provides a proof of Theorem 1.4. As for the above regularity
result in the one dimensional setting, we need a generalization of the DuBois-
Reymond Lemma, which in this case is the following.

######  Lemma 4.3.

Let $k$ be a non-negative integer. If $\mu\colon[a,b]\to TM$ is a vector field
of class $L^{1}$ along an absolute continuous curve $\gamma\colon[a,b]\to M$
such that

$\int_{a}^{b}g\big{(}\mu(t),\eta(t)\big{)}\mathrm{d}t=0,$

for any $\eta\in C^{\infty}_{c}((a,b),\mathbb{R})$ that satisfies

$\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}t^{\ell}\eta(t)\mathrm{d}t=0,\qquad\forall\ell=0,\dots,k,$

then there exist $k+1$ constants parallel vector fields along $\gamma$,
denoted by $\zeta_{0},\dots,\zeta_{k}$, such that

$\mu(t)=t^{k}\zeta_{k}(t)+\dots+t\zeta_{1}(t)+\zeta_{0}(t)=\sum_{\ell=0}^{k}t^{\ell}\zeta_{\ell}(t),\qquad\text{a.e.
in }(a,b).$ (4.3)

###### Proof.

The proof follows the same lines of the one of Lemma 4.1, namely it uses an
induction argument and a direct construction of the first term, namely of the
parallel vector field that appears with the higher order in (4.3).

If $k=0$, then the thesis is the DuBois-Reymond Lemma in the Riemannian
setting. Thus, let us assume that the thesis holds for $k$ and let us prove it
for $k+1$.

Let $\xi\colon[a,b]\to TM$ be a smooth vector field along $\gamma$ with
compact support in $(a,b)$ such that

$\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}t^{\ell}\xi(t)\mathrm{d}t=0,\qquad\forall\ell=0,\dots,k.$

Let us fix a function $\phi\in C^{\infty}_{c}((a,b),\mathbb{R})$ such that
$\int_{a}^{b}\phi(t)\mathrm{d}t=1$. Let $c_{\xi}\colon[a,b]\to TM$ be the
parallel vector field along $\gamma$ such that

$c_{\xi}(b)=\frac{(-1)^{k}}{(k+1)!}\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}t^{k+1}\xi(t)\mathrm{d}t\in
T_{\gamma(b)}M.$

With this notation, let us define the vector field $\eta\colon[a,b]\to TM$ as
follows:

$\eta(t)\coloneqq\xi(t)+\phi^{(k+1)}(t)c_{\xi}(t).$

For any $\ell=0,\dots,k$, we have

$\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}t^{\ell}\eta(t)\mathrm{d}t=\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}t^{\ell}\xi(t)\mathrm{d}t+\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}\phi^{(k+1)}(t)t^{\ell}c_{\xi}(t)\mathrm{d}t=(-1)^{\ell}\,\ell!\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}\phi^{(k+1-\ell)}(t)c_{\xi}(t)=0,$

where, since $c_{\xi}$ is a parallel vector field, we have
$\mathrm{D}_{t}\big{(}t^{\ell}c_{\xi}(t)\big{)}=\ell\,t^{\ell-1}c_{\xi}(t)$.
Moreover, we have

$\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}t^{k+1}\eta(t)\mathrm{d}t=\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}t^{k+1}\xi(t)\mathrm{d}t+(-1)^{k+1}(k+1)!\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}\phi(t)c_{\xi}(t)\mathrm{d}t\\\
=\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}t^{k+1}\xi(t)\mathrm{d}t+(-1)^{k+1}(k+1)!c_{\xi}(b)=\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}t^{k+1}\xi(t)\mathrm{d}t-\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}t^{k+1}\xi(t)\mathrm{d}t=0.$

Therefore, by hypothesis, we have

$\int_{a}^{b}g(\mu(t),\eta(t))\mathrm{d}t=0,$

hence

$\int_{a}^{b}g\big{(}\mu(t),\xi(t)+\phi^{(k+1)}(t)c_{\xi}(t)\big{)}=0.$ (4.4)

Let $\zeta_{k+1}\colon[a,b]\to TM$ be the parallel vector field such that

$\zeta_{k+1}(b)=-\frac{(-1)^{k}}{(k+1)!}\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}\phi^{(k+1)}(t)\mu(t)\mathrm{d}t,$

and let us notice that

$\int_{a}^{b}g\big{(}\mu(t),\phi^{k+1}(t)c_{\xi}(t)\big{)}\mathrm{d}t=\int_{a}^{b}g\big{(}\phi^{k+1}(t)\mu(t),c_{\xi}(t)\big{)}\mathrm{d}t\\\
=g\Big{(}\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}\phi^{(k+1)}(t)\mu(t)\mathrm{d}t,c_{\xi}(b)\Big{)}=g\Big{(}\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}\phi^{(k+1)}(t)\mu(t)\mathrm{d}t,\frac{(-1)^{k}}{(k+1)!}\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}t^{k+1}\xi(t)\mathrm{d}t\Big{)}\\\
=-g\Big{(}\zeta_{k+1}(b),\int\\!\\!\\!\\!\\!\\!\int_{a}^{b}t^{k+1}\xi(t)\Big{)}=-\int_{a}^{b}g\big{(}\zeta_{k+1}(t),t^{k+1}\xi(t)\big{)}\mathrm{d}t=-\int_{a}^{b}g\big{(}t^{k+1}\zeta_{k+1}(t),\xi(t)\big{)}\mathrm{d}t.$

Therefore, from (4.4) we obtain the following identity:

$\int_{a}^{b}g\big{(}\mu(t)-t^{k+1}\zeta_{k+1}(t),\xi(t)\big{)}=0.$

By the arbitrariness of $\xi$ and the inductive hypothesis, there exist $k+1$
constant vector fields along $\gamma$, say $\zeta_{0},\dots,\zeta_{k}$ such
that

$\mu(t)=t^{k+1}\zeta_{k+1}(t)+\dots+t\zeta_{1}(t)+\zeta_{0}(t),\qquad\text{a.e.
in }[a,b],$

and we are done. ∎

###### Proof of Theorem 1.4.

In the setting provided by Theorem 1.4, for every $\gamma\in\Gamma_{k}$ the
set of admissible variations is

