# Bounded Palais-Smale sequences with Morse type information for some
constrained functionals

Jack<EMAIL_ADDRESS>Laboratoire de Mathématiques
(CNRS UMR 6623), Université de Bourgogne Franche-Comté, Besançon 25030, France
Xiaojun Chang<EMAIL_ADDRESS>School of Mathematics and Statistics &
Center for Mathematics and Interdisciplinary Sciences, Northeast Normal
University, Changchun 130024, Jilin, PR China  Louis
<EMAIL_ADDRESS>Laboratoire de Mathématiques (CNRS
UMR 6623), Université de Bourgogne Franche-Comté, Besançon 25030, France
Nicola<EMAIL_ADDRESS>Dipartimento di Matematica, Politecnico
di Milano, Via Bonardi 9, 20133, Milano, Italy

###### Abstract

In this paper, we study, for functionals having a mountain pass geometry on a
constraint, the existence of bounded Palais-Smale sequences carrying Morse
index type information.

Keywords: Critical point theory; Palais-Smale sequences; Morse type
information; Variational methods.
Mathematics Subject Classification: 35J60, 47J30

Acknowledgements: X. J. Chang is partially supported by NSFC (11971095). N.
Soave is partially supported by the INDAM-GNAMPA group.
J. Borthwick gratefully acknowledges that part of this work was supported by
the French “Investissements d’Avenir” program, project ISITE-BFC (contract
ANR- 15-IDEX-0003).

##  1 Introduction and main results

In recent years, the search of critical points for constrained functionals
having, restricted to a constraint, a mountain pass geometry has been an
active direction of research. A core motivation for this is the search for
solutions having a prescribed $L^{2}$-norm to some stationary nonlinear
Schrödinger equations whose nonlinearity is so-called mass supercritical. As a
simple model one may consider

$-\Delta u+\lambda u+V(x)u=f(u),\quad u\in H^{1}(\mathbb{R}^{N}),$ (1.1)

where $V(x)$ is a given potential and $f(u)$ is modeled on $f(u)=|u|^{p-2}u$
where $2+\frac{N}{4}<p<2^{*}$ with $2^{*}=\frac{2N}{N-2}$ and $2^{*}=+\infty$
if $N=1,2$. A critical point of the associated Energy functional

$E(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}\Big{(}|\nabla
u|^{2}+(\lambda+V(x))|u|^{2}\Big{)}\,dx-\frac{1}{p}\int_{\mathbb{R}^{N}}|u|^{p}\,dx$

restricted to the constraint

$S_{\mu}=\\{u\in
H^{1}(\mathbb{R}^{N})\,|\,\int_{\mathbb{R}^{N}}|u|^{2}\,dx=\mu\\}$

for some $\mu>0$ is a solution to Equation (1.1) with norm $\sqrt{\mu}$ ; the
$\lambda\in\mathbb{R}$ arising then as a Lagrange parameter. Assuming, for
simplicity, that $f(u)=|u|^{p-2}u$ and since $p>2+\frac{4}{N}$, the functional
$E$ is unbounded from below on $S_{\mu}$, for any $\mu>0$ but it enjoys a
mountain-pass geometry. Namely, there exist $w_{1},w_{2}\in S_{\mu}$ such
that, setting

$\Gamma=\left\\{\gamma\in C([0,1],S_{\mu})\,|\,\
\gamma(0)=w_{1},\quad\gamma(1)=w_{2}\right\\},$

we have

$c:=\inf_{\gamma\in\Gamma}\
\max_{t\in[0,1]}E(\gamma(t))>\max\\{E(w_{1}),E(w_{2})\\}.$ (1.2)

It is a direct consequence of Ekeland’s variational principle that a
functional with a mountain pass geometry admits Palais-Smale sequences at the
mountain pass level $c$. If one can prove that one of these sequences
converges then one obtains a critical point at the mountain pass level.
However, for constrained functionals, proving the boundedness of such
sequences is already a major difficulty. Furthermore, boundedness is of course
not sufficient to obtain convergence, and one must also overcome another
delicate step consisting of, using the terminology introduced by P.L. Lions
[19, 20], showing that the sequence is not vanishing.

Starting with [16], several approaches have been developed to overcome these
issues but ultimately they all seem to rely on the fact that critical points
satisfy a natural constraint on which the functional can be shown to be
coercive and for which, additionally, sequences close to the constraint do not
vanish, see for example [7, 8]. In practice, this constraint is often provided
by a Pohozaev type identity, or some connected identity, see for example [7,
8, 11, 24, 29] in that direction. Nevertheless, to prove useful, this Pohozaev
type constraint must have a simple expression which, in most cases, requires
that the underlying functional enjoys some scaling properties. Typically when
dealing with problems of the type given by Equation (1.1), these equations
must be autonomous or some smallness assumptions on the potential $V(x)$ or
its derivatives, which somehow guarantee that the problem keeps some key
features of the autonomous case must be assumed. As a consequence, general
non-autonomous equations and, more generally, problems where scaling is not
possible – such as $L^{2}$-prescribed solutions for mass supercritical
equations of type (1.1) set on bounded domains or on graphs – remain
essentially unexplored. See nevertheless [2, 6, 25, 26] for interesting
contributions in that direction.

The aim of our paper is to present a general abstract tool, Theorem 1, to deal
with cases where a Pohozaev type identity either does not exist or does not
provide useful information. Roughly speaking, we shall show that one can
expect generically, for a functional with a mountain-pass geometry on a
constraint, to find a bounded Palais-Smale sequence which carries Morse type
information. Along with that of boundedness, this second-order information
will be key in the proof that the sequence converges. In fact, our Theorem 1
has already proved decisive in the papers [12, 13] in which we study mass
prescribed solutions for mass-superlinear problem set on a graph. The fact
that the underlying equation is set on a graph prevents the existence of a
Pohozaev identity.

### Setting and statement of Theorem 1

We now introduce the setting in which our main result, Theorem 1, will be
stated. This setting is borrowed from [10, Section 8].
Let $(E,\langle\cdot,\cdot\rangle)$ and $(H,(\cdot,\cdot))$ be two _infinite-
dimensional_ Hilbert spaces and assume that:

$E\hookrightarrow H\hookrightarrow E^{\prime},$

with continuous555In the applications, these injections will also be dense but
we have not used this property. injections. For simplicity, we assume that the
continuous injection $E\hookrightarrow H$ has norm at most $1$ and identify
$E$ with its image in $H$. We also introduce:

$\begin{cases}\|u\|^{2}=\langle u,u\rangle, \\\ |u|^{2}=(u,u),\end{cases}\quad
u\in E.$

Define for $\mu\in(0,+\infty)$:

$S_{\mu}=\\{u\in E\,|\,|u|^{2}=\mu\\}.$

Note that $S_{\mu}$ is a submanifold of $E$ of codimension $1$ and that its
tangent space at a given point $u\in S_{\mu}$ can be considered as the closed
codimension $1$ subspace of $E$ given by:

$T_{u}S_{\mu}=\\{v\in E\,|\,(u,v)=0\\}.$

In order to state Theorem 1 we need some definitions.

######  Definition 1.

Let $\phi:E\rightarrow\mathbb{R}$ be a $C^{2}$-functional on $E$ and
$\alpha\in(0,1]$ we shall say that $\phi^{\prime}$ and $\phi^{\prime\prime}$
are:

  1. HOL 1

: (globally) $\alpha$-Hölder continuous if there is $M>0$ such that for any
$u_{1},u_{2}\in E$,

$||\phi^{\prime}(u_{1})-\phi^{\prime}(u_{2})|| \leq
M||u_{2}-u_{1}||^{\alpha},\quad||\phi^{\prime\prime}(u_{1})-\phi^{\prime\prime}(u_{2})||
\leq M||u_{1}-u_{2}||^{\alpha}, $

  2. HOL 2

: $\alpha$-Hölder continuous on bounded sets if for any $R\in
\mathbb{R}_{+}^{*}$ one can find $M=M(R)>0$ such that for any $u_{1},u_{2}\in
B(0,R)$:

$||\phi^{\prime}(u_{1})-\phi^{\prime}(u_{2})|| \leq
M||u_{2}-u_{1}||^{\alpha},\quad||\phi^{\prime\prime}(u_{1})-\phi^{\prime\prime}(u_{2})||
\leq M||u_{1}-u_{2}||^{\alpha},$ (1.3)

  3. HOL 3

: locally $\alpha$-Hölder continuous if for any $u\in E$ one can find an open
neighbourhood $U$ of $u$ such that the restriction of $\phi$ to $U$,
$\phi|_{U}$, satisfies HOL 1.

######  Remark 1.1.

Clearly: HOL 1 $\Rightarrow$ HOL 2 $\Rightarrow$ HOL 3. In _finite_
dimensions, using local compactness and the Lebesgue number lemma, HOL 3 is
equivalent to:

  * HOL 2’

: for every $R>0$ one can find $\delta(R)>0,M(R)>0$ such that, if
$u_{1},u_{2}\in B(0,R)$ _and_ $||u_{1}-u_{2}||<\delta(R)$, Equation (1.3)
holds.

Since the estimates in Equation (1.3) are most interesting when $u_{1}$ and
$u_{2}$ are close, in a finite dimensional case one could imagine replacing
HOL 2 and HOL 3 by HOL 2’. This last remark does _not_ extend to the infinite
dimensional setting, due to the failure of local compactness.

######  Definition 2.

Let $\phi$ be a $C^{2}$-functional on $E$, for any $u\in E$ define the
continuous bilinear map:

$D^{2}\phi(u)=\phi^{\prime\prime}(u)-\frac{\phi^{\prime}(u)\cdot
u}{|u|^{2}}(\cdot,\cdot). $

######  Remark 1.2.

The geometric relevance of Definition 2 will be explained in Section 2.2.

######  Definition 3.

Let $\phi$ be a $C^{2}$-functional on $E$, for any $u\in S_{\mu}$ and
$\theta>0$, we define the _approximate Morse index_ by

$\tilde{m}_{\theta}(u)=\sup\left\\{\dim\,L\left|~{}L\text{ is a subspace of
$T_{u}S_{\mu}$ such that:~{}$\forall\varphi\in L,\,$
}D^{2}\phi(u)[\varphi,\varphi]<-\theta\|\varphi\|^{2}\right.\right\\}.$

Our main result is the following.

######  Theorem 1.

Let $I\subset(0,+\infty)$ be an interval and consider a family of $C^{2}$
functionals $\Phi_{\rho}:E\to\mathbb{R}$ of the form

$\Phi_{\rho}(u)=A(u)-\rho B(u),\qquad\rho\in I,$

where $B(u)\geq 0$ for every $u\in E$, and

either $A(u)\to+\infty$ or $B(u)\to+\infty$ as $u\in E$ and $\|u\|\to+\infty$.
(1.4)

Suppose moreover that $\Phi_{\rho}^{\prime}$ and $\Phi_{\rho}^{\prime\prime}$
are $\alpha$-Hölder continuous on bounded sets in the sense of Definition 1
(HOL 2) for some $\alpha\in(0,1]$. Finally, suppose that there exist
$w_{1},w_{2}\in S_{\mu}$ (independent of $\rho$) such that, setting

$\Gamma=\left\\{\gamma\in
C([0,1],S_{\mu})\,|\,\gamma(0)=w_{1},\quad\gamma(1)=w_{2}\right\\},$

we have

$c_{\rho}:=\inf_{\gamma\in\Gamma}\
\max_{t\in[0,1]}\Phi_{\rho}(\gamma(t))>\max\\{\Phi_{\rho}(w_{1}),\Phi_{\rho}(w_{2})\\},\quad\rho\in
I.$ (1.5)

Then, for almost every $\rho\in I$, there exist sequences $\\{u_{n}\\}\subset
S_{\mu}$ and $\zeta_{n}\to 0^{+}$ such that, as $n\to+\infty$,

  * (i)

$\Phi_{\rho}(u_{n})\to c_{\rho}$;

  * (ii)

$||\Phi^{\prime}_{\rho}|_{S_{\mu}}(u_{n})||\to 0$;

  * (iii)

$\\{u_{n}\\}$ is bounded in $E$;

  * (iv)

$\tilde{m}_{\zeta_{n}}(u_{n})\leq 1$.

######  Remark 1.3.

For any $u\in S_{\mu}$ we denote by: $\Phi^{\prime}_{\rho}|_{S_{\mu}}(u)$ the
differential at $u$ of the restriction of $\Phi_{\rho}$ to $S_{\mu}$, it is
therefore a linear map: $T_{u}S_{\mu}\rightarrow
T_{\Phi_{\rho}(u)}\mathbb{R}\cong\mathbb{R}$. The notation
$||\Phi^{\prime}_{\rho}|_{S_{\mu}}(u)||$ introduced in Theorem 1 (ii) is
understood to refer to the dual norm induced by the norm of $T_{u}S_{\mu}$
which is inherited from $E$. It is known from [10, Lemma 3] that, if a
sequence $\\{v_{n}\\}\subset S_{\mu}$ is bounded, then the following are
equivalent:

  * •

$\|\Phi^{\prime}_{\rho}|_{S_{\mu}}(v_{n})\|\to 0,\,$ as $n\to+\infty$.

  * •

$\Phi^{\prime}_{\rho}(v_{n})-\displaystyle\frac{1}{\mu}(\Phi’_{\rho}(v_{n})\cdot
v_{n})(v_{n},\cdot)\to 0\,$ in $E^{\prime}\,$, as $n\to+\infty$.

Thus, in particular, Theorem 1 (ii) implies that

$\Phi^{\prime}_{\rho}(u_{n})-\displaystyle\frac{1}{\mu}(\Phi’_{\rho}(u_{n})\cdot
u_{n})(u_{n},\cdot)\to 0\,\mbox{ in }E^{\prime}\mbox{ as }n\to+\infty.$ (1.6)

######  Remark 1.4.

From Theorem 1 (iv) we immediately get that if there exists a subspace
$W_{n}\subset T_{u_{n}}S_{\mu}$ such that

$\Phi^{\prime\prime}_{\rho}(u_{n})[w,w]-\frac{1}{\mu}(\Phi’_{\rho}(u_{n})\cdot
u_{n})(w,w)<-\zeta_{n}||w||^{2},\quad\mbox{for all }w\in W_{n},$

then necessarily $\dim W_{n}\leq 1$.