$V_{k,\gamma}\coloneqq\left\\{\begin{aligned} \xi\in H^{k}([0,1],TM):\xi(t)\in
T_{\gamma(t)}M,\,\forall t\in[0,1],&\quad\xi(t_{i})=0,\,\forall
i=0,\dots,N,\\\
&\quad\mathrm{D}_{t}^{\ell}\xi(t_{j})=0,\,\forall\ell=1,\dots,k-1\end{aligned}\right\\},$
(4.5)

and the differential $\mathrm{d}f_{k}(\gamma)\colon V_{k,\gamma}\to\mathbb{R}$
is given by the following formula (cf. [7, Proposition 3.1]):

$\mathrm{d}f_{k}(\gamma)[\xi]=\int_{0}^{1}g\Big{(}\mathrm{D}_{t}^{k-1}\dot{\gamma},\mathrm{D}_{t}^{k}\xi+\sum_{\ell=0}^{k-2}\mathrm{D}_{t}^{\ell}\big{(}R(\xi,\dot{\gamma})\mathrm{D}_{t}^{k-\ell-2}\dot{\gamma}\big{)}\Big{)}\mathrm{d}t.$
(4.6)

Due to the lengthy computations and heavy notation involving the covariant
derivatives of the curvature tensor $R$, we will first show the regularity
result for the case $k=3$. This approach allows us to clearly present the key
steps of the proof without being overwhelmed by excessive notation. We will
then extend the results to higher values of $k$.

Let $k=3$; as usual, for a given $i=1,\dots,N$, let us consider a variation
$\xi\in V_{k,\gamma}$ with compact support in $(t_{i-1},t_{i})$. Then, by
applying the symmetric and skew-symmetric properties of the curvature tensor
$R$ and of its covariant derivative, we have

$\displaystyle\mathrm{d}f_{k}(\gamma)[\xi]$
$\displaystyle=\int_{t_{i-1}}^{t_{i}}g\big{(}\mathrm{D}_{t}^{2}\dot{\gamma},\mathrm{D}_{t}^{3}\xi\big{)}\mathrm{d}t+\int_{t_{i-1}}^{t_{i}}g\big{(}\mathrm{D}_{t}^{2}\dot{\gamma},R(\xi,\dot{\gamma})\mathrm{D}_{t}\dot{\gamma}\big{)}\mathrm{d}t+\int_{t_{i-1}}^{t_{i}}g\big{(}\mathrm{D}_{t}^{2}\dot{\gamma},\mathrm{D}_{t}\big{(}R(\xi,\dot{\gamma})\mathrm{D}_{t}\dot{\gamma}\big{)}\big{)}\mathrm{d}t$
$\displaystyle=\int_{t_{i-1}}^{t_{i}}g\big{(}\mathrm{D}_{t}^{2}\dot{\gamma},\mathrm{D}_{t}^{3}\xi\big{)}\mathrm{d}t+\int_{t_{i-1}}^{t_{i}}g\big{(}\mathrm{D}_{t}^{2}\dot{\gamma},R(\xi,\dot{\gamma})\mathrm{D}_{t}\dot{\gamma}\big{)}\mathrm{d}t$
$\displaystyle+\int_{t_{i-1}}^{t_{i}}g\Big{(}\mathrm{D}_{t}^{2}\dot{\gamma},(\mathrm{D}_{t}R)(\xi,\dot{\gamma})\dot{\gamma}+R(\mathrm{D}_{t}\xi,\dot{\gamma})\dot{\gamma}+R(\xi,\mathrm{D}_{t}\dot{\gamma})\dot{\gamma}+R(\xi,\dot{\gamma})\mathrm{D}_{t}\dot{\gamma}\Big{)}\mathrm{d}t$
$\displaystyle=\int_{t_{i-1}}^{t_{i}}g\big{(}\mathrm{D}_{t}^{2}\dot{\gamma},\mathrm{D}_{t}^{3}\xi\big{)}\mathrm{d}t-\int_{t_{i-1}}^{t_{i}}g\big{(}R(\mathrm{D}_{t}\dot{\gamma},\mathrm{D}_{t}^{2}\dot{\gamma})\dot{\gamma},\xi\big{)}\mathrm{d}t$
$\displaystyle-\int_{t_{i-1}}^{t_{i}}g\big{(}(\mathrm{D}_{t}R)(\dot{\gamma},\mathrm{D}_{t}^{2}\dot{\gamma})\dot{\gamma}+R(\dot{\gamma},\mathrm{D}_{t}^{2}\dot{\gamma})\mathrm{D}_{t}\dot{\gamma}+R(\mathrm{D}_{t}\dot{\gamma},\mathrm{D}_{t}^{2}\dot{\gamma})\dot{\gamma},\xi\big{)}\mathrm{d}t$
$\displaystyle-\int_{t_{i-1}}^{t_{i}}g\big{(}R(\dot{\gamma},\mathrm{D}_{t}^{2}\dot{\gamma})\dot{\gamma},\mathrm{D}_{t}\xi\big{)}\mathrm{d}t.$

For the sake of simplicity, we can rewrite the previous equation as follows:

$\mathrm{d}f_{k}(\gamma)[\xi]=\int_{t_{i-1}}^{t_{i}}g\big{(}\mathrm{D}_{t}^{2}\dot{\gamma},\mathrm{D}_{t}^{3}\xi\big{)}\mathrm{d}t-\int_{t_{i-1}}^{t_{i}}g\big{(}h_{3}(\dot{\gamma},\mathrm{D}_{t}\dot{\gamma},\mathrm{D}_{t}^{2}\dot{\gamma}),\xi\big{)}\mathrm{d}t-\int_{t_{i-1}}^{t_{i}}g\big{(}h_{2}(\dot{\gamma},\mathrm{D}_{t}^{2}\dot{\gamma}),\mathrm{D}_{t}\xi\big{)}\mathrm{d}t,$

where both $h_{3}$ and $h_{2}$ are vector fields along $\gamma$ of class
$L^{2}$. As a consequence, setting