######  Remark 1.5.

Typically, Theorem 1 will be used with a space $E$ which consists of functions
defined on $\mathbb{R}^{N}$, some subset of $\mathbb{R}^{N}$, or graphs as in
[12, 13]. For instance, to deal with Equation (1.1) we shall set
$E=H^{1}(\mathbb{R}^{N})$ and $H=L^{2}(\mathbb{R}^{N})$.
In such situations, denoting by $|u|_{*}$ the modulus of a function $u\in E$,
we shall see that it is possible to choose $\\{u_{n}\\}\subset S_{\mu}$ with
the property that $u_{n}\geq 0$ on $E$ if all of the following are satisfied:
$u\in E\ \mapsto\ |u|_{*}\in E$, $w_{1},w_{2}\geq 0$, the map
$u\mapsto|u|_{*}$ is continuous, and $\Phi_{\rho}(u)=\Phi_{\rho}(|u|_{*})$.

Under the assumptions of Theorem 1, one can readily observe that the function
$\rho\mapsto c_{\rho}$ is non-increasing. Therefore, its derivative
$c_{\rho}^{\prime}$ is well defined for almost every $\rho\in I$. Following
the ideas of Struwe’s monotonicity trick developed in [17], we shall show that
the existence of $c_{\rho}^{\prime}$ ensures, for such a value of $\rho$, the
existence of a sequence of minimising paths in $\Gamma$ whose tops lie in a
given ball. From there we shall prove the existence of the desired Palais-
Smale sequence.

The first author to emphasise the importance of second order conditions, such
as ($iv$) in Theorem 1, in compactness problems was P.-L. Lions [21], see also
[4, 5]. The main idea in these papers was to use Morse type information on
exact critical points. When one deal with almost critical points, as it is the
case in a Palais-Smale sequence, a major contribution is due to Fang and
Ghoussoub, see [14, Theorem 1]. The authors proved (along with other valuable
properties) that an unconstrained functional having a mountain-pass geometry
possesses, at the mountain-pass level, a Palais-Smale sequence whose elements
have approximately Morse index 1.

Unfortunately, [14, Theorem 1] requires that the first and second derivative
of the functional satisfy the Hölder continuity assumption HOL 1 in a
neighbourhood of the level set corresponding to the mountain pass value.
Actually, inspecting the proof of [14, Theorem 1], one observe that this
condition only needs to be required around some dual set. However, without
precise information on the functional, one has to check the full global
condition HOL 1 and this has limited the possibility to use directly [14,
Theorem 1]. The ideas of [14] nevertheless played a crucial role in the works
[9] and [22] in which the authors combining the ideas of [14, 17], managed to
derive unconstrained versions of our Theorem 1, under the weaker assumption
HOL 2.

Actually, in [15], [14, Theorem 1] was extended, directly under the assumption
that the Hölder condition HOL 1 holds, to a much larger setting. In
particular, in [15] one can handle general minimax structures, as defined in
Definition 4, instead of just a mountain pass setting. Moreover, it is stated
in [15] that the result of [14] holds for functionals defined on a
complete666This term is perhaps a little ambiguous due to the failure of the
Hopf-Rinow theorem in infinite dimensions [3]. $C^{2}$\- Riemannian manifold
modeled on a Hilbert space $E$, see [15, Theorem 1.7 and Theorem 3.1] for the
precise statements. However, the proof of these two theorems is only carried
out for an (unconstrained) functional defined on a Hilbert space. In [15, page
1609], the authors indicate that the general case follows by a direct
adaptation of the proof given in the Hilbert case.

Whilst we agree that the arguments used to prove the various lemmata ([15,
Lemma 3.(3-7)]) that build up to the proof of the main result [15, Theorem
3.1] are local in nature and can be made sense of in a local chart, one cannot
substitute the _local_ Hölder assumption HOL 3 – which one could expect to
extend in a standard fashion to manifolds – for HOL 1 directly in the
statement of [15, Theorem 3.1]. More specifically, inspecting the proof of
[15, Theorem 3.1], one can observe that although the inequalities need only be
satisfied locally on a small ball around a point $B(u_{0},r)$, one must be
able to choose the size of the ball and the constant $M$ in HOL 1 _uniformly_
, i.e. independently of the point $u_{0}$. The condition therefore requires at
least a uniform structure, like that provided by a distance, to speak
uniformly of the size of neighbourhoods.

On a Riemannian manifold $(X,g)$ modeled on a Hilbert space $E$, the natural
distance at hand is the Riemannian distance [18, Chapter VII, §6]:

$\textrm{dist}_{g}(x,y)=\inf\\{L(\gamma)\,|\,\gamma:[a,b]\rightarrow
M,\gamma(a)=x,\gamma(b)=y\\},$

where $\displaystyle
L(\gamma)=\int_{a}^{b}\sqrt{g(\gamma(t))(\gamma^{\prime}(t),\gamma^{\prime}(t))}\textrm{d}t$
is the length of the curve. However, to our knowledge, there is no consensus
on how to formulate a meaningful _coordinate-invariant_ version of HOL 1, and
the authors do not define it in the statement of their theorem.

### Remarks on possible notions of Hölder continuity on Riemannian manifolds

A natural way to do this could be to introduce a distance on the tangent and
double tangent bundles, $TX$ and $TTX$ by means of the Sasaki metric [28],
constructed from the metric $g$ on $X$ in a local chart777$U$ open in $X$,
$\psi:U\mapsto\psi U$ a homeomorphism onto an open subset of $E$. $(U,\psi)$
according to:

${g_{Sasaki}}(x,v)((u_{1},w_{1}),(u_{2},w_{2}))=g(x)(u_{1},u_{2})+g(x)(w_{1}-B_{U}(x;u_{1},v),w_{2}-B_{U}(x;u_{2},v)),$
(1.7)

where $g:\psi U\rightarrow L^{2}_{sym}(E)$ is the local representative of the
metric, and $B_{U}(x;v,w)$ the symmetric bilinear map associated with the
spray of the Levi-Civita connection of $g$ over the chart $(U,\psi)$. We refer
to [18, Chapter IV:§3, Chapter VII:§2,§4] for more details on these objects.
Iterating the construction – as much as the regularity of the manifold permits
– leads to natural Riemannian metrics and distances on higher-order tangent
bundles.

Equipped with these distances, the usual notion of $\alpha$-Hölder continuity
($\alpha\in(0,1]$) between metric spaces carries over to the tangent and
double tangent maps $\phi_{*}:TX\rightarrow T\mathbb{R}$ and
$\phi_{**}:TTX\rightarrow TT\mathbb{R}$. For example, using the canonical
identification $T\mathbb{R}=\mathbb{R}\times\mathbb{R}$, the condition on
$\phi_{*}$ would read in a local chart $(U,\psi)$ on $X$ that there is some
$M\in\mathbb{R}_{+}^{*}$ (independent of the chart) such that for any
$(x,v_{1}),(y,v_{2})\in U\times E$:

$|\phi_{U}(x)-\phi_{U}(y)|+|\phi_{U}^{\prime}(x)\cdot
v_{1}-\phi_{U}^{\prime}(y)\cdot v_{2}|\leq
Md_{TM}(\psi^{-1}_{*x}v_{1},\psi^{-1}_{*y}v_{2}))^{\alpha},$ (1.8)

where we have defined $\phi_{U}=\phi\circ\psi^{-1}.$ This condition is however
stronger than HOL 1. For instance, if $U$ is also a _convex_ neighbourhood
(see [18, p214]) such that any two points can be joined by a length-minimising
geodesic (completely contained in $U$), then Equation (1.8) implies that
$\phi$ satisfies888To see this let $v_{1}=v_{2}=0$. The local representation
of a minimising geodesic in $TX$ that joins the two points is
$t\mapsto(\gamma(t),0)$, where $\gamma$ is the image of the length minimising
geodesic joining $\phi^{-1}(x)$, $\phi^{-1}(y)$ in the chart.:

$ |\phi_{U}(u_{1})-\phi_{U}(u_{2})|\leq M{\rm
dist}_{g}(\psi^{-1}(u_{1}),\psi^{-1}(u_{2})). $

With this in mind, a more appropriate condition would be to project directly
the result of $\phi_{*}$ onto the last factor of
$T\mathbb{R}=\mathbb{R}\times\mathbb{R}$, so that the condition reads locally:

$|\phi_{U}^{\prime}(u_{1})\cdot v_{1}-\phi_{U}^{\prime}(u_{2})\cdot v_{2}|\leq
Md_{TX}(\psi^{-1}_{*u_{1}}v_{1},\psi^{-1}_{*u_{2}}v_{2}))^{\alpha}.$ (1.9)

Unfortunately, this still does not replicate entirely the first part of HOL 1
and, in fact, is too strong a condition to be useful. The problem is that the
distance induced by the Sasaki metric (1.7) does not scale with $v_{1}$ and
$v_{2}$. To illustrate this point it is sufficient to consider the case where
$X=E$ is a Hilbert space. Working in the canonical global chart, $g$ is then
the inner product of $E$ and the bilinear form $B_{U}$ vanishes identically.
The Sasaki metric therefore reduces to the canonical product inner product on
$TE\cong E\times E$ and $d_{TE}((x,v_{1}),(y,v_{2}))=||x-y||+||v_{1}-v_{2}||$.
Taking $v_{1}=v_{2}=v\in E$, Equation (1.9) yields:

$|\phi^{\prime}(x)v-\phi^{\prime}(y)v|\leq||x-y||^{\alpha}. $

Since the right-hand side is homogeneous in $v$ and the left-hand side not,
this condition can only be satisfied if $\phi^{\prime}(x)=\phi^{\prime}(y)$
for all $x,y\in E$. One way one might overcome this is, returning to the
general situation of a Riemannian manifold, to impose the following condition:

$|\phi_{*x}(v_{x})-\phi_{*y}(v_{y})|
\leq\frac{M}{2}\left(\sqrt{g(v_{x},v_{x})}+\sqrt{g(v_{y},v_{y})}\right)d_{TX}(v_{x},v_{y})^{\alpha},\quad
v_{x},v_{y}\in TX. $ (1.10)

In a Hilbert space, this would imply the usual condition:

$ |\phi^{\prime}(x)v-\phi^{\prime}(y)v| \leq M||v||\cdot||x-y||^{\alpha}.$

On a Riemannian manifold $(X,g)$, if $x,y\in X$ are joined by a minimising
geodesic $\gamma$, and $v_{y}=T^{\gamma}_{y}v_{x}$ is the vector obtained by
parallel transport of $v_{x}$ along $\gamma$, then we would have:

$|\phi_{*}(v_{x})-\phi_{*y}(T^{\gamma}_{y}v_{x})|\leq
M\sqrt{g(v_{x},v_{x})}\textrm{dist}_{g}(x,y)^{\alpha}. $ (1.11)

Despite this proposal, that would replicate partly on a Riemannian manifold
HOL 1 and could possibly be adapted to replicate HOL 2, it is apparent that
working with the Riemannian distances of $X$,$TX$ and even more so $TTX$, will
be a difficult task. In practice, we would need to work on convex
neighbourhoods of points, where the existence of unique length-minimising
geodesics between arbitrary pairs of points is guaranteed. Such neighbourhoods
are provided by Whitehead’s theorem [18, Theorem 5.7, Chapter VIII §5] which
is, however, a _local_ statement. Hence, without further knowledge of the
geometry of the manifold, and as one cannot expect to control uniformly the
size of such neighbourhoods, this will destroy any (relative) uniformity in
our arguments.

### Strategy of the proof of Theorem 1: the use of two different sprays

It transpires from the above discussion that one should not hope to generalise
[15, Theorem 3.1] to Riemannian manifolds without more careful consideration
of the geometry. Our aim in this work is to develop an approach to the ideas
in [15] based solely on geometric objects that are easy to handle. As the idea
of constraint implies the existence of an ambient space, we restrict to the
framework of a submanifold $X$ in a Hilbert space $E$. This has the advantage
of providing a univocal and convenient notion of Hölder continuity through the
natural induced distance on $X$, negating the need to work with the Riemannian
distances. It also enables us to present our results in a simpler, more
streamlined language.

The geometric structure at the heart of our considerations is that of a
_spray_ on $X$ and its geodesics. This general notion provides us with
sufficient tools to explore the submanifold $X$ and associates to any
functional $\phi$ a natural second-order differential quantity we denote by
$D^{2}\phi$; the covariant derivative of the one-form999The tangent map
$\phi_{*}$ of a functional $\phi$ can naturally be identified with a section
of the cotangent bundle $T^{\vee}M$, that we note $\textrm{d}\phi$. In a local
chart, this is just the map $x\mapsto\phi^{\prime}_{U}(x)$. $\textrm{d}\phi$
restricted to $X$. Just as the possibility to use information on the second-
order derivative to perform deformations that decrease the value of the
functional is key to the proof of [15, Theorem 1.7 and Theorem 3.1], we will
do this with information on $D^{2}\phi$. This is accomplished in Lemma 13,
however it is more involved than the analogous statement in [15], as we are
naturally confronted with the effects of curvature, for example, in the form
of holonomy.

Although the general philosophy of our method is, in principle, applicable in
this general framework, it still has the disadvantage of requiring rather
detailed knowledge of the geodesics; curves that are defined by a _non_
-linear second-order ordinary differential equation. More precisely, in view
of our result, the essential object we need to be able to understand in a
_quantitative_ way is the exponential map “$\exp$” of the spray. In general,
it is only defined, but nevertheless well understood, locally. In particular,
$\exp$ enables the construction of small convex neighbourhoods [18, Theorem
5.7, Chapter VIII §5] around each point and the “size” of the neighbourhood on
which it is defined is related to the existence time of solutions. However, as
we have mentioned, the nature of our result requires a certain uniformity (on
a bounded set) that is slightly beyond these local considerations. To
alleviate us of this difficulty, we have chosen in this work to restrict to
the setting introduced by Berestycki and Lions [10, Section 8]. This already
provides us with an interesting framework that allows us to handle several
applications.