$\eta_{3}^{i}(t)=-\int\\!\\!\\!\\!\\!\\!\int_{t_{i-1}}^{t}\left(\int\\!\\!\\!\\!\\!\\!\int_{t_{i-1}}^{\tau}\Big{(}\int\\!\\!\\!\\!\\!\\!\int_{t_{i-1}}^{s}h_{3}(\dot{\gamma},\mathrm{D}_{\sigma}\dot{\gamma},\mathrm{D}_{\sigma}^{2}\dot{\gamma})\mathrm{d}\sigma\Big{)}\mathrm{d}s\right)\mathrm{d}\tau,\qquad\forall
t\in[t_{i-1},t_{i}],$

and

$\eta_{2}^{i}(t)=\int\\!\\!\\!\\!\\!\\!\int_{t_{i-1}}^{t}\left(\int\\!\\!\\!\\!\\!\\!\int_{t_{i-1}}^{\tau}w(\dot{\gamma},\mathrm{D}_{s}^{2}\dot{\gamma})\mathrm{d}s\right)\mathrm{d}\tau,\qquad\forall
t\in[t_{i-1},t_{i}],$

by integrating by parts we obtain

$\mathrm{d}f(\gamma)[\xi]=\int_{t_{i-1}}^{t_{i}}g\big{(}\mathrm{D}_{t}^{2}\dot{\gamma}(t)-\eta_{3}^{i}(t)-\eta_{2}^{i}(t),\mathrm{D}_{t}^{3}\xi(t)\big{)}\mathrm{d}t=0.$

As a consequence, by Lemma 4.3 we obtain the existence of three parallel
vector fields along $\gamma|_{[t_{i-1},t_{i}]}$, say
$\zeta_{0}^{i},\zeta_{1}^{i}$, and $\zeta_{2}^{i}$, such that

$\mathrm{D}_{t}^{2}\dot{\gamma}(t)=\eta_{3}^{i}(t)+\eta_{2}^{i}(t)+t^{2}\zeta_{2}^{i}(t)+t\zeta_{1}^{i}(t)+\zeta_{0}^{i}(t),\qquad\forall
t\in[t_{i-1},t_{i}].$

At this step, the bootstrap iterative procedure can start: since
$\eta_{2}^{i}$ is of class $H^{2}$ and $\eta_{3}^{i}$ is of class $H^{3}$, we
obtain that $\mathrm{D}_{t}^{2}\gamma$ is of class $H^{2}$, so $\gamma$ is of
class $H^{4}$ in $[t_{i-1},t_{i}]$; but then, $\eta_{2}^{i}$ is of class
$H^{3}$, and so $\gamma^{i}$ is of class $H^{5}$, and so forth. By this
iteration, we have that $\gamma$ is actually smooth in $[t_{i-1},t_{i}]$.

For higher-order derivatives, a similar procedure will lead to the desired
regularity results: let $k\geq 2$; for every $\ell=0,\dots,k-2$, we denote by
$\Omega_{\ell}\subset\mathbb{N}^{4}$ the set of (ordered) quadruples of non-
negative integers whose sum is equal to $\ell$. Hence,
$\omega\in\Omega_{\ell}$ if
$\omega=(\omega^{1},\omega^{2},\omega^{3},\omega^{4})\in\mathbb{N}^{4}$ and
$\omega^{1}+\omega^{2}+\omega^{3}+\omega^{4}=\ell$. With this notation, for
any $\xi\in V_{k,\gamma}$ with compact support in $(t_{i-1},t_{i})$, from
(4.6) we infer the following equation

$\displaystyle\mathrm{d}f_{k}(\gamma)[\xi]$
$\displaystyle=\int_{t_{i-1}}^{t_{i}}g\Big{(}\mathrm{D}_{t}^{k-1}\dot{\gamma},\mathrm{D}_{t}^{k}\xi+\sum_{\ell=0}^{k-2}\mathrm{D}_{t}^{\ell}\big{(}R(\xi,\dot{\gamma})\mathrm{D}_{t}^{k-\ell-2}\dot{\gamma}\big{)}\Big{)}\mathrm{d}t$
$\displaystyle=\int_{t_{i-1}}^{t_{i}}g\big{(}\mathrm{D}_{t}^{k-1}\dot{\gamma},\mathrm{D}_{t}^{k}\xi\big{)}\mathrm{d}t+\int_{t_{i-1}}^{t_{i}}g\Big{(}\mathrm{D}_{t}^{k-1}\dot{\gamma},\sum_{\ell=0}^{k-2}\sum_{\omega\in\Omega_{\ell}}\big{(}(\mathrm{D}_{t}^{\omega^{1}}R)(\mathrm{D}_{t}^{\omega^{2}}\xi,\mathrm{D}_{t}^{\omega^{3}}\dot{\gamma})\mathrm{D}_{t}^{k-\ell-2+\omega^{4}}\dot{\gamma}\big{)}\Big{)}\mathrm{d}t.$

By the linearity of the integral operator and the symmetric and skew-symmetric
properties of the curvature tensor $R$, we then obtain

$\displaystyle\mathrm{d}f_{k}(\gamma)[\xi]$
$\displaystyle=\int_{t_{i-1}}^{t_{i}}g\big{(}\mathrm{D}_{t}^{k-1}\dot{\gamma},\mathrm{D}_{t}^{k}\xi\big{)}\mathrm{d}t+\sum_{\ell=0}^{k-2}\sum_{\omega\in\Omega_{\ell}}\int_{t_{i-1}}^{t_{i}}g\Big{(}\mathrm{D}_{t}^{k-1}\dot{\gamma},(\mathrm{D}_{t}^{\omega^{1}}R)(\mathrm{D}_{t}^{\omega^{2}}\xi,\mathrm{D}_{t}^{\omega^{3}}\dot{\gamma})\mathrm{D}_{t}^{k-\ell-2+\omega^{4}}\dot{\gamma}\Big{)}\mathrm{d}t$
$\displaystyle=\int_{t_{i-1}}^{t_{i}}g\big{(}\mathrm{D}_{t}^{k-1}\dot{\gamma},\mathrm{D}_{t}^{k}\xi\big{)}\mathrm{d}t-\sum_{\ell=0}^{k-2}\sum_{\omega\in\Omega_{\ell}}\int_{t_{i-1}}^{t_{i}}g\Big{(}(\mathrm{D}_{t}^{\omega^{1}}R)(\mathrm{D}_{t}^{k-\ell-2+\omega^{4}}\dot{\gamma},\mathrm{D}_{t}^{k-1}\dot{\gamma})\mathrm{D}_{t}^{\omega^{3}}\dot{\gamma},\mathrm{D}_{t}^{\omega^{2}}\xi\Big{)}\mathrm{d}t.$

As a consequence, by rearranging the terms we obtain $k-1$ vector fields along
$\gamma$, say $h_{2},h_{3},\dots,h_{k}$, each depending on $\dot{\gamma}$ and
its covariant derivatives up to order $k-1$ and, consequently, they are of
class $L^{2}$, such that

$\mathrm{d}f_{k}(\gamma)[\xi]=\int_{t_{i-1}}^{t_{i}}g\big{(}\mathrm{D}_{t}^{k-1}\dot{\gamma},\mathrm{D}_{t}^{k}\xi\big{)}\mathrm{d}t+\sum_{\ell=2}^{k}\int_{t_{i-1}}^{t_{i}}g\big{(}h_{\ell},\mathrm{D}_{t}^{k-\ell}\xi\big{)}\mathrm{d}t.$

Now, for each $\ell=2,\dots,k$ we apply the integration by part formula $\ell$
times, obtaining a vector field $\eta^{i}_{\ell}$ of class $H^{\ell}$, that
with opportune changes of signs lead to the following equation

$\displaystyle\mathrm{d}f_{k}(\gamma)[\xi]$
$\displaystyle=\int_{t_{i-1}}^{t_{i}}g\big{(}\mathrm{D}_{t}^{k-1}\dot{\gamma},\mathrm{D}_{t}^{k}\xi\big{)}\mathrm{d}t-\sum_{\ell=2}^{k}\int_{t_{i-1}}^{t_{i}}g\big{(}\eta^{i}_{\ell},\mathrm{D}_{t}^{k}\xi\big{)}\mathrm{d}t$
$\displaystyle=\int_{t_{i-1}}^{t_{i}}g\Big{(}\mathrm{D}_{t}^{k-1}\dot{\gamma}-\sum_{\ell=2}^{k}\eta^{i}_{\ell},\mathrm{D}_{t}^{k}\xi\Big{)}=0.$

Now, the bootstrap iterative procedure can start: by Lemma 4.3 we obtain $k$
parallel vector fields along $\gamma$, say
$\zeta^{i}_{0},\dots,\zeta_{k-1}^{i}$, such that

$\mathrm{D}_{t}^{k-1}\dot{\gamma}(t)=\sum_{\ell=2}^{k}\eta^{i}_{\ell}(t)+\sum_{l=0}^{k-1}t^{l}\zeta^{i}_{l}(t),\qquad\forall
t\in[t_{i-1},t_{i}].$

As a consequence, $\mathrm{D}_{t}^{k-1}\dot{\gamma}$ has the same regularity
as the least regular vector field on the right-hand side of the previous
equation, namely $\eta^{i}_{2}\in H^{2}([t_{i-1},t_{i}],TM)$. But then
$\eta^{i}_{2}$ is of class $H^{4}$ and so on. Therefore, by this iterative
procedure we conclude that $\gamma|_{[t_{i-1},t_{i}]}$ is a smooth function,
for every $i=1,\dots,N$.

From this regularity, one can integrate by parts the first variation formula
(4.6) and then apply the fundamental lemma of the calculus of variations to
obtain (1.7).

Finally, the $C^{2k-2}$ regularity of $\gamma|_{[0,t_{j}]}$ and
$\gamma|_{[t_{j},1]}$ and the conditions at the extrema can be obtained from
iterative integration by parts of (4.6), when applied to a general $\xi\in
V_{k,\gamma}$ and set to zero by the criticality condition of $\gamma$, and by
(1.7), using a similar procedure employed in the proofs of Theorem 1.1 and
Proposition 4.2. ∎

##  5 Existence of minimizers

As stated in the introduction, the existence result we provide stems from a
direct inspection of the counter-example provided as a proof of [9, Lemma
2.15], which states that a minimizer of $f$ can not exist if only the
interpolation points are prescribed. For the reader’s convenience, and because
of its importance in the subsequent discussion, we report here the above
counterexample.

######  Example 5.1.

Let $M\subset\mathbb{R}^{3}$ be the following $2$–dimensional cylinder,
endowed with the standard Euclidean metric induced by $\mathbb{R}^{3}$:

$M=\left\\{(x,y,z)\in\mathbb{R}^{3}:x^{2}+y^{2}=\frac{1}{4\pi^{2}}\right\\},$

so that the “perimeter” of $M$ has length equal to one. Let us set
$(t_{0},t_{1},t_{2})=(0,r,1)$, where $r\in(0,1)$ is an _irrational_ number,
and let $p_{0}=p_{2}=\left(\frac{1}{2\pi},0,0\right)$,
$p_{1}=\left(-\frac{1}{2\pi},0,0\right)$. This setting is illustrated in
Figure 2. Now, let us prove that a minimizer of the spline energy functional
among the smooth curves passing through that points at these times doesn’t
exist. This can be done by showing the existence of a minimizing sequence,
i.e., a sequence whose functional converges to its infimum, but the infimum
cannot be achieved.

$x$$y$$z$$p_{0}=p_{2}$$p_{1}$$\gamma$$\tilde{\gamma}$$M$

$t$$r$$(1,2)$$(1,4)$$(1,k)$$3/2$$5/2$$m+\frac{1}{2}$

Figure 2: The setting of Example 5.1 (left) and its representation in the
plane (right): On the plane, it can be observed how the straight lines (dashed
in the figure) joining $(0,0)$ with points $(0,\mathbb{Z})$ can pass
arbitrarily close to points in $(r,\frac{1}{2}+\mathbb{Z})$, thus requiring
small perturbations to pass through those points, which implies a small
increase in the spline functional.