A novel point that arises in this setting, illustrating a certain flexibility
in our approach, is that we will work with two distinct, but nevertheless
closely related, sprays. The first is chosen so that we have a nice manageable
second-order quantity $D^{2}\phi$ from which it is effectively possible to
extract interesting information, and the second being induced from the
structure of $E$, we will develop this point further in Section 2. Finally,
since we work essentially with sprays and their geodesics, one could certainly
imagine natural extensions to Finsler submanifolds of Banach spaces with
sufficiently regular norms.

The paper is organized as follows. In Section 2 we present our key technical
result, Theorem 2, which is based on the deformations constructed in Lemma 13.
Theorem 1 is proved in Section 3 where we also explore some of its
consequences when the sequence $\\{u_{n}\\}\subset S_{\mu}$ converges. In
particular we derive information on the Morse index of the limiting critical
point, see Theorem 3.

##  2 Statement and proof of Theorem 2

This section, which is the heart of the paper, is devoted to the proof of
Theorem 2. In fact, in this theorem we deal with a more general situation than
the one necessary to establish Theorem 1: we consider the case of a general
homotopic family in the sense of Definition 4 below. Our hope is that this
degree of generality will be useful for possible extensions of Theorem 1.

We will work quite intensively with the induced topology on $S_{\mu}$, in
order to simplify notations we will write
$B(S_{\mu};u_{0},\delta)=B(u_{0},\delta)\cap S_{\mu}$ for any $u_{0}\in
S_{\mu}$.

######  Definition 4.

A family $\mathcal{F}$ of subsets of $S_{\mu}$ is said to be homotopy-stable
family of dimension $n$ with boundary $B$ if there exist a compact subset $D$
of $\mathbb{R}^{n}$, containing a closed subset $D_{0},$ and a continuous
function $\eta_{0}$ from $D_{0}$ onto $B$ such that

$\mathcal{F}=\\{A\subset S_{\mu}\,|\,A=f(D)\mbox{ for some }f\in
C(D;S_{\mu})\mbox{ with }f=\eta_{0}\mbox{ on }D_{0}\\}.$

######  Theorem 2.

Let $\phi$ be a $C^{2}$-functional on $E$, satisfying HOL 2 for some
$\alpha\in(0,1]$ and some $R>1$. Let $\mathcal{F}$ be a homotopy-stable family
of $S_{\mu}$ of dimension $n$ with boundary $B$ such that

$c:=\inf_{A\in\mathcal{F}}\max_{u\in A}\phi(u)>\max_{B}\phi$

is finite. Let $0<\alpha_{1}\leq\frac{\alpha}{2(\alpha+2)}<1$ and
$\varepsilon>0$. Then for any $A\in\mathcal{F}$ with $\max_{u\in A}\phi(u)\leq
c+\varepsilon$ satisfying

$K:=\\{u\in A\,|\,\phi(u)\geq c-\varepsilon\\}\subset B(0,R-1),$ (2.1)

assuming that $\varepsilon>0$ is sufficiently small there exists
$u_{\varepsilon}\in S_{\mu}$ such that

  1. 1.

$c-\varepsilon\leq\phi(u_{\varepsilon})\leq c+\varepsilon$ ;

  2. 2.

$||\phi^{\prime}|_{S_{\mu}}(u_{\varepsilon})||\leq 3\varepsilon^{\alpha_{1}}$
;

  3. 3.

$u_{\varepsilon}\in A$ ;

  4. 4.

If $D^{2}\phi(u_{\varepsilon})[w,w]<-\varepsilon^{\alpha_{1}}\|w\|^{2}$ for
all $w$ in a subspace $W$ of $T_{u_{\varepsilon}}S_{\mu}$, then $\dim\,W\leq
n$.

In order to prove Theorem 2 we recall, for the convenience of the reader, some
results presented in [15]; the first is [15, Lemma 2.5].

######  Lemma 1.

For each $n\in\mathbb{N}^{*},$ there is an integer
$N(n)\leq(2\sqrt{n+1}+2)^{n}$ such that for any compact subset
$D\subset\mathbb{R}^{n}$ and any $\varepsilon>0$, there exist a finite number
of distinct points $\\{x_{i}:1\leq i\leq k\\}$ with the following properties :

  1. 1.

$\displaystyle D\subset\bigcup_{i=1}^{k}B(x_{i},\frac{\varepsilon}{4})\subset
N_{\frac{\varepsilon}{2}}(D)$;

  2. 2.

The intersection of any distinct $N(n)$ elements of the cover
$(\overline{B}(x_{i},\frac{\varepsilon}{2}))_{i=1}^{k}$ is empty.

The following result is [15, Corollary 2.4].

######  Lemma 2.

Let $V$ be a Banach space, let $K$ be a closed subset of $\mathbb{R}^{n}$, and
let $\theta$ be a continuous mapping from $K$ into the unit sphere $S_{V}$ of
$V$. If $n<\dim V$, then $\theta$ can be extended to a continuous mapping from
$\mathbb{R}^{n}$ into $S_{V}.$

###  2.1 The geodesic spray coming from the scalar product of $H$

We will now introduce the geometric structure at the heart of our discussion:
sprays. For completeness, we recall the general definition, and we refer the
reader to [18] for all the basic notation and terminology. Let $X$ be a
smooth101010for simplicity manifold modeled on a Hilbert space $E$. Let $TX$
denote the tangent bundle of $X$ and $\pi:TX\rightarrow X$ the canonical
projection111111Over a chart, where $TX$ looks like a product, this is just
projection onto the first factor.. A _second-order vector field_ $F$ is a
section of $\pi$ that satisfies the condition:

$\pi_{*}\circ F=\textrm{id}_{TX}. $

In a local chart, F is represented by a map: $F_{U}:U\times E\rightarrow
U\times E\times E\times E$, such that:

$F_{U}(x,v)=(x,v,v,f(x,v)).$

A _spray_ is a second order vector field that satisfies an additional
quadratic condition that we describe as follows. For every $s\in\mathbb{R}$,
one has a canonical vector bundle morphism that we shall abusively denote by
$s:TX\rightarrow TX$, obtained by multiplying by $s$ in the fibres. The
tangent map of $s$ gives a map $s_{*}:TTX\rightarrow TTX$. $F$ is a spray if
it satisfies:

$ F(sv)=s_{*}(sF(v)),v\in TX,s\in\mathbb{R}.$

In a local chart, this translates to:

$F(x,sv)=(x,sv,sv,f(x,sv))=(x,sv,sv,s^{2}f(x,v))=s_{*}(sF(v)).$

It follows from this that a spray is locally determined by a symmetric
bilinear map such that:

$f(x,v)=B_{U}(x;v,v).$

A spray defines a second-order differential equation on $X$, namely:

$ \ddot{\alpha}(t)=F(\dot{\alpha}(t)),$

where $\alpha:I\rightarrow X$ is a curve on $X$, $\dot{\alpha}:I\rightarrow
TX$ the canonical lift of $\alpha$, and $\ddot{\alpha}:I\rightarrow TTX$ that
of $\dot{\alpha}$. Solutions of this equation are known as the _geodesics_ of
$F$.

Let us now return to our framework. $S_{\mu}=\\{u\in E,(u,u)=|u|^{2}=\mu\\}$
naturally inherits from $E$ the structure a smooth Riemannian manifold,
however, the scalar product $(\cdot,\cdot)$ of $H$ does not necessarily equip
it with a Riemannian structure. In fact the tangent spaces, in general, will
not be complete under the induced norm. Nevertheless, it does equip $S_{\mu}$
with a spray. We describe this as follows, let $F_{1}:TE\cong E\times
E\rightarrow TTE\cong E\times E\times E\times E$ be defined by:

$F_{1}(u,v)=(u,v,v,-\frac{|v|^{2}}{|u|^{2}}u).$

It is clear that this is a spray on $E$ (the last term is $2$-homogeneous with
respect to $v$).

######  Lemma 3.

The spray $F_{1}$ restricts to a spray on $S_{\mu}$, i.e.
$F_{1}:TS_{\mu}\rightarrow TTS_{\mu}$.

###### Proof.

This is a direct consequence of the following characterisations of $TS_{\mu}$
and $TTS_{\mu}$:

$\begin{gathered}TS_{\mu}=\\{(u,v) \in TE,u\in S_{\mu},v \in
T_{u}S_{\mu}\\}=\\{(u,v)\in TE,u\in S_{\mu},(u,v)=0\\},\\\ TTS_{\mu}=
\\{(u,v,v^{\prime},w)\in TTE,u\in
S_{\mu},(u,v)=0,(u,v^{\prime})=0,(u,w)+(v^{\prime},v)=0\\}. \end{gathered} $

These characterisations result directly from the fact that $TS_{\mu}$ is
defined in $TE=E\times E$ by the equations:

$ (u,v) \in TS_{\mu}\Leftrightarrow u\in S_{\mu},v\in
T_{u}S_{\mu}\Leftrightarrow(u,v)\in E\times E,|u|^{2}=\mu,(u,v)=0.$

Consequently, if $\psi:E\times E\rightarrow\mathbb{R}^{2}$ is defined by
$\psi(u,v)=(|u|^{2}-\mu,(u,v))$, $\psi^{-1}(\\{(0,0)\\})=TS_{\mu}$ and hence
$T_{(u,v)}TS_{\mu}=\ker\psi^{\prime}(u,v).$ ∎

The geodesics of $F_{1}$ are the projection onto the first component of the
integral curves of $F_{1}$ in $TE$; these are curves $\sigma:I\rightarrow E$
that satisfy the equation:

$\sigma^{\prime\prime}(t)+\frac{|\sigma^{\prime}(t)|^{2}}{|\sigma(t)|^{2}}\sigma(t)=0.
$ (2.2)

######  Remark 2.1.

Lemma 3 is the differential expression of the fact that if the initial data of
Equation (2.2) is taken in $TS_{\mu}$ then the solution lives on $S_{\mu}$,
i.e. for any $t\in I$, $|\sigma(t)|^{2}=\mu$.

The principal particularity of this spray is that:

######  Lemma 4.

Let $\sigma:I\rightarrow S_{\mu}$ be a geodesic of the spray $F_{1}$ on
$S_{\mu}$, then:

$\forall t\in I,\quad|\sigma^{\prime}(t)|^{2}=|\sigma^{\prime}(0)|^{2}.$

A major advantage of the spray $F_{1}$ is that the geodesics, with initial
data $(u,v)\in TS_{\mu}$, are explicitly known:

$\sigma(t,u,v)=\cos(\omega t)u+\frac{\sin(\omega
t)}{\omega}v,\qquad\omega=\frac{|v|}{\sqrt{\mu}},\quad t\in\mathbb{R}.$ (2.3)

These curves are globally defined and, as we observed in Lemma 4, have
constant speed with respect to the norm of $H$, but _not_ with that of $E$.
They also possess the following homogeneity property of geodesics
$a\in\mathbb{R}$:

$\sigma(at,u,v)=\sigma(t,u,av). $

Let us introduce and state a few results about the exponential map associated
with the spray:

######  Lemma 5.

For $(u,v)\in TS_{\mu}$, define:

$\exp_{u}(v)=\sigma(1,u,v),$

and $\exp:TS_{\mu}\rightarrow S_{\mu}\times S_{\mu}$,

$\exp(u,v)=(u,\exp_{u}(v)).$

Then:

$\textrm{d}\exp_{(u,v)}(v^{\prime},w)=\underbrace{\left(v^{\prime},\cos\left(\frac{|v|}{\sqrt{\mu}}\right)\left[v^{\prime}+\frac{(w,v)}{|v|^{2}}v\right]+\sin\left(\frac{|v|}{\sqrt{\mu}}\right)\sqrt{\mu}\left[\frac{w}{|v|}-\frac{(w,v)}{|v|^{3}}v-\frac{(w,v)}{\mu
|v|}u\right]\right)}_{\in T_{(u,\exp_{u}(v))}S_{\mu}\cong T_{u}S_{\mu}\times
T_{\exp_{u}(v)}S_{\mu}.}$

Let $\mathcal{U}=\\{(u,v)\in TS_{\mu},|v|<\sqrt{\mu}\pi\\}$, then
$\exp\mathcal{U}$ is open and $\exp$ is a diffeomorphism onto its image, which
is $\\{(u_{1},u_{2})\in S_{\mu}\times S_{\mu},u_{2}\neq-u_{1}\\}$. For fixed
$u_{0}\in S_{\mu}$:

$\exp_{u_{0}}^{-1}(u)=\frac{\arccos\frac{(u,u_{0})}{\mu}}{\sqrt{1-\frac{(u,u_{0})^{2}}{\mu^{2}}}}\left(u-\frac{(u,u_{0})}{\mu}u_{0}\right),\quad
u\neq-u_{0}$ (2.4)

and:

$|\exp_{u_{0}}^{-1}(u)|=\sqrt{\mu}\arccos\frac{(u,u_{0})}{\mu}.$ (2.5)

###### Proof.

$\mathcal{U}=\\{(u,v)\in TE=E\times E,|v|<\sqrt{\mu\pi} \\}\cap TS_{\mu}$ is
open in $TS_{\mu}$ and, given the expression of the differential, one can see
that it is (continuously) invertible at each point in $\mathcal{U}$. Let us
briefly justify Equation (2.4), we solve directly for $v$:

$u=\cos(\frac{|v|}{\sqrt{\mu}})u_{0}+\operatorname{sinc}(\frac{|v|}{\sqrt{\mu}})v,
\qquad v\in T_{u_{0}}S_{\mu},|v|<\sqrt{\mu}\pi,$

where $u\neq-u_{0}$. Since $v\in T_{u_{0}}S_{\mu}$, it follows that:

$(u,u_{0})=\mu\cos(\frac{|v|}{\sqrt{\mu}})\Rightarrow|v|=\sqrt{\mu}\arccos\frac{(u,u_{0})}{\mu}.$

Then:

$\operatorname{sinc}\frac{|v|}{\sqrt{\mu}}=\left\\{
\begin{array}[]{lc}\sqrt{1-\frac{(u,u_{0})^{2}}{\mu^{2}}}\left(\arccos\frac{(u,u_{0})}{\mu}\right)^{-1}&\textrm{if
$u\neq u_{0},$}\\\ 1&\textrm{if $u=u_{0}$}.\end{array}\right.$

Reinjecting these expressions into the original equation provides the
expression given by (2.4) for $v$. ∎

######  Remark 2.2.

It will be convenient to note that, due to the homogeneity property, for any
$t\in\mathbb{R}$,

$\exp_{u}(tv)=\sigma(1,u,tv)=\sigma(t,u,v).$ (2.6)

###  2.2 Geodesics and $D^{2}\phi$

We shall now explain the relevance of Definition 2. In short, it is the
covariant derivative of $\textrm{d}\phi$ (viewed as a one-form) on $S_{\mu}$
induced by $F_{1}$, but we shall not develop this point of view further. For
us, the important point is given by the following lemma.

######  Lemma 6.

Let $\phi$ be a $C^{2}$-functional on $E$ and
$\sigma:(-\varepsilon,\varepsilon)\rightarrow E$ an arbitrary curve satisfying
Equation (2.2), then for any $t\in(-\varepsilon,\varepsilon)$:

$\frac{\textrm{d}^{2}}{\textrm{d}t^{2}}\phi(\sigma(t))=D^{2}\phi(\sigma(t))[\sigma^{\prime}(t),\sigma^{\prime}(t)].
$

###### Proof.

$\frac{\textrm{d}}{\textrm{d}t}\phi(\sigma(t))=\phi^{\prime}(\sigma(t))\cdot\sigma^{\prime}(t).
$

Therefore:

$\displaystyle\frac{\textrm{d}^{2}}{\textrm{d}t^{2}}\phi(\sigma(t))$
$\displaystyle=\phi^{\prime\prime}(\sigma(t))[\sigma^{\prime}(t),\sigma^{\prime}(t)]+\phi^{\prime}(\sigma(t))\cdot\sigma^{\prime\prime}(t)$
$\displaystyle=\phi^{\prime\prime}(\sigma(t))[\sigma^{\prime}(t),\sigma^{\prime}(t)]-\frac{|\sigma^{\prime}(t)|^{2}}{|\sigma(t)|^{2}}\phi^{\prime}(\sigma(t))\cdot\sigma(t).$

∎

######  Remark 2.3.

If $u$ is a critical point of the functional $\phi|_{S_{\mu}}$, then the above
also shows that the restriction of $D^{2}\phi(u)$ to $T_{u}S_{\mu}$ coincides
with the _Hessian_ of $\phi|_{S_{\mu}}$ at $u$ (as defined, for instance, in
[27, p307]); this follows, for instance, from [23, Proposition 5.2.3]. This is
also a direct consequence of the fact that the restriction of $D^{2}\phi(u)$
to $T_{u}S_{\mu}$ is the covariant derivative associated with the spray
$F_{1}$ of the one-form $d\phi|_{S_{\mu}}$.

######  Lemma 7.

Let $\phi$ be a $C^{2}$-functional on $E$, $\alpha\in(0,1]$ such that
$\phi,\phi^{\prime\prime}$ are $\alpha$-Hölder continuous on bounded sets (HOL
2). Then for any $R\geq 0$, one can find $C\in\mathbb{R}_{+}^{*}$ such that
for any $u_{1},u_{2} \in S_{\mu}\cap B(0,R)$:

$\begin{cases}||D^{2}\phi(u_{1})-D^{2}\phi(u_{2})|| \leq
C||u_{1}-u_{2}||^{\alpha}&\textrm{if $||u_{1}-u_{2}||\leq 1$},\\\
||D^{2}\phi(u_{1})-D^{2}\phi(u_{2})|| \leq C||u_{1}-u_{2}||&\textrm{if
$||u_{1}-u_{2}||>1$}.\end{cases} $

###### Proof.

Let us estimate $|\phi^{\prime}(u_{1})u_{1}-\phi^{\prime}(u_{2})u_{2}|$:

$\displaystyle|\phi^{\prime}(u_{1})u_{1}-\phi^{\prime}(u_{2})u_{2}|$
$\displaystyle\leq|(\phi^{\prime}(u_{1})-\phi^{\prime}(u_{2}))\cdot u_{1}|
+|\phi^{\prime}(u_{2})\cdot(u_{2}-u_{1})|$ $\displaystyle\leq
MR||u_{1}-u_{2}||^{\alpha}+MR^{\alpha}||u_{1}-u_{2}||.$

It then follows that:

$||D^{2}\phi(u_{1})-D^{2}\phi(u_{2})||\leq
M\left(1+\frac{R}{\mu}\right)||u_{1}-u_{2}||^{\alpha}+\frac{MR^{\alpha}}{\mu}||u_{1}-u_{2}||.\qed$

######  Corollary 1.

Let $\phi$ be a $C^{2}$-functional on $E$ such that $\phi^{\prime}$ and
$\phi^{\prime\prime}$ are $\alpha$-Hölder continuous on bounded sets. Assume
that for some $R>1$, $u_{0}\in S_{\mu}\cap B(0,R-1)$, one can find a constant
$\beta\in(0,1)$ and a subspace $W\in T_{u_{0}}S_{\mu}$ such that for any $w\in
W:$

$D^{2}\phi(u_{0})[w,w]< -\beta||w||^{2}. $

Then, for any $u\in B(S_{\mu};u_{0},\delta_{1})$ and $w\in W$:

$ D^{2}\phi(u)[w,w]<-\frac{3\beta}{4}||w||^{2},$ (2.7)

where

$\delta_{1}=\min\left(1,\left(\frac{\beta}{4C}\right)^{\frac{1}{\alpha}}\right),\quad\mbox{with}\quad
C=M\left(1+\frac{R+R^{\alpha}}{\mu}\right).$ (2.8)

###### Proof.

This follows directly from Lemma 7 since if $u\in B(u_{0},\delta),$
$\delta\leq 1$, $u\in B(0,R)$. ∎

For future reference, let us fix $K(R)\geq 1$ such that

$||D^{2}\phi(u)||\leq K(R)\quad\mbox{and}\quad||\phi^{\prime}(u)||\leq
K(R)\quad\mbox{for any }u\in B(0,R)\cap S_{\mu}.$ (2.9)

###  2.3 Transporting $W$ to other points

The condition expressed in Equation (2.7) in the result of Corollary 1 is not
an intrinsic condition on $S_{\mu}$. The problem is that in general, $W$ is
not tangent to the manifold at points $u\neq u_{0}$ on $S_{\mu}$. To overcome
this, $W$ should also move with the point to remain tangent to the manifold.
In other words, for each point $u$ in a small neighbourhood of $u_{0}$, we
should exhibit a subspace121212In more geometric terms, we need to construct a
_distribution_ on this neighbourhood. $W(u)$ of $T_{u}S_{\mu}$, which
satisfies a condition of the same nature as (2.7).

Our approach to this will be based on the notion of _parallel transport_ of
vectors along curves associated, however, to a second spray $F_{2}\neq F_{1}$
. Parallel transport of a given vector $w_{0}\in T_{u_{0}}S_{\mu}$ along
geodesics of $F_{1}$ originating from $u_{0}$, we will obtain a vector field
on a ball $B(S_{\mu};u_{0},\delta)$. The $F_{2}$ spray, introduced below, is,
contrary to $F_{1}$, naturally associated with the Riemannian structure on
$S_{\mu}$ induced by $E$; it is known as the _canonical spray_ , we refer the
reader to [18, Chapter VII, §7, Chapter VIII, §4]) for the detailed
construction.

######  Remark 2.4.

An alternative approach could be to construct a local coordinate chart
$(U,\phi)$ near $u_{0}$ and set $W(u)=\phi^{-1}_{*\phi(u)}\phi_{*{u_{0}}}W$.
In fact, if our geodesics and parallel transport where associated with the
same spray, _normal coordinates_ at $u_{0}$ would do just that. We feel,
however, in order to discuss uniformity with $u_{0}$ in a bounded set, that it
is more convenient to describe the construction in a coordinate invariant
manner. We can also observe from Equation (1.11) that in our proposition of an
invariant Hölder condition based on the Riemannian distance, parallel
transport intervenes naturally.

To describe the spray $F_{2}$ in our setting, introduce the injective linear
map $G:E\rightarrow E$, such that for any $u\in E$, $Gu$ is the unique vector
in $E$ that satisfies:

$\forall h\in E,(u,h)=\langle Gu,h\rangle. $ (2.10)

Note that:

$ ||Gu||^{2}=\langle Gu,Gu\rangle=(u,Gu)\leq||Gu||\cdot|u|,$

therefore, for $u\neq 0$,

$ ||Gu||\leq|u|\leq||u||.$

Furthermore, if $u\in S_{\mu}$ then $\langle Gu,u\rangle=|u|^{2}=\mu$, thus
$\mu\leq||Gu||\cdot||u||$. We collect these observations in the following
lemma for future reference:

######  Lemma 8.

The injective linear map $G$ has norm $||G|| \leq 1$ and for any $u\in
S_{\mu}:$

$||Gu|| \geq\frac{\mu}{||u||}.$ (2.11)

In terms of $G$, the canonical spray $F_{2}:TE\rightarrow TTE$ on $E$ can be
defined by:

$F_{2}(u,v)=f(u,v,v,-|v|^{2}\frac{Gu}{||Gu||^{2}}).$

One can check in a straightforward manner that it restricts to a spray on
$S_{\mu}$; its particularity is that it preserves the induced Riemannian
structure on $S_{\mu}$ (see Lemma 9 below.)

On this specific example, let us describe how a spray is used to define
parallel transport of vectors along curves. The discussion applies to
arbitrary curves, however in our situation, they will systematically be
geodesics of the spray $F_{1}$, more precisely: let $u_{0}\in S_{\mu}$ and
$u\in B(S_{\mu};u_{0},\delta)$; assume $\delta<2\sqrt{\mu}$ so that
$-u_{0}\notin B(S_{\mu};u_{0},\delta)$. Set $v=\exp_{u_{0}}^{-1}(u)$
and131313See (2.6) in Remark 2.2., for any $t\in[0,1]$,
$\sigma(t)=\exp_{u_{0}}(tv)=\sigma(t,u_{0},v)$. If $w_{0}\in T_{u_{0}}S$, we
define a vector field on $S_{\mu}$ along the curve $\sigma$ by the following
differential equation constructed from the symmetric bilinear operator
associated with the spray $F_{2}$:

$\varphi^{\prime}(t)+(\varphi(t),\sigma^{\prime}(t))\frac{G\sigma(t)}{||G\sigma(t)||^{2}}=0,\quad\varphi(0)=w_{0}.$
(2.12)

Roughly, the spray induces a notion of parallelism between two vectors in
tangent spaces at _different_ points of $S_{\mu}$. The vector field along
$\sigma$ defined by Equation (2.12) remains parallel in this sense at each
point of the curve; in some sense it is “constant”. We quote the following
properties of this parallel transport:

######  Lemma 9.

Let $u_{0},u\in S_{\mu}$ such that $v=\exp_{u_{0}}^{-1}(u)$ and define
$\sigma(t)=\exp_{u_{0}}(tv),t\in[0,1]$. Define a map:
$T_{u_{0},u,\sigma}:T_{u_{0}}S_{\mu}\rightarrow T_{u}S_{\mu}$ by
$T_{u_{0},u,\sigma}w_{0}=\varphi(1)$ where $t\mapsto\varphi(t)$ is the
solution to Equation (2.12). Then:

  * •

$T_{u_{0},u,\sigma}$ is a linear isometry (for the induced Hilbert space
structure on the tangent spaces),

  * •

The inverse is given by: $T^{-1}_{u_{0},u,\sigma}=T_{u,u_{0},\tilde{\sigma}}$
where $\tilde{\sigma}(t)=\sigma(1-t),t\in[0,1]$.

###### Proof.

We shall only prove that it is an isometry; differentiating
$(\varphi,\sigma)$:

$(\varphi^{\prime}(t),\sigma(t))+(\varphi(t),\sigma^{\prime}(t))=-\frac{(G\sigma(t),\sigma(t))}{||G\sigma(t)||^{2}}(\varphi(t),\sigma^{\prime}(t))+(\varphi(t),\sigma^{\prime}(t))=0.$

Since $(G\sigma(t),\sigma(t))=||G\sigma(t)||^{2}.$ Similarly differentiating
$\langle\varphi(t),\varphi(t)\rangle$ yields:

$\begin{split}\langle\varphi(t),\varphi(t)\rangle^{\prime}=2\langle\varphi^{\prime}(t),\varphi(t)\rangle&=-2\frac{(\varphi(t),\sigma^{\prime}(t))}{||G\sigma(t)||^{2}}\langle
G\sigma(t),\varphi(t)\rangle\\\
&=-2\frac{(\varphi(t),\sigma^{\prime}(t))}{||G\sigma(t)||^{2}}(\sigma(t),\varphi(t))=0.\end{split}$

∎

######  Remark 2.5.

If $F$ is spray on a manifold $X$, and $\gamma$ a geodesic of the spray, then
$\gamma^{\prime}(0)$ is automatically parallel transported (in the sense
defined by $F$) along $\gamma$.

Since all our curves will be geodesics of $F_{1}$ that originate from a fixed
point $u_{0}\in S_{\mu}$, we abbreviate: $T_{u_{0},u,\sigma}$ to $T_{u}$ and
say that we _parallel transport radially_ from $u_{0}$. Given $w_{0}\in W$, we
shall define a vector field on $B(S_{\mu};u_{0},\delta)$ by:

$w(u)=T_{u}w_{0},\quad u\in B(S_{\mu};u_{0},\delta). $

As we mentioned above, the spray on $S_{\mu}$ allows us to compare vectors at
different points of $S_{\mu}$ and, the vector field defined by Equation (2.12)
is “constant” along the curve $\sigma$. However, viewing things in $E$, it
must change to remain tangent to $S_{\mu}$; the next lemma estimates this
change.