First of all, let us notice that for any curve on $M$, its projection on the
plane $z\equiv 0$ has lower spline energy. More formally, for any curve
$\gamma\colon[0,1]\to M$,
$\gamma(t)=(\gamma_{1}(t),\gamma_{2}(t),\gamma_{3}(t))$, setting
$\tilde{\gamma}(t)=(\gamma_{1}(t),\gamma_{2}(t),0)$, we have
$f(\tilde{\gamma})\leq f(\gamma)$. So, in our interpolation problem, we can
restrict our analysis on the curves that lie on the $z\equiv 0$ plane (see
Figure 2).

Secondly, using the standard identification of $S^{1}$ with
$\mathbb{R}/\mathbb{Z}$, the problem can be cast in the following setting:

$\inf\left\\{\int_{0}^{1}\ddot{\gamma}^{2}(t)\mathrm{d}t:\gamma\in
H^{2}([0,1],\mathbb{R}),\gamma(0)=0,\gamma(r)\in\frac{1}{2}+\mathbb{Z},\gamma(1)\in\mathbb{Z}\right\\},$

as represented in Figure 2. By the irrationality of $r$, any straight line
joining the point $(0,0)$ with a point in $(0,\mathbb{Z})$ doesn’t pass
through $(r,\frac{1}{2}+\mathbb{Z})$. Hence, the spline energy is strictly
greater than zero for any curve that satisfies the prescribed interpolation
conditions, meaning that the infimum of the above problem is greater then or
equal to $0$, and that the minimum, if it exists, can’t be zero. However, if
we consider the sequence of parabolas
$(\gamma_{k})_{k\in\mathbb{N}}\colon[0,1]\to\mathbb{R}$ such that
$\gamma_{k}(0)=0$, $\gamma_{k}(1)=k$ and $\gamma_{k}(r)=m+1/2$, where
$m\in\mathbb{Z}$ will be determined later, a direct computation shows that

$\int_{0}^{1}\ddot{\gamma}_{k}^{2}(t)\mathrm{d}t=4\frac{\left(m+\frac{1}{2}-kr\right)^{2}}{(r^{2}-r)^{2}}.$

By Dirichlet’s approximating theorem, there exists a subsequence
$(\gamma_{k_{j}})_{j\in\mathbb{N}}$ and a sequence $m_{j}\in\mathbb{N}$ such
that

$\lim_{j\to\infty}\Big{(}m_{j}+\frac{1}{2}-k_{j}r\Big{)}=0,$

see Figure 2. Hence, we have that $\lim_{j\to\infty}f(\gamma_{k_{j}})=0$, so
that $(\gamma_{k_{j}})_{j\in\mathbb{N}}$ is a minimizing sequence and the
infimum is indeed $0$, which can’t be a minimum. Therefore, a minimizer for
the spline energy functional with the above interpolation constraints does not
exist.

######  Remark 5.2.

As one can infer by a direct inspection of Example 5.1, the construction of
the minimizing sequence is obtained by a small perturbation of straight lines
that are approximating the interpolation conditions, and such a sequence of
maps can exist only if no velocity is prescribed, since the initial velocity
of such a sequence of functions has to diverge. Indeed, we have

$\lim_{j\to\infty}|\dot{\gamma}_{k_{j}}|=\lim_{j\to\infty}\left(k_{j}-\frac{2m_{j}+1-2k_{j}r}{r^{2}-r}\right)=+\infty.$

Remark 5.2 suggests that, if at least one of the velocities of the curves is
prescribed, for example the initial one, then in Example 5.1 the set of points
$(r,\mathbb{Z}+1/2)$ that can be achieved without increasing “too much” the
spline energy functional is finite. This is the heuristically observation that
suggested the existence result formally given by Theorem 1.6.

######  Remark 5.3.

One can be easily convinced that if natural or periodic boundary conditions
are prescribed, then a minimizer of $f$ can not exist when $\sigma=0$.
Considering once again the problem given in Example 5.1, this can be
understood by observing that suitable small perturbations of the straight
lines passing through $(0,0)$ and $(1,k)$ lead to a sequence of curves
$(\gamma_{k})_{k\in\mathbb{N}}$ satisfying the interpolation conditions and
both the natural and periodic boundary conditions, all while keeping the
increase in the spline energy functional relatively low. Consequently, we
still have $\lim_{k\to\infty}f(\gamma_{k})=0$, indicating the absence of a
minimizer.

For the sake of precision, let us explicitly construct one of these sequences.
Let $(\omega_{k_{j}})_{j\in\mathbb{N}}$ be the sequence of straight lines
passing through $(0,0)$ and $(1,k_{j})$; as previously done, let us also
consider a sequence $(m_{j})_{j\in\mathbb{N}}\subset N$ such that

$\alpha_{j}\coloneqq m_{j}+\frac{1}{2}-k_{j}r\to 0,\qquad\text{as
}j\to\infty.$

As a consequence, the distance between the point $(r,m_{j}+\frac{1}{2})$ and
the line $\omega_{k_{j}}$ is converging to zero. Let us choose $\delta>0$ such
that $[r-\delta,r+\delta]\subset(0,1)$ and a function $\phi\in
C^{2}([0,1],\mathbb{R})$ such that
$\operatorname{supp}\phi\subset(r-\delta,r+\delta)$, and $\phi(r)=1$. Then,
every function $\gamma_{j}\in C^{2}([0,1],\mathbb{R})$ defined by
$\gamma_{j}=\omega_{k_{j}}+\alpha_{j}\phi$ satisfies both the natural and the
periodic boundary conditions, since
$\ddot{\gamma}_{j}(0)=\ddot{\gamma}_{j}(1)=0$ and
$\dot{\gamma}_{j}(0)=\dot{\gamma}_{j}(1)=k_{j}$, respectively, and it
satisfies the interpolation condition at $r$ by construction. Moreover,

$\lim_{j\to\infty}f(\gamma_{j})=\lim_{j\to\infty}\alpha_{j}^{2}\int_{0}^{1}\ddot{\phi}^{2}(t)\mathrm{d}t=0,$

hence $(\gamma_{j})_{j\in\mathbb{N}}$ is a minimizing sequence, but the
problem doesn’t admit a minimum.