######  Lemma 10.

Let $w_{0}\in T_{u_{0}}S_{\mu}$, $u_{0}\in B(0,R-1)\cap S_{\mu}$, $u\in
B(S_{\mu};u_{0},\delta_{0})$ with $\delta_{0}:=\min(1,\sqrt{\mu})$. Then, if
$v=\exp_{u_{0}}^{-1}(u)$:

$||T_{u}w_{0}-w_{0}||\leq C(R,\mu)\,|v|\cdot||w_{0}|| $

where we have set

$C(R,\mu)=\frac{1}{\mu}\big{(}R+\frac{(R-1)}{\sqrt{\mu}}\big{)}.$ (2.13)

###### Proof.

By definition of $T_{u}w_{0}$, it is sufficient to estimate
$||\varphi(1)-\varphi(0)||$, where $\varphi$ is defined by Equation (2.12). By
the mean-value theorem one has:

$\displaystyle||\varphi^{\prime}(t)||\leq\frac{||\varphi(0)||\cdot|\sigma^{\prime}(0)|}{||G\sigma(t)||}\leq\frac{||w_{0}||\cdot|v|\cdot||\sigma(t)||}{\mu},$

where we have used Equation (2.11) and the fact that
$|\sigma^{\prime}(t)|=|\sigma^{\prime}(0)|=|v|$ by Lemma 4.

Now, taking into account that $\delta_{0}\leq\sqrt{\mu}$ guarantees that
$(u,u_{0})\geq 0$, we estimate $||\sigma(t)||$ as follows:

$\begin{aligned} ||\sigma(t)||&\leq
R-1+\sqrt{\mu}\sin\left(\frac{|v|}{\sqrt{\mu}}t\right)\frac{||v||}{|v|}\\\
&\leq
R-1+\sqrt{\mu}\sin\left(\frac{|v|}{\sqrt{\mu}}\right)\frac{||v||}{|v|}.\end{aligned}
$

From Equations (2.4), (2.5) from Lemma 5:

$
|v|=\sqrt{\mu}\arccos\frac{(u,u_{0})}{\mu},\frac{v}{|v|}=\frac{\sqrt{\mu}}{\sqrt{\mu^{2}-(u,u_{0})^{2}}}(u-\frac{(u,u_{0})}{\mu}u_{0}).$

Therefore:

$\begin{aligned} ||\sigma(t)|| &\leq R-1+||u-\frac{(u,u_{0})}{\mu}u_{0}||\\\
&=R-1+||u-u_{0}-\frac{(u-u_{0},u_{0})}{\mu}u_{0}||\\\ &\leq
R-1+\delta_{0}+\frac{(R-1)\delta}{\sqrt{\mu}}.\end{aligned} $

The result follows as $\delta_{0}\leq 1$. ∎

We can now prove the main result of this section:

######  Lemma 11.

Let $\phi:E\rightarrow\mathbb{R}$ such that $\phi^{\prime}$ and
$\phi^{\prime\prime}$ are $\alpha$-Hölder continuous on bounded sets,
$u_{0}\in B(0,R-1)\cap S_{\mu}$, $R>1$, and suppose that one can find
$\beta\in(0,1)$ and a subspace $W\equiv W(u_{0})\subset T_{u_{0}}S_{\mu}$ of
dimension $n+1$ such that:

$D^{2}\phi(u_{0})[w,w]<-\beta||w||^{2}.$

For any $u\in B(S_{\mu};u_{0},2\sqrt{\mu})$ set $W(u)=T_{u}W$ the image of $W$
under radial parallel transport, then for any $u\in
B(S_{\mu};u_{0},\delta_{2})$ and $w\in W(u)$,

$D^{2}\phi(u)[w,w]<-\frac{\beta}{2}||w||^{2},$

where

$\delta_{2}=\begin{cases}\displaystyle\min\left(\sqrt{\mu}\left(1-\cos\left(\frac{\beta}{8\sqrt{\mu}\,K(R)\
C(R,\mu)}\right)\right),\delta_{1},\delta_{0}\right)&\textrm{if
}\displaystyle\frac{\beta}{8\sqrt{\mu}\,K(R)\ C(R,\mu)}<\pi,\\\
\min(\delta_{1},\delta_{0})&\textrm{otherwise}\end{cases}$ (2.14)

with $\delta_{0}>0$ being defined in Lemma 10, $\delta_{1}>0$ in Corollary 1,
$K(R)>0$ and $C(R,\mu)>0$ in (2.9) and (2.13) respectively.

###### Proof.

According to Corollary 1, one can already find $\delta_{1}>0$ satisfying
Equation (2.8) such that for any $u\in B(S_{\mu};u_{0},\delta_{1})$ and any
$w\in W(u_{0})$:

$D^{2}\phi(u)[w,w]<-\frac{3\beta}{4}||w||^{2}.$

Now, fix $u\in B(S_{\mu};u_{0},\delta_{2})$, where $\delta_{2}>0$ satisfies
(2.14), and choose $(\tilde{w}_{i},\dots,\tilde{w}_{n})$ an orthonormal basis
of $T_{u}W$ that diagonalises the bilinear form $D^{2}\phi(u)$. Set for each
$i\in\\{1,\dots,n+1\\}$, $w_{i}=T_{u}^{-1}\tilde{w}_{i}$; this defines an
orthonormal basis of $W$. Using Lemma 10, and the definition of $K(R)$, we
have for any $i\in\\{1,\dots,n+1\\}$:

$\displaystyle|D^{2}\phi(u)[w_{i},w_{i}]-D^{2}\phi(u)[T_{u}w_{i},T_{u}w_{i}]|$
$\displaystyle=|D^{2}\phi(u)[w_{i}-T_{u}w_{i},w_{i}+T_{u}w_{i}]|$
$\displaystyle\leq\left(R+\frac{R-1}{\sqrt{\mu}}\right)\frac{2K(R)}{\mu}|\exp^{-1}_{u_{0}}(u)|$
$\displaystyle=2K(R)C(R,\mu)|\exp^{-1}_{u_{0}}(u)|.$

Note that:

$\frac{(u,u_{0})}{\mu}=\frac{(u-u_{0},u_{0})}{\mu}+1\geq
1-\frac{\delta_{2}}{\sqrt{\mu}}, $

thus, by Equation (2.5):

$|\exp_{u_{0}}^{-1}(u)|\leq\sqrt{\mu}\arccos(1-\frac{\delta_{2}}{\sqrt{\mu}}).$

Thus, for any $u\in B(S_{\mu};u_{0},\delta_{2})$, and any
$i\in\\{1,\dots,n+1\\}$, in view of the condition on $\delta_{2}>0$ given in
(2.14), if $\frac{\beta}{8\sqrt{\mu}\,K(R)\ C(R,\mu)}<\pi$, then

$2K(R)C(R,\mu)|\exp^{-1}_{u_{0}}(u)|\leq\frac{\beta}{4};$

otherwise,

$2K(R)C(R,\mu)|\exp^{-1}_{u_{0}}(u)|\leq
2K(R)C(R,\mu)\frac{\sqrt{\mu}\pi}{2}\leq\frac{\beta}{4}.$

Hence

$D^{2}\phi(u)[T_{u}w_{i},T_{u}w_{i}]<-\frac{\beta}{2}=-\frac{\beta}{2}||T_{u}w_{i}||^{2}.$

Since $T_{u}w_{i}=\tilde{w}_{i}$ by definition, the inequality extends to all
of $W(u)$ by bilinearity. ∎

###  2.4 Fang-Ghoussoub’s result

The remainder of this section is devoted to the proof of Theorem 2, which will
be divided into several lemmata. Our initial goal is to show how to use the
second-order information on a functional $\phi$ in the conclusion of Lemma 11
to find appropriate deformations of an element $A$ of the $n$-dimensional
family $\mathcal{F}$ so as to locally decrease the value of the functional
$\phi$. As in [15], we shall probe $S_{\mu}$ by running out along specific
curves from points in a neighbourhood of a point $u_{0}\in A$; in our case
these curves will be geodesics of the spray $F_{1}$.

Our first lemma identifies, uniformly on a bounded set, a maximal travel time
that does not take us too far on $S_{\mu}$.

######  Lemma 12.

Let $\delta>0$ and assume $u_{0}\in B(0,R)\cap S_{\mu}$, $u\in
B(S_{\mu};u_{0},\frac{\delta}{2})$, $v\in T_{u}S_{\mu}\setminus\\{0\\}$,
$||v|| \leq 1$. Set $t_{max}=\min\\{\frac{2\mu}{R},\frac{\delta}{4} \\}$,
then, for any $0\leq t<t_{max}$:

$\sigma(t,u,v)\in B(u_{0},\delta).$

###### Proof.

Note that since $0<|v|\leq||v||\leq 1$ it follows that
$\sqrt{\mu}\leq\frac{1}{\omega}$, so
$t_{max}\leq\frac{2\mu}{R}\leq\frac{2}{R\omega^{2}}$. Then:

$\displaystyle||\sigma(t,u,v)-u_{0}||$ $\displaystyle<|\cos(\omega
t)|\frac{\delta}{2}+|1-\cos(\omega t)|R+\frac{|\sin(\omega t)|}{\omega}$
$\displaystyle<\frac{\delta}{2}+2|\sin\frac{\omega
t}{2}|\left(|\sin\frac{\omega t}{2}|R+\frac{1}{\omega}\right)$
$\displaystyle<\frac{\delta}{2}+\frac{4}{\omega}|\sin\frac{\omega
t}{2}|\leq\delta.$

∎

We will encounter two geometric issues in the sequel; the first comes from the
fact that we use two different sprays. Since the geodesics of $F_{1}$ differ
from those of $F_{2}$ they do not parallel transport their initial velocity
vector according to the spray $F_{2}$; therefore in general even if
$\sigma^{\prime}(0,u,w)\in W(u)$, $\sigma^{\prime}(1,u,w)\notin
W(\sigma(1,u,w))$. The second is the phenomenon of _holonomy_ 141414The
correct notion is in fact “holonomy of a connection”, but for simplicity we
have not introduced the underlying connection (induced by the spray) in our
text. Therefore we will describe the general idea in different terms than
customary.; this is a global expression of “curvature”. In fact, if we
consider parallel transport along a closed curve $\gamma:[0,1]\rightarrow
S_{\mu}$, $\gamma(0)=\gamma(1)=u_{0}$, one can define a linear operator:
$A:T_{u_{0}}S_{\mu}\rightarrow T_{u_{0}}S_{\mu}$ and, in general,
$A\neq\textrm{id}$. Instead, it is an element of the _holonomy group_ , see
Figure 1.

Figure 1:  General parallel transport of a vector $v_{0}$ along a closed
curve; the result $v_{f}$ depends linearly on $v_{0}$.

This means in particular that parallel transport of a vector along a geodesic
of $F_{1}$ from a point $u_{0}\in S_{\mu}$ to $u_{1}\in S_{\mu}$ is not
equivalent to parallel transport along a geodesic path going from $u_{0}$ to
$u_{1}$ via $u$; thus even if we parallel transport (in the sense defined by
$F_{2}$) a vector in $W(u)$ along a geodesic of $F_{1}$ joining $u$ et $u_{1}$
we cannot be sure that result will lie in $W(u_{1})$.

We shall deal with these issues as follows. Assume that $\phi$ satisfies the
hypotheses of Lemma 11 and let $u_{0}\in B(0,R-1)\cap S_{\mu}$, $u\in
B(S_{\mu};u_{0},\frac{\delta}{2})$ where $\delta\leq\delta_{2}$ with
$\delta_{2}>0$ also given by Lemma 11. Choose $w\in
W(u)=\textrm{span}~{}\\{T_{u}e_{1},\dots,T_{u}e_{n+1}\\}$, where
$(e_{1},\dots,e_{n+1})$ is an orthonormal basis of $W\subset
T_{u_{0}}S_{\mu}$, and assume that $||w||\leq 1$. Consider the curve:
$\sigma(t,u,w)=\exp_{u}(tw)$, for $t\in[0,t_{max})$ where $t_{max}$ is given
by Lemma 12. Let $\tau\in(0,t_{max})$ and set:

$w_{1}=\sigma^{\prime}(\tau,u,w),\quad u_{1}=\sigma(\tau,u,w).$

We begin by an estimate of $||w_{1}-w||$; by assumption:
$\sigma^{\prime\prime}(t,u,w)=-\frac{|w|^{2}}{\mu}\sigma(t,u,w)$, thus:

$||\sigma^{\prime\prime}(t,u,w)||\leq\frac{|w|^{2}}{\mu}R.$

Therefore:

$||w_{1}-w|| \leq\frac{Rt_{\max}}{\mu}.$ (2.15)

As we mentioned above $w_{1}$ may not lie in $W(u_{1})$ so we shall consider:

$\displaystyle Pw_{1}=\sum_{i=1}^{n+1}\langle w_{1},T_{u_{1}}e_{i}\rangle
T_{u_{1}}e_{i},$

the orthogonal projection of $w_{1}$ onto $W(u_{1})$. In order to estimate
$||Pw_{1}-w_{1}||$, we introduce two intermediate vectors. First, let
$w_{0}\in W$ be such that $w=T_{u}w_{0}$, we have:

$w_{0}=T_{u}^{-1}w=\sum_{i=1}^{n+1}\langle w,T_{u}e_{i}\rangle e_{i}.$

Secondly, set $(Pw_{1})_{0}=T_{u_{1}}^{-1}Pw_{1}\in W$, and introduce
$v_{0}=\exp^{-1}_{u_{0}}(u)$, $v_{1}=\exp^{-1}_{u_{0}}(u_{1})$. Using Lemma 10
and Equation (2.15):