######  Remark 5.4.

Before giving the proof of Theorem 1.6, let us make an important observation
about the number of constraints and the number of parameters we have. We give
it having in mind the one dimensional setting, but the same arguments hold for
the more general Riemannian setting. From the regularity result (i.e.,Theorem
1.1), we know that a minimizer is a cubic spline that, in each interval
$[t_{i-1},t_{i}]$ can be uniquely determined by $4$ parameters. Therefore,
each cubic spline is characterized by $4N$ parameters.

On the other hand, the constraints we impose on the problem give a total of
$4N$ constraints on the parameters of the cubic splines, independently on
where the velocity is imposed. Let us explicitly count them in the case of
prescribed initial velocity. Since the curve has to pass through the points
$p_{0},\ldots,p_{N}\in M$, each cubic polynomial is constraints on the extreme
points, and this provides $2N$ constraints. Moreover, the first one has
prescribed velocity, while the last one, because of the natural boundary
condition $\ddot{\gamma}(1)=0$, has another constraint, so that we have $2N+2$
constrained parameters. Moreover, by Theorem 1.1, the minimizer is a function
of class $C^{2}$, hence on each of the points $p_{1},\ldots,p_{N-1}$ we must
impose other $2$ constraints, i.e.,
$\dot{\gamma}(t_{i}^{-})=\dot{\gamma}(t_{i}^{+})$ and
$\ddot{\gamma}(t_{i}^{-})=\ddot{\gamma}(t_{i}^{+})$, and this regularity
condition provides $2(N-1)$ constraints. Summing up, the total number of the
constraints is $2N+2+2(N-1)=4N$, so the minimizer is uniquely determined.

If the velocity is prescribed at one interior time $t_{j}\in(0,1)$, then two
natural boundary conditions have to be taken into account, so that the number
of constraints at the extrema doesn’t change; because both
$\dot{\gamma}(t_{j}^{-})$ and $\dot{\gamma}(t_{j}^{+})$ must be equal to $v$,
we have an additional constraint, but at the same time we lose one constraint
on the regularity, since in $t_{j}$ the minimizer has only the
$C^{1}$–regularity. Therefore, even in this case, we have a total of $4N$
constraints, and the minimizer is unique.

###  5.1 Proof of Theorem 1.6

The proof of Theorem 1.6 follows the direct method of the calculus of
variations. Specifically, we will prove the coercivity of the functional
$f\colon\Gamma\to\mathbb{R}$, which ensures the weak convergence of a
minimizing sequence (more formally, of one of its subsequences). Additionally,
we will demonstrate that the functional is weakly lower semicontinuous,
meaning that if $\gamma_{k}$ weakly converges to $\tilde{\gamma}$, then
$\liminf_{k\to\infty}f(\gamma_{k})\geq f(\tilde{\gamma})$, so that
$\tilde{\gamma}$ is a minimizer. These results have already been obtained in
[6, Section 3] for the case of initial and final prescribed points and
velocities, but a close inspection of the arguments shows that the same
procedure applies in our setting. For the reader’s convenience, and for a
self-contained work, we provide the proof here with the necessary
modifications.

Let $\Gamma\subset H^{2}([0,1],M)$ be given as in Theorem 1.6. We start with
the following coercivity result, whose proof relies on the boundary condition
on the velocity at the initial time, i.e., $\dot{\gamma}(0)=v$ for every
$\gamma\in\Gamma$.

######  Lemma 5.5.

Let $(\gamma_{k})_{k\in N}\subset\Gamma$ be such that $\sup_{k\in
N}f(\gamma_{k})<+\infty$. Then, $g(\dot{\gamma}_{k},\dot{\gamma}_{k})$ is
uniformly bounded and $(\gamma_{k})_{k\in N}$ admits a subsequence that weakly
converges, with respect to the $H^{2}$–norm, to a curve in $\Gamma$.

###### Proof.

The proof follows along the lines of [6, Lemma 3.1]. Let
$c^{2}=\sup_{k\in\mathbb{N}}f(\gamma_{k})$ and define, for every $k$,
$H_{k}\colon[0,1]\to\mathbb{R}$ as $H_{k}(t)\coloneqq
g(\dot{\gamma}_{k}(t),\dot{\gamma}_{k}(t))$. Since $\dot{\gamma}_{k}(0)=v$ for
every $k$, setting $h_{0}=g(v,v)$, we obtain the following estimate, which
also uses the classical Hölder inequality:

$H_{k}(t)=h_{0}+\int_{0}^{t}\frac{\mathrm{d}}{\mathrm{d}\tau}H_{k}(\tau)\mathrm{d}\tau\leq
h_{0}+2\int_{0}^{1}\Big{|}g\big{(}\mathrm{D}_{\tau}\dot{\gamma}_{k},\dot{\gamma}_{k}\big{)}\Big{|}\mathrm{d}\tau\\\
\leq
h_{0}+2\big{(}f(\gamma_{k})\big{)}^{\frac{1}{2}}\Big{(}\int_{0}^{1}g(\dot{\gamma}_{k},\dot{\gamma}_{k})\mathrm{d}\tau\Big{)}^{\frac{1}{2}}\leq
h_{0}+2c\lVert H_{k}\rVert_{L^{\infty}}^{\frac{1}{2}},\qquad\forall
t\in[0,1].$

By the above inequality, taking the supremum over $t\in[0,1]$, we obtain that
$\lVert H_{k}\rVert_{L^{\infty}}$ is uniformly bounded.

Let us show the existence of a subsequence that weakly converges to a curve in
$\Gamma$. As a first step, let us notice that the uniform boundness of $\lVert
H_{k}\rVert_{L^{\infty}}$ implies the weak convergence with respect to the
$H^{1}$–norm of a subsequence of $(\gamma_{k})_{k\in\mathbb{N}}$, which we
denote again by $(\gamma_{k})_{k\in\mathbb{N}}$, to a curve $\tilde{\gamma}$.
Moreover, by the Ascoli-Arzelà Theorem, this convergence is also uniform.