$\displaystyle||Pw_{1}-w_{1}||$
$\displaystyle\leq||Pw_{1}-(Pw_{1})_{0}||+||(Pw_{1})_{0}-w_{0}||+||w_{0}-w||+||w-w_{1}||$
$\displaystyle\leq
C(R,\mu)|v_{1}|||Pw_{1}||+||(Pw_{1})_{0}-w_{0}||+C(R,\mu)|v_{0}|\cdot||w||+R\frac{t_{max}}{\mu}$
$\displaystyle\leq
C(R,\mu)(|v_{1}|(1+\frac{Rt_{max}}{\mu})+|v_{0}|)+R\frac{t_{max}}{\mu}+||(Pw_{1})_{0}-w_{0}||.$

Due to holonomy, generally $(Pw_{1})_{0}\neq w_{0}$, but we can estimate:

$\begin{aligned} ||(Pw_{1})_{0}-w_{0}||^{2}&=\sum_{i=1}^{n+1}(\langle
w_{1},T_{u_{1}}e_{i})\rangle-\langle w,T_{u}e_{i}\rangle)^{2}\\\
&=\sum_{i=1}^{n+1}\left(\langle w_{1}-w,T_{u_{1}}e_{i}\rangle+\langle
w,T_{u_{1}}e_{i}-e_{i}\rangle+\langle w,e_{i}-T_{u}e_{i}\rangle\right)^{2}\\\
&\leq(n+1)\left(||w_{1}-w||+(|v_{1}|+|v_{0}|)C(R,\mu)\right)^{2}\\\
&\leq(n+1)\left(\frac{Rt_{max}}{\mu}+(|v_{1}|+|v_{0}|)C(R,\mu)\right)^{2}.\end{aligned}
$

So overall:

$ \begin{aligned} ||Pw_{1}-w_{1}||&\leq
C(R,\mu)(|v_{1}|(1+\frac{Rt_{max}}{\mu})+|v_{0}|)+R\frac{t_{max}}{\mu}+\sqrt{n+1}\left(\frac{Rt_{max}}{\mu}+(|v_{1}|+|v_{0}|)C(R,\mu)\right)\\\
&\leq
C(R,\mu)(3|v_{1}|+|v_{0}|)+R\frac{t_{max}}{\mu}+\sqrt{n+1}\left(\frac{Rt_{max}}{\mu}+(|v_{1}|+|v_{0}|)C(R,\mu)\right)\\\
&\leq(3+\sqrt{n+1})C(R,\mu)(|v_{0}|+|v_{1}|)+(1+\sqrt{n+1})\frac{Rt_{max}}{\mu}.\end{aligned}$

As in the proof of Lemma 11 :

$|v_{0}|\leq\sqrt{\mu}\arccos(1-\frac{\delta}{2\sqrt{\mu}}),|v_{1}|\leq\sqrt{\mu}\arccos(1-\frac{\delta}{\sqrt{\mu}}).$

Hence:

$||Pw_{1}-w_{1}|| \leq
2\sqrt{\mu}(3+\sqrt{n+1})C(R,\mu)\arccos(1-\frac{\delta}{\sqrt{\mu}})+(1+\sqrt{n+1})\frac{R}{\mu}\frac{\delta}{4}.
$

Now let:

$\displaystyle\delta_{3}=\begin{cases}\displaystyle\min\left(\sqrt{\mu}\left(1-\cos\left(\frac{\beta}{\hat{C}}\right)\right),\frac{\beta\mu}{12K(R)R(1+\sqrt{n+1})},\delta_{2}\right)&\textrm{if
$\displaystyle\frac{\beta}{\hat{C}}<\pi,$}\\\
\displaystyle\min\left(\frac{\beta\mu}{12K(R)R(1+\sqrt{n+1})},\delta_{2}\right)&\textrm{otherwise},\end{cases}$
(2.16)

where $\hat{C}:=96\sqrt{\mu}K(R)(3+\sqrt{n+1})C(R,\mu)$. With this choice, for
any $u\in B(S_{\mu};u_{0},\delta_{3})$, if $\frac{\beta}{\hat{C}}<\pi$ we have
setting $K:=K(R)$,

$\displaystyle
2\sqrt{\mu}(3+\sqrt{n+1})C(R,\mu)\arccos(1-\frac{\delta_{3}}{\sqrt{\mu}})$
$\displaystyle\leq$ $\displaystyle
2\sqrt{\mu}(3+\sqrt{n+1})C(R,\mu)\cdot\frac{\beta}{96\sqrt{\mu}K(R)(3+\sqrt{n+1})C(R,\mu)}$
$\displaystyle=$ $\displaystyle\frac{\beta}{48K},$

and

$(1+\sqrt{n+1})\frac{R}{\mu}\frac{\delta_{3}}{4}\leq(1+\sqrt{n+1})\frac{R}{\mu}\frac{1}{4}\cdot\frac{\beta\mu}{12K(R)R(1+\sqrt{n+1})}=\frac{\beta}{36K}.$

Otherwise, we have

$\displaystyle
2\sqrt{\mu}(3+\sqrt{n+1})C(R,\mu)\arccos(1-\frac{\delta_{3}}{\sqrt{\mu}})$
$\displaystyle\leq$ $\displaystyle
2\sqrt{\mu}(3+\sqrt{n+1})C(R,\mu)\cdot\frac{\pi}{2}$ $\displaystyle\leq$
$\displaystyle\sqrt{\mu}(3+\sqrt{n+1})C(R,\mu)\cdot\frac{\beta}{96\sqrt{\mu}K(R)(3+\sqrt{n+1})C(R,\mu)}$
$\displaystyle=$ $\displaystyle\frac{\beta}{96K},$

and $(1+\sqrt{n+1})\frac{R}{\mu}\frac{\delta_{3}}{4}\leq\frac{\beta}{36K}.$
Thus,

$||Pw_{1}-w_{1}|| \leq \frac{\beta}{24K}.$

It follows then that:

$|D^{2}\phi(u_{1})[w_{1},w_{1}]-D^{2}\phi(u_{1})[Pw_{1},Pw_{1}]|
\leq||D^{2}\phi(u)||\cdot||Pw_{1}-w_{1}||\underbrace{||Pw_{1}+w_{1}||}_{\leq
6}\leq 6K\frac{\beta}{24K}=\frac{\beta}{4}.$

Hence:

$D^{2}\phi(u_{1})[w_{1},w_{1}]<-\frac{\beta}{2}||Pw_{1}||^{2}+\frac{\beta}{4}.$

We will now assume that $||w||=1$ and avail ourselves of the assumption that
$\beta\in(0,1)$,

$\displaystyle ||Pw_{1}||$ $\displaystyle\geq||w_{1}||-\frac{\beta}{24K}$
$\displaystyle\geq 1-\frac{Rt_{max}}{\mu}-\frac{\beta}{24K}$
$\displaystyle\geq 1-\frac{\beta}{48K}\left(\frac{1}{1+\sqrt{n+1}}+2\right).$

Since $K\geq 1$, $\frac{1}{1+\sqrt{n+1}}+2\leq 3\leq 3K$, therefore:

$ ||Pw_{1}||\geq 1-\frac{\beta}{16}.$

Hence, using that $\beta\in(0,1)$:

$\displaystyle-\frac{\beta}{2}||Pw_{1}||^{2}+\frac{\beta}{4}$
$\displaystyle\leq-\frac{\beta}{2}\left(1-\frac{\beta}{16}\right)^{2}+\frac{\beta}{4}$
$\displaystyle=-\frac{\beta}{4}+\frac{\beta^{2}}{16}-\frac{\beta^{3}}{512}$
$\displaystyle\leq-\frac{\beta}{6}.$

Overall we arrive at:

$D^{2}\phi(u_{1})[w_{1},w_{1}]<-\frac{\beta}{6}. $

We can now formulate the following adaptation of [15, Lemma 3.3], key to the
proof of Theorem 2.

######  Lemma 13.

Let $\phi:E\rightarrow\mathbb{R}$ a $C^{2}$-functional such that
$\phi^{\prime},\phi^{\prime\prime}$ are $\alpha$-Hölder continuous
($\alpha\in(0,1]$) on bounded sets. Fix $R>1$ and $u_{0}\in B(0,R-1)\cap
S_{\mu}$. Assume that one can find $W\subset T_{u_{0}}S_{\mu}$ of finite
dimension $n+1$ and $\beta\in(0,1)$ such that for any $w\in W$:

$ D^{2}\phi(u_{0})[w,w]<-\beta||w||^{2}.$

Let

$\hat{w}(u)=\begin{cases}-\frac{P_{u}\nabla\phi(u)}{||P_{u}\nabla\phi(u)||}&\textrm{if
$\,\nabla\phi(u)\neq 0$},\\\ 0& \textrm{otherwise},\end{cases}$

where $\nabla\phi$ is the gradient151515The metric dual of the one-form
$\textrm{d}\phi|_{S_{\mu}}$. At each point $u\in S_{\mu}$, this is simply the
orthogonal projection (in the sense of $E$) of the usual gradient
$\nabla^{E}\phi$ in $E$ onto $T_{u}S_{\mu}$, i.e.
$\nabla\phi(u)=\nabla^{E}\phi(u)-(\phi^{\prime}(u)\cdot
Gu)\frac{Gu}{||Gu||^{2}}\in T_{u}S_{\mu}={Gu}^{\perp}.$ of $\phi$ and $P_{u}$
denotes the orthogonal projection (in $T_{u}S_{\mu}$) onto $T_{u}W$.

Then, if $u\in B(S_{\mu};u_{0},\frac{\delta_{3}}{2})$, where $\delta_{3}>0$
satisfies (2.16), and $t_{max}=\min(\frac{2\mu}{R},\frac{\delta_{3}}{4})$
either:

  1. 1.

$\hat{w}(u)\neq 0$ and for any $t\in(0,t_{max})$, we have:

$\phi(\exp_{u}(t\hat{w}(u)))<\phi(u)-\frac{\beta}{12}t^{2}.$

  2. 2.

$\hat{w}(u)=0$, and for any $w\in T_{u}W$, $||w||=1$ and any
$t\in(0,t_{max})$,

$\phi(\exp_{u}(tw))<\phi(u)-\frac{\beta}{12}t^{2}.$

Furthermore, in this case, for any $t_{0}\in(0,t_{max})$, one can find
$0<\delta_{u}<\frac{\delta_{3}}{2}-|u-u_{0}|$ such that for any
$z\in\overline{B}(u,\delta_{u})\cap S_{\mu}$, $w\in T_{z}W$ and
$t\in[t_{0},t_{max})$ we have:

$\phi(\exp_{z}(tw))<\phi(z)-\frac{\beta}{24}t^{2}.$

###### Proof.

For every $w\in T_{u}W$, and any $t\in(0,t_{max})$ the Taylor-Lagrange theorem
guarantees that there is $\tau\in(0,t)$:

$\phi(\exp_{u}(tw))=\phi(u)+t\langle\nabla\phi(u),w\rangle+\frac{t^{2}}{2}D^{2}\phi(\exp_{u}(\tau
w))\left[\left.\frac{d}{dt}(\exp_{u}(tw))\right|_{t=\tau},\left.\frac{d}{dt}(\exp_{u}(tw))\right|_{t=\tau}\right].$

  1. 1.

If $\hat{w}(u)\neq 0$, substitute in the above $w=\hat{w}(u)$ and notice that:

$\langle\nabla\phi(u),\hat{w}(u)\rangle=-1.$

Since $t>0$ it follows that:

$\phi(\exp_{u}(t\hat{w}(u)))<\phi(u)+\frac{t^{2}}{2}D^{2}\phi(\exp_{u}(\tau\hat{w}(u)))\left[\left.\frac{d}{dt}(\exp_{u}(t\hat{w}(u)))\right|_{t=\tau},\left.\frac{d}{dt}(\exp_{u}(t\hat{w}(u)))\right|_{t=\tau}\right].$

  2. 2.

If $\hat{w}(u)=0$, then take any $w\in T_{u}W$ with $||w||=1$, and notice that
$\langle\nabla\phi(u),w\rangle=0$, therefore:

$\phi(\exp_{u}(tw))=\phi(u)+\frac{t^{2}}{2}D^{2}\phi(\exp_{u}(\tau
w))\left[\left.\frac{d}{dt}(\exp_{u}(tw))\right|_{t=\tau},\left.\frac{d}{dt}(\exp_{u}(tw))\right|_{t=\tau}\right].$

In order to conclude the proof in both cases we apply the discussion preceding
the lemma which implies that, for any $u\in
B(S_{\mu};u_{0},\frac{\delta_{3}}{2})$, $w\in T_{u}W$, $||w||=1$ and any
$\tau\in(0,t_{max})$:

$D^{2}\phi(\exp_{u}(\tau
w))\left[\left.\frac{d}{dt}(\exp_{u}(tw))\right|_{t=\tau},\left.\frac{d}{dt}(\exp_{u}(tw))\right|_{t=\tau}\right]<-\frac{\beta}{6}.$
(2.17)

The first two points ensue.

For the final point, by continuity of $\phi^{\prime}$ at $u$, choose
$\delta_{u}<\frac{\delta_{3}}{2}-|u-u_{0}|$ such that for any
$z\in\overline{B}(u,\delta_{u})\cap S_{\mu}$:

$||\phi^{\prime}(u)-\phi^{\prime}(z)||<\frac{\beta}{48}t_{0}.$ (2.18)

We must however take into account the fact that $T_{u}W\neq T_{z}W$ so we need
to approximate $w_{z}\in T_{z}W$ by $w_{u}\in T_{u}W$. To this end, observe
that for every $u$, $T_{u}$ can be viewed as a continuous map
$T_{u}:T_{u_{0}}S_{\mu}\rightarrow T_{u}S_{\mu}\subset E$. The restriction of
$T_{u}$ to $W$, denoted by $T_{u}|_{W}$, can be thought of as an element of
$L(W,E)$. We claim that the map $u\mapsto T_{u}|_{W}$ is continuous.

For this we shall use the fact that $W$ is finite dimensional; choose an
orthonormal basis $(e_{1},\dots,e_{n+1})$ of $W$, if for each
$i\in\\{1,\dots,n+1\\}$: $u\mapsto T_{u}e_{i}$ is continuous, it will follow
that $u\mapsto T_{u}|_{W}$ is continuous. Indeed, let $\varepsilon>0$ and
observe that $||\cdot||$ and161616$(e_{i}^{*})$ is the dual basis to $(e_{i})$
$|w|_{\infty}=\max_{i\in\\{1,\dots,n+1\\}}|e^{*}_{i}(w)|$ define equivalent
norms on $W$. Therefore, if $u\in B(S_{\mu};u_{0},\frac{\delta_{3}}{2})$ is
fixed and $\tilde{\delta}$ is chosen small enough such that for every
$i\in\\{1,\dots,n+1\\}$ $z\in\overline{B}(u,\tilde{\delta})\cap S_{\mu}$:

$||T_{u}e_{i}-T_{z}e_{i}|| \leq\frac{\varepsilon}{n+1},$

we have

$||T_{u}w-T_{z}w||\leq\varepsilon|w|_{\infty}\leq
C\varepsilon||w||,~{}~{}~{}\textrm{for all $w\in W$,}$

where $C>0$ is a constant independent of $w\in W$. Therefore $u\mapsto
T_{u}|_{W}$ is continuous. It remains to justify that the map $u\mapsto
T_{u}e_{i}$ is continuous for each $i\in\\{1,\dots,n+1\\}$. Fix such an $i$,
and recall that $T_{u}e_{i}$ is defined to be $\varphi(1)$, where $\varphi$ is
the solution to the Cauchy problem:

$\varphi^{\prime}(t)+(\varphi(t),\frac{d}{dt}(\exp_{u_{0}}(tv))\frac{G\exp_{u_{0}}(tv)}{||G\exp_{u_{0}}(tv)||^{2}}=0,\quad\varphi(0)=e_{i},$

with $v=\exp^{-1}_{u_{0}}u$. Denoting the differential of the map
$v\mapsto\exp_{u_{0}}v$ at a point $v$ by ${\exp_{u_{0}}}_{*v}$, the above can
be rewritten as a system:

$\begin{cases}\varphi^{\prime}(t)+(\varphi(t),{\exp_{u_{0}}}_{*t\psi(t)}\cdot\psi(t))\frac{G\exp_{u_{0}}(t\psi(t))}{||G\exp_{u_{0}}(t\psi(t))||^{2}}=0,\\\
\psi^{\prime}(t)=0,\end{cases}$

with initial conditions: $\psi(0)=v$, $\varphi(0)=e_{i}$.

By continuity with respect to initial conditions, the solution, and a
fortiori, $\varphi(1)$, depend continuously on $v$, however
$v=\exp_{u_{0}}^{-1}(u)$, so $\varphi(1)$ and thus $T_{u}e_{i}$ depend
continuously on $u$. This proves our claim. Consequently, reducing further, if
necessary, $\delta_{u}$ we can assume that for any
$z\in\overline{B}(u,\delta_{u})\cap S_{\mu}$:

$||T_{u}|_{W}-T_{z}|_{W}||<\frac{\beta\,t_{0}}{48K(R)},$ (2.19)

where $K(R)$ is defined in (2.9). Recall now that, for any $w\in T_{z}W$,
$||w||=1$, $z\in\overline{B}(u,\delta_{u})\cap S_{\mu}\subset
B(S_{\mu};u_{0},\frac{\delta_{3}}{2})$, $t\in[t_{0},t_{max})$ one can find
$\tau\in(0,t)$ such that:

$\phi(\exp_{z}(tw))=\phi(z)+t\phi^{\prime}(z)\cdot
w+\frac{t^{2}}{2}D^{2}\phi(\exp_{z}(\tau
w))\left[\left.\frac{d}{dt}(\exp_{z}(tw))\right|_{t=\tau},\left.\frac{d}{dt}(\exp_{z}(tw))\right|_{t=\tau}\right].$

If we write $w_{0}=T_{z}^{-1}w$, then $\phi^{\prime}(u)\cdot T_{u}w_{0}=0$,
and, using Lemma 9, Equations (2.9), (2.18) and (2.19):

$\displaystyle|\phi^{\prime}(z)\cdot w|$ $\displaystyle=|\phi^{\prime}(z)\cdot
T_{z}w_{0}-\phi^{\prime}(u)\cdot T_{u}w_{0}|$
$\displaystyle\leq|\phi^{\prime}(z)\cdot T_{z}w_{0}-\phi^{\prime}(u)\cdot
T_{z}w_{0}|+|\phi^{\prime}(u)\cdot T_{z}w_{0}-\phi^{\prime}(u)\cdot
T_{u}w_{0}|$
$\displaystyle\leq||\phi^{\prime}(z)-\phi^{\prime}(u)||\cdot||T_{z}w_{0}||+||\phi^{\prime}(u)||\cdot||T_{z}w_{0}-T_{u}w_{0}||$
$\displaystyle\leq\frac{\beta}{48}t_{0}+K(R)\cdot\frac{\beta t_{0}}{48K(R)}$
$\displaystyle=\frac{\beta}{24}t_{0}.$

Since $z\in\overline{B}(u_{0},\frac{\delta_{3}}{2})\cap S_{\mu}$, we can
estimate the second order term using Equation (2.17) and thus, for
$t\in[t_{0},t_{\max})$:

$\displaystyle\phi(\exp_{z}(tw))$
$\displaystyle<\phi(z)+t\frac{\beta}{24}t_{0}-\frac{\beta}{12}t^{2}$
$\displaystyle<\phi(z)-\frac{\beta}{24}t^{2}.$

∎

Having proved the key Lemma 13, in our next two lemmas we follow closely [15,
Lemma 3.4 and Lemma 3.5].

######  Lemma 14.

Assume that the hypotheses of Lemma 13 are satisfied and let $f$ be a
continuous map from a closed subset $D\subset\mathbb{R}^{n}$ into $S_{\mu}$.
Suppose that $K_{1}\subset D$ is a compact subset of $D$ such that
$f(K_{1})\subset B(S_{\mu};u_{0},\frac{\delta_{3}}{2})$. Then for sufficiently
small $\nu>0$ and $t_{0}\in(0,t_{max})$ there is a continuous map
$\eta:[0,t_{max}]\times D\rightarrow S_{\mu}$ that satisfies:

  1. 1.

$\eta(t,x)=f(x)$, if $(t,x)\in(\\{0\\}\times D) \cup([0,t_{max}]\times
D\setminus N_{\nu}(K_{1}))$;

  2. 2.

$\phi(\eta(t,x))\leq\phi(f(x))$ if $(t,x)\in[0,t_{max})\times D$;

  3. 3.

$\phi(\eta(t,x))<\phi(f(x))-\frac{\beta}{24}t^{2}$ if
$(t,x)\in[t_{0},t_{max})\times K_{1}$;

  4. 4.

$||\eta(t,x)-f(x)||\leq 3t$ for all $(t,x)\in[0,t_{max}]\times D$.

###### Proof.

Let $N_{\nu}(K_{1})$ denote the $\nu$-neighbourhood of $K_{1}$ in
$\mathbb{R}^{n}$, namely,

$N_{\nu}(K_{1})=\\{x\in\mathbb{R}^{n}\,|\,d(x,K_{1})<\nu\\}.$

Assume that $\nu$ is small enough so that, $f(N_{\nu}(K_{1})\cap D)\subset
B(S_{\mu};u_{0},\frac{\delta_{3}}{2})$ and define:

$T=\\{x\in N_{\nu}(K_{1})\cap D\,|\,w(f(x))=0\\}.$

By the last point in Lemma 13, for every $y\in T$ one can find $\nu^{y}>0$
such that for any $z\in B(y,\nu^{y})\cap D$, and any $w\in T_{f(z)}W$,
$||w||=1$ we have the inequality:

$\phi(\exp_{f(z)}(tw))<\phi(f(z))-\frac{\beta}{24}t^{2},\quad
t\in[t_{0},t_{max}).$

Put $O=\cup_{y\in T}B(y,\frac{\nu^{y}}{2})$ and let
$g:\mathbb{R}^{n}\rightarrow[0,1]$ be a continuous function such that:

$g(x)=\begin{cases}1&x\in K_{1},\\\ 0&x\in\mathbb{R}^{n}\setminus
N_{\nu}(K_{1}).\end{cases}$

Next, choose an orthonormal basis $(e_{1},\dots,e_{n})$ of $W \subset
T_{u_{0}}S_{\mu}$ and let $f_{1}:D\setminus O\rightarrow
S^{n}\subset\mathbb{R}^{n+1}$ be defined by:

$f_{1}(x)=(\langle T_{f(x)}e_{1},w(f(x))\rangle,\dots,\langle
T_{f(x)}e_{n+1},w(f(x))\rangle). $

As in the proof of Lemma 13, the continuity of the map: $u\mapsto T_{u}|_{W}$
guarantees that $f_{1}$ is itself continuous. Since $D\setminus O$ is closed
in $\mathbb{R}^{n}$ one can extend $f_{1}$ to a continuous map
$f_{2}:\mathbb{R}^{n}\rightarrow S^{n}$ by Lemma 2. Define now a continuous
map $f_{3}$ on $D$:

$f_{3}(x)=\sum_{i=1}^{n+1}E_{i}^{*}(f_{2}(x))T_{f(x)}e_{i},\in T_{f(x)}W, $

where we denote $(E_{i}^{*})$ the dual basis of the canonical basis
$(E_{1},\dots,E_{n+1})$ of $\mathbb{R}^{n+1}$, then set

$\eta(t,x)=\exp_{f(x)}(tg(x)f_{3}(x)),x\in D,t\in[0,t_{max}].$

It is straight forward to check that $\eta$ satisfies the required conditions.
The last point follows from Equation (2.15) and the mean value theorem. ∎

######  Lemma 15.

Let $\phi:E\rightarrow\mathbb{R}$ a $C^{2}$-functional such that
$\phi^{\prime},\phi^{\prime\prime}$ are $\alpha$-Hölder continuous
($\alpha\in(0,1]$) on bounded sets and let $f$ be a continuous map from a
closed subset $D$ of $\,\mathbb{R}^{n}$ into $S_{\mu}$. Suppose $K_{2}$ is a
compact subset of $D$ with the following properties:

  * •

There exists $R>1$ such that $f(K_{2})\subset B(0,R-1)\cap S_{\mu}$.

  * •

There exists a constant $\beta>0$ such, that for all $y\in K_{2}$, there is a
subspace $W_{y}$, of $T_{f(y)}S_{\mu}$ with $\dim\,W_{y}\geq n+1$ so that

$D^{2}\phi(f(y))[w,w]<-\beta||w||^{2},\quad\mbox{for all }\,w\in W_{y}.$
(2.20)

Then for any $0<\delta\leq\delta_{3}$ where $\delta_{3}>0$ satisfies (2.16)
and $\nu>0$ there is a continuous map $\hat{f}:D\rightarrow S_{\mu}$ such that
if $N:=N(n)$ is the number given in Lemma 1, we have :

  * (i)

$\hat{f}(x)=f(x)$ for $u\in D\backslash N_{\nu}(K_{2})$;

  * (ii)

$\phi(\hat{f}(x))\leq\phi(f(x))$ for all $x\in D$;

  * (iii)

If $x\in K_{2}$, then
$\phi(\hat{f}(x))<\displaystyle\phi(f(x))-\frac{\beta\,\delta^{2}}{864N^{2}};$

  * (iv)

$||\hat{f}(x)-f(x)||\leq\frac{\delta}{2}$ for all $x\in D$.

###### Proof.

Let $0<\delta\leq\delta_{3}$ be fixed. For any $y\in K_{2}$ we can choose a
ball $B_{y}$ in $\mathbb{R}^{n}$ such that $\overline{B}_{y}\subset
N_{\nu}(K_{2})$ and $f(x)\in B(S_{\mu};f(y),\frac{\delta}{8N})$ for all
$x\in\overline{B}_{y}\cap D$ by continuity of $f$. Since $K_{2}$ is compact,
there is a finite subcovering $B_{y_{1}},\cdots,B_{y_{n}}$ of $K_{2}$.
Choose $0<\tau<\nu$ small enough such that
$\cup_{i=1}^{m}N_{\tau}(B_{y_{i}})\subset N_{\nu}(K_{2}),$ and for $1\leq
1\leq m,$

$f(x)\in B\Big{(}S_{\mu};f(y_{i}),\frac{\delta}{4N}\Big{)},\quad
x\in\overline{N}_{\tau}(B_{y_{i}})\cap D.$

Using Lemma 1, we may assume that any $N$ possible distinct
$\overline{N}_{r}(B_{y_{i}})^{\prime}s$ have an empty intersection.
We shall now define by induction, continuous functions
$f_{0},f_{1},\cdots,f_{m}:D\rightarrow S_{\mu}$ such that for all $1\leq i\leq
m$ we have that

$\phi(f_{i}(x))<\phi(f_{i-1}(x))-\frac{\beta\,\delta^{2}}{864N^{2}}\quad\mbox{if
}x\in\overline{B}_{y_{i}}\cap D,$ (2.21)
$\phi(f_{i}(x))\leq\phi(f_{i-1}(x))\quad\mbox{if }x\in D,$ (2.22)

and

$||f_{i}(x)-f_{i-1}(x)||\leq\begin{cases}0\quad\mbox{if }x\in D\backslash
N_{\tau}(\overline{B}_{y_{i}}),\\\ \frac{\delta}{2N}\quad\mbox{if }x\in
N_{\tau}(\overline{B}_{y_{i}})\cap D.\end{cases}$ (2.23)

Let $f_{0}=f$ and suppose that $f_{0},f_{1},\cdot,f_{k}$ are well-defined and
satisfy inequalities (2.21), (2.22) and (2.23) for $k<m.$ Clearly

$||f_{i}(x)-f(x)||\leq\frac{i\delta}{2N}\quad\mbox{if
}x\in\bigcap_{j=1}^{i}N_{\tau}(\overline{B}_{y_{i}})\cap D.$