Now, using a finite cover of $\tilde{\gamma}$, we will prove the weak
$H^{2}$–convergence. Let $I_{1},I_{2},\dots,I_{L}\subset[0,1]$ a finite
collection of closed interval such that $[0,1]=\cup_{\ell=1}^{L}I_{\ell}$, and
each $\tilde{\gamma}|_{I_{\ell}}$ is inside a (open) local chart
$(U_{\ell},\varphi_{\ell})$ for $k=1,\dots,K$. More precisely, this means
that, for every $k$, $U_{\ell}$ is an open connected subset of
$\mathbb{R}^{n}$, $\varphi_{\ell}\colon U_{\ell}\to M$ is a (smooth)
diffeomorphism and $\tilde{\gamma}(I_{\ell})\subset\varphi_{\ell}(U_{\ell})$.
Since $\gamma_{n}$ is uniformly convergent to $\tilde{\gamma}$, without loss
of generality we can assume that also
$\gamma_{k}(I_{\ell})\subset\varphi_{\ell}(U_{\ell})$ for every $\ell$ and for
every $k$. Let us denote by $(x_{k})_{k\in\mathbb{N}}$ and $\tilde{x}$ the
parametrizations of $(\gamma_{k})_{k\in\mathbb{N}}$ and $\tilde{\gamma}$,
respectively, in these local charts, independently on $\ell$, namely
$\gamma_{k}(t)=\varphi_{\ell}(x_{k}(t))$ for every $k$ and for every $t\in
I_{\ell}$, and similarly for $\tilde{\gamma}$. We will prove that, for every
$\ell$, $(x_{k})_{k\in\mathbb{N}}$ weakly converges in the $H^{2}$–norm to
$\tilde{x}$, up to subsequences, and this will end the proof. Hence, it
suffices to prove that $\lVert\ddot{x}_{k}\rVert_{L^{2}(I_{\ell})}$ is
uniformly bounded, independently on $\ell$.

By the uniform convergence of $(\gamma_{k})_{k\in\mathbb{N}}$ to
$\tilde{\gamma}_{k}$, there exists a compact subset of $M$ that contains all
the images of $\gamma_{k}$. This implies that, setting

$g_{\ell}(v,w)=g(\mathrm{d}\varphi_{\ell}(v),\mathrm{d}\varphi_{\ell}(w)),\qquad\forall
v,w\in\mathbb{R}^{n},$

we can assume the existence of two constants $\alpha,\beta>0$ such that

$\alpha\lVert\ddot{x}_{k}\rVert^{2}\leq
g_{\ell}(\ddot{x}_{k},\ddot{x}_{k})\leq\beta\lVert\ddot{x}_{k}\rVert^{2},\qquad\forall\ell=1,\dots,L.$

As a consequence, we need to prove that

$\int_{I_{\ell}}g_{k}(\ddot{x}_{k},\ddot{x}_{k})\mathrm{d}t$

is uniformly bounded, independently on $k$. In every local chart, the
components of the $\mathrm{D}_{t}\dot{\gamma}_{k}$ are given by a formula of
the following kind

$\ddot{x}_{k}+\Gamma_{\ell}(x_{k},\dot{x}_{k}),$

where $\Gamma_{\ell}\colon U_{\ell}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ is a
continuous map homogeneous of degree $2$ with respect to the second variable,
which can be explicitly computed from the Christoffel symbols of the
connection in the local coordinates. By the homogeneity of degree $2$ of the
maps $\Gamma_{\ell}$, and by the uniform boundness of
$g(\dot{\gamma}_{k},\dot{\gamma_{k}})=g_{\ell}(\dot{x}_{k},\dot{x}_{k})$, we
infer the existence of a constant $a>0$ such that

$\int_{I_{\ell}}g_{\ell}\big{(}\Gamma_{\ell}(x_{k},\dot{x}_{k},),\Gamma_{\ell}(x_{k},\dot{x}_{k})\big{)}\mathrm{d}t\leq
a^{2},\qquad\forall\ell=1,\dots,L.$

Since $f(\gamma_{k})\leq c^{2}$ for every $k\in\mathbb{N}$, we then obtain the
following estimate

$c^{2}\geq
f(\gamma_{k})\geq\frac{1}{2}\int_{I_{\ell}}g\big{(}\mathrm{D}_{t}\dot{\gamma}_{k},\mathrm{D}_{t}\dot{\gamma}_{k}\big{)}\mathrm{d}t=\frac{1}{2}\int_{I_{\ell}}g_{\ell}\big{(}\ddot{x}_{k}+\Gamma_{\ell}(x_{k},\dot{x}_{k}),\ddot{x}_{k}+\Gamma_{\ell}(x_{k},\dot{x}_{k})\big{)}\mathrm{d}t\\\
=\frac{1}{2}\int_{I_{\ell}}g_{\ell}\big{(}\ddot{x}_{k},\ddot{x}_{k}\big{)}\mathrm{d}t+\int_{I_{\ell}}g_{\ell}\big{(}\ddot{x}_{k},\Gamma_{\ell}(x_{k},\dot{x}_{k})\big{)}\mathrm{d}t+\frac{1}{2}\int_{I_{\ell}}g_{\ell}\big{(}\Gamma_{\ell}(x_{k},\dot{x}_{k}),\Gamma_{\ell}(x_{k},\dot{x}_{k})\big{)}\mathrm{d}t\\\
\geq\frac{1}{2}\int_{I_{\ell}}g_{\ell}\big{(}\ddot{x}_{k},\ddot{x}_{k}\big{)}\mathrm{d}t-\left(\int_{I_{\ell}}g_{\ell}\big{(}\ddot{x}_{k},\ddot{x}_{k}\big{)}\mathrm{d}t\right)^{\frac{1}{2}}\left(\int_{I_{\ell}}g_{\ell}\big{(}\Gamma_{\ell}(x_{k},\dot{x}_{k}),\Gamma_{\ell}(x_{k},\dot{x}_{k})\big{)}\mathrm{d}t\right)^{\frac{1}{2}}\\\
\geq\frac{1}{2}\int_{I_{\ell}}g_{\ell}\big{(}\ddot{x}_{k},\ddot{x}_{k}\big{)}\mathrm{d}t-a\left(\int_{I_{\ell}}g_{\ell}\big{(}\ddot{x}_{k},\ddot{x}_{k}\big{)}\mathrm{d}t\right)^{\frac{1}{2}},$

from we we infer

$\int_{I_{\ell}}g_{\ell}\big{(}\ddot{x}_{k},\ddot{x}_{k}\big{)}\mathrm{d}t\leq\big{(}a+\sqrt{a^{2}+2c^{2}}\big{)}^{2}.$