Since any intersection of $N$ distinct sets $N_{\tau}(B_{y_{i}})$ is empty, we
have that

$||f_{k}(x)-f(x)||\leq\frac{\delta(N-1)}{2N}\quad\mbox{if }x\in D.$

Since $f:\overline{B}_{y_{k+1}}\cap D\rightarrow
B(S_{\mu};f(y_{k+1}),\frac{\delta}{4N})$, we see that $f_{k}$ maps
$\overline{B}_{u_{k+1}}\cap D$ into
$B(S_{\mu};f(y_{k+1}),\frac{\delta}{2}(1-\frac{1}{N}+\frac{\delta}{4N})\subset
B(S_{\mu};f(y_{k+1}),\frac{\delta}{2}).$ By assumption (2.20), there is some
subspace $W_{y_{k+1}}$ of $E$ with $\dim W_{y_{k+1}}\geq n+1$ such that for
any $w\in W_{y_{k+1}},$ with $||w||=1,$ we have that
$D^{2}\phi(f(y_{k+1}))[w,w]<-\beta$. Hence, we may apply Lemma 14 with $f_{k}$
and any $t_{0}\in(0,t_{max})$ to obtain a continuous deformation $\eta(t,x)$
satisfying the conclusion of that lemma. Define now
$f_{k+1}(x)=\eta(\frac{\delta}{6N},x)$ to get a continuous function
$f_{k+1}:D\rightarrow S_{\mu}$ satisfying

$\phi(f_{k+1}(x))<\phi(f_{k}(x))-\frac{\beta\,\delta^{2}}{864N^{2}}\quad\mbox{for
}x\in\overline{B}_{y_{k+1}}\cap D,$
$\phi(f_{k+1}(x))\leq\phi(f(x))\quad\mbox{for }x\in D,$

and

$||f_{k+1}(x)-f_{k}(x)||\leq\begin{cases}0\quad\mbox{if }x\in D\backslash
N_{\tau}(\overline{B}_{y_{k+1}}),\\\ \frac{\delta}{2N}\quad\mbox{if }x\in
N_{\tau}(\overline{B}_{y_{k+1}})\cap D.\end{cases}$

By induction we see that $f_{0},\cdots,f_{m}$ are well-defined. Clearly
$\hat{f}=f_{m}$ verifies the claims of the lemma. ∎

Finally, we shall need the following lemma which follows directly from [30,
Lemma 5.15] used with $S=f(K_{3})$ and $V=S_{\mu}$.

######  Lemma 16.

Let $\phi$ be a $C^{1}$ functional on $E$ and let $f$ be a continuous map from
a closed subset $D$ of $\,\mathbb{R}^{n}$ into $S_{\mu}$. Let $\tilde{c}$,
$\tilde{\varepsilon}$, $\tilde{\mu}>0$ be three constants. Suppose $K_{3}$ is
a compact subset of $D$ satisfying

$\tilde{c}-\tilde{\varepsilon}\leq\phi(f(x))\leq\tilde{c}+\tilde{\varepsilon},\quad\mbox{for
}x\in K_{3}.$

Assume that, for all $x\in K_{3}$,

$||\phi^{\prime}|_{S_{\mu}}(u)||\geq\frac{8\tilde{\varepsilon}}{\tilde{\mu}},\quad\mbox{for
}u\in B(S_{\mu};f(x),2\tilde{\mu}),$

then there is a continuous map $\hat{f}:D\rightarrow S_{\mu}$ such that

  * (i)

$\hat{f}(x)=f(x)$ if $\,\phi(f(x))\leq\tilde{c}-2\tilde{\varepsilon}$;

  * (ii)

$\phi(\hat{f}(x)))\leq\phi(f(x))$ for all $x\in D$;

  * (iii)

If $x\in K_{3}$, then $\phi(\hat{f}(x))\leq\tilde{c}-\tilde{\varepsilon}.$

We now have all the ingredients to prove the main result of this section.

###### Proof of Theorem 2.

Suppose $\max_{u\in A}\phi(u)\leq c+\varepsilon$ where $A$ is a set in
$\mathcal{F}$ satisfying (2.1). There exists a continuous function $f$ from
$D\subset\mathbb{R}^{n}$ into $S_{\mu}$, which is equal to $\sigma$ on $D_{0}$
and such that $A=f(D)$.

Let $\delta_{3}>0$ satisfy the equality in (2.16). Observing that it is not
restrictive to assume that $\alpha\leq\frac{1}{2}$ in condition (1.3) we can
find a constant $\gamma>0$ such that
$\gamma\beta^{\frac{1}{\alpha}}\leq\delta_{3}$ for any $\beta>0$ small enough.
Now let

$\delta=\delta(\varepsilon)=\frac{1}{2}\min\Big{(}\frac{\varepsilon^{\alpha_{1}}}{M},\gamma\,\varepsilon^{\frac{\alpha_{1}}{\alpha}}\Big{)}$
(2.24)

and observe that $\delta\leq\delta_{3}$ when $\beta>0$ is given by
$\beta=\varepsilon^{\alpha_{1}}$.

Consider the closed set

$K=\\{x\in D\,|\,\phi(f(x))\geq c-\varepsilon\\}.$ (2.25)

Since $c-\max_{B}\phi>0$, taking $\varepsilon>0$ sufficiently small, we can
assume that $K$ is a compact subset of $D\backslash D_{0}$ and that
$c-2\varepsilon>\max_{B}\phi$.
Now suppose that the conclusion of the theorem does not hold, then for all
$x\in K$, we have either
$||\phi^{\prime}|_{S_{\mu}}(f(x))||>3\varepsilon^{\alpha_{1}}$ or there is a
subspace $W_{x}$ of $T_{f(x)}S_{\mu}$ with $\dim W_{x}\geq n+1$ such that for
all $w\in W_{x}$, we have
$D^{2}\phi(f(x))[w,w]<-\varepsilon^{\alpha_{1}}||w||^{2}$.

In view of the assumption HOL 2 on $\phi^{\prime}$ and of the definition of
$\delta>0$ given in (2.24), setting $\hat{\delta}=2\delta$ we deduce that for
all $x\in K$ such that
$||\phi^{\prime}|_{S_{\mu}}(f(x))||>3\varepsilon^{\alpha_{1}}$ we have
$||\phi^{\prime}|_{S_{\mu}}(u)||>\varepsilon^{\alpha_{1}}$ for all $u\in
B(f(x),\hat{\delta})$. Let

$T_{1}=\\{x\in
K\,|\,||\phi^{\prime}|_{S_{\mu}}(u)||>\varepsilon^{\alpha_{1}},\mbox{ for all
}u\in B(S_{\mu};f(x),\hat{\delta})\\}$

and $T_{2}=K\backslash T_{1}$. Note that $\overline{T}_{1}$,
$\overline{T}_{2}$ are compact, and $K=\overline{T}_{1}\cup\overline{T}_{2}$.
Now apply Lemma 15 with $K_{2}=\overline{T}_{2}$,
$\beta=\varepsilon^{\alpha_{1}}$ and $\nu=\frac{1}{2}{\rm dist}(D_{0},K)>0$ to
obtain a continuous map $g:D\rightarrow S_{\mu}$ such that

$g(x)=f(x)\quad\mbox{for }x\in D\backslash
N_{\nu}(\overline{T}_{2})\quad\mbox{and}\quad\phi(g(x))\leq\phi(f(x)),\quad\mbox{for
}x\in D,$ (2.26)
$\phi(g(x))\leq\phi(f(x))-\frac{\varepsilon^{\alpha_{1}}\delta^{2}}{864N^{2}},\quad\mbox{for
}x\in\overline{T}_{2},$ (2.27)
$||g(x)-f(x)||\leq\frac{\delta}{2},\quad\mbox{for }x\in
N_{\nu}(\overline{T}_{2})\cap D.$ (2.28)

Observe that Inequality (2.28) yields for $x\in\overline{T}_{1},$
$B(S_{\mu};g(x),\delta)\subset B(S_{\mu};f(x),\hat{\delta})$. Now we apply
Lemma 16 with $\tilde{c}=c$, $\tilde{\varepsilon}=\varepsilon$ and
$\tilde{\mu}=\frac{\delta}{2}$. Observing that since
$\alpha_{1}<\frac{\alpha}{\alpha+1}$ we have that

$\varepsilon^{\alpha_{1}}\geq\frac{16\varepsilon}{\delta},\quad\mbox{if
}\varepsilon>0,\mbox{ is sufficiently small.}$

we deduce that there exists a continuous map $\hat{f}:D\rightarrow S_{\mu}$
such that

$\hat{f}(x)=g(x)\quad\mbox{if }\phi(g(x))\leq
c-2\varepsilon\quad\mbox{and}\quad\phi(\hat{f}(x))\leq\phi(f(x)),\quad\mbox{for}\quad
x\in D,$ (2.29) $\phi(\hat{f}(x))\leq c-\varepsilon,\quad\mbox{for}\quad
x\in\overline{T}_{1},$ (2.30)

Note that $\hat{f}(D)\in\mathcal{F}$. In view of (2.27), (2.29), (2.30) and of
the definition of $K$ given in (2.25) to get a contradiction we just need to
show that

$\frac{\varepsilon^{\alpha_{1}}\delta^{2}}{864N^{2}}>2\varepsilon,\quad\mbox{for}\quad
x\in\overline{T}_{2}.$ (2.31)

But, for $\varepsilon>0$ small enough,

$\delta^{2}=\frac{4}{4}\gamma^{2}\varepsilon^{\frac{2\alpha_{1}}{\alpha}}$

and since $\alpha_{1}<\frac{\alpha}{\alpha+2}$, Equation (2.31) will hold if
$\varepsilon>0$ is small enough.

Summarizing, we managed to construct, assuming that the conclusion of the
theorem does not hold, a path in $\mathcal{F}$ for which the maximum of $\phi$
on this path is strictly below the value of $c$. This contradiction ends the
proof. ∎

##  3 Proof of Theorem 1

This section is devoted to the

###### Proof of Theorem 1.

Since, for each $u\in E$, the function $\rho\mapsto\Phi_{\rho}(u)$ is non-
increasing, the function $\rho\mapsto c_{\rho}$ is non-increasing as well.
Therefore, its derivative $c_{\rho}^{\prime}$ is well defined for almost every
$\rho\in I$. We show that the existence of $c_{\rho}^{\prime}$ ensures that of
the desired Palais-Smale sequence. Let then $\rho\in J^{\circ}$ (the interior
of $J$) be such that $c_{\rho}^{\prime}$ exists, and let
$\\{\rho_{n}\\}\subset J$ be a monotone increasing sequence converging to
$\rho$.

_Step 1)_ There exist $\\{\gamma_{n}\\}\subset\Gamma$ and
$K=K(c_{\rho}^{\prime})>0$ such that:

  * (i)

$\|\gamma_{n}(t)\|\leq K$ whenever

$\Phi_{\rho}(\gamma_{n}(t))\geq
c_{\rho}-(2-c_{\rho}^{\prime})(\rho-\rho_{n});$

  * (ii)

$\displaystyle\max_{t\in[0,1]}\Phi_{\rho}(\gamma_{n}(t))\leq
c_{\rho}+(2-c_{\rho}^{\prime})(\rho-\rho_{n}).$

The proof of this step is completely analogue to the one of [17, Proposition
2.1], and hence is omitted.

_Step 2)_ Let $K=K(c_{\rho}^{\prime})$ be the constant found in Step 1, and
let $M\geq 1$ be the constant of the assumption HOL 2 corresponding to the
value $R=K+1$. Finally, let
$\varepsilon_{n}:=(2-c_{\rho}^{\prime})(\rho-\rho_{n})\to 0^{+}$, (where
$\\{\rho_{n}\\}$ is the sequence introduced at Step 1). For each
$n\in\mathbb{N}$ large enough, Theorem 2 guarantees the existence of $u_{n}\in
S_{\mu}$ with the following properties:

  * (a)

$c_{\rho}-\varepsilon_{n}\leq\Phi_{\rho}(u_{n})\leq c_{\rho}+\varepsilon_{n}$;

  * (b)

$\|\Phi^{\prime}_{\rho}|_{S_{\mu}}(u_{n})\|\leq
3\varepsilon_{n}^{\alpha_{1}}$;

  * (c)

$u_{n}\in\gamma_{n}([0,1])$;

  * (d)

$\tilde{m}_{\varepsilon_{n}^{\alpha_{1}}}(u_{n})\leq 1$.

_Step 3) Conclusion of the proof._ Recalling that
$\varepsilon_{n}=(2-c_{\rho}^{\prime})(\rho-\rho_{n})\to 0^{+}$, we claim that
$\\{u_{n}\\}\subset S_{\mu}$ with $u_{n}\in S_{\mu}$ satisfying (a)-(d) of
Step 2 is the Palais-Smale sequence we are looking for in Theorem 1.
From (a) and (b) of Step 2, it is a Palais-Smale sequence at the level
$c_{\rho}$. From (a) and (c) we also have

$u_{n}\in\gamma_{n}([0,1])\cap\\{\Phi_{\rho}\geq
c_{\rho}-(2-c_{\rho}^{\prime})(\rho-\rho_{n})\\},$

so that $\|u_{n}\|\leq K$, thanks to Step 1; that is, $\\{u_{n}\\}\subset
S_{\mu}$ is bounded. Finally, we observe that, since
$\varepsilon_{n}^{\alpha_{1}}\to 0^{+}$, (d) of Step 2 implies (iv) of Theorem
1. At this point the proof of the theorem is completed. ∎

###### Proof of Remark 1.5.

The remark follows directly from the observation that it is possible to
replace in Step 1 of the proof of Theorem 1, $\gamma_{n}\in\Gamma$ by
$|\gamma_{n}|_{*}\in\Gamma$. Since $u_{n}\in|\gamma_{n}|_{*}([0,1])$, this
gives the additional property $u_{n}\geq 0$. ∎

When the sequence $\\{u_{n}\\}\subset S_{\mu}$ obtained in Theorem 1 converges
its limit is a critical point of $\Phi_{\rho}|_{S_{\mu