Since the right hand term is independent on $k\in\mathbb{N}$ and $\ell\in
1,\dots,L$, we are done. ∎

###### Proof of Theorem 1.6.

Since $f$ is lower bounded, it admits a minimizing sequence
$(\gamma_{k})_{k\in N}\subset\Gamma$ such that

$m\coloneqq\inf_{\gamma\in\Gamma}f(\gamma)=\lim_{k\to\infty}f(\gamma_{k})\in\mathbb{R}.$

Therefore, without loss of generality, we can assume that

$f(\gamma_{k})\leq m+1,\qquad\forall k\in\mathbb{N}.$

By Lemma 5.5, $(\gamma_{k})_{k\in\mathbb{N}}$ weakly converges with respect to
the $H^{2}$–norm to a curve $\tilde{\gamma}\in\Gamma$, up to subsequences.
Therefore, it suffices to prove that $f$ is weakly lower semicontinuous with
respect to this norm, so that

$f(\tilde{\gamma})\leq\liminf_{k\to\infty}f(\gamma_{k})=m,$ (5.1)

namely, $\tilde{\gamma}$ is a minimizer of $f$.

Using again the localization argument employed in the proof of Lemma 5.5,
together with its notation, let $(U_{\ell},\varphi_{\ell})_{\ell=1,\dots,L}$ a
collection of local charts that covers $\tilde{\gamma}([0,1])$. Taking a
subsequence if necessary, we can also assume that
$(U_{\ell},\varphi_{\ell})_{\ell=1,\dots,L}$ covers also the image of each
$\gamma_{k}$. Then, (5.1) is obtained by proving that

$\int_{I_{\ell}}g\big{(}\mathrm{D}_{t}\dot{\tilde{\gamma}},\mathrm{D}_{t}\dot{\tilde{\gamma}}\big{)}\mathrm{d}t\leq\liminf_{k\to\infty}\int_{I_{\ell}}g\big{(}\mathrm{D}_{t}\dot{\gamma}_{k},\mathrm{D}_{t}\dot{\gamma}_{k}\big{)}\mathrm{d}t,\qquad\forall\ell=1,\dots,L.$
(5.2)

Since $(x_{k})_{k\in\mathbb{N}}$ $x_{k}\rightharpoonup\tilde{x}$ in
$H^{2}(I_{\ell},U_{\ell})$, it converges to $x$ with respect to $C^{1}$–norm
and, as a consequence, $\Gamma_{\ell}(x_{k},\dot{x}_{k})$ converges to
$\Gamma_{\ell}(\tilde{x},\dot{\tilde{x}})$ with respect to the $L^{\infty}$
norm, so strongly with respect to $L^{2}$, meaning that

$\lim_{k\to\infty}\int_{I_{l}}g_{\ell}\big{(}\Gamma_{\ell}(x_{k},\dot{x}_{k}),\Gamma_{\ell}(x_{k},\dot{x}_{k})\big{)}\mathrm{d}t=\int_{I_{l}}g_{\ell}\big{(}\Gamma_{\ell}(\tilde{x},\dot{\tilde{x}}),\Gamma_{\ell}(\tilde{x},\dot{\tilde{x}})\big{)}\mathrm{d}t,\qquad\forall\ell=1,\ldots,L.$

Moreover, thanks to the weak convergence of $\ddot{x}_{k}$ to
$\ddot{\tilde{x}}$ in the $L^{2}$–norm, we have also that

$\lim_{k\to\infty}\int_{I_{\ell}}g_{\ell}\big{(}\ddot{x}_{k},\Gamma_{\ell}(x_{k},\dot{x}_{k})\big{)}\mathrm{d}t=\lim_{k\to\infty}\int_{I_{\ell}}g_{\ell}\big{(}\ddot{x}_{k},\Gamma_{\ell}(\tilde{x},\dot{\tilde{x}})\big{)}\mathrm{d}t=\int_{I_{\ell}}g_{\ell}\big{(}\ddot{\tilde{x}},\Gamma_{\ell}(\tilde{x},\dot{\tilde{x}})\big{)}\mathrm{d}t,$

for every $\ell=1,\dots,L$. Summing up, (5.2) reduces as follows:

$\int_{I_{\ell}}g_{\ell}\big{(}\ddot{\tilde{x}},\ddot{\tilde{x}}\big{)}\mathrm{d}t\leq\liminf_{k\to\infty}\int_{I_{\ell}}g_{\ell}\big{(}\ddot{x}_{k},\ddot{x}_{k}\big{)}\mathrm{d}t,\qquad\forall\ell=1,\dots,L,$

which is nothing but the weak lower semicontinuity of the norm in $L^{2}$,
hence we are done. ∎

Acknowledgements

D. Corona thanks the partial support of INdAM (Italian National Institute of
High Mathematics), Grant: “mensilità di borse di studio per l’estero per
l’a.a. 2023-2024”.

D. Corona and R. Giambò thank the partial support of GNAMPA INdAM (Italian
National Institute of High Mathematics), Project: CUP-E53C22001930001.

D. Corona and P. Piccione thank the partial support of FAPESP (Projeto
Temático Fapesp 2022/16097-2).

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D. Corona |  | R. Giambò
---|---|---
Università degli Studi di Camerino |  | Università degli Studi di Camerino
School of Science and Technology |  | School of Science and Technology
Via Madonna delle Carceri 9 |  | Via Madonna delle Carceri 9
62032 – Camerino (MC), Italy |  | 62032 – Camerino (MC), Italy
_E-mail_<EMAIL_ADDRESS>|  |  _E-mail_<EMAIL_ADDRESS>
P. Piccione |  |
Zhejiang Normal University, |  |
School of Mathematical Sciences, |  |
321004, Jinhua-ZJ, |  |
People’s Republic of China. |  |
_Permanent Address_ : |  |
Universidade de São Paulo |  |
Departamento de Matemática |  |
Rua do Matão 1010 |  |
São Paulo, SP 05508–090, Brazil |  |
_E-mail_<EMAIL_ADDRESS>|  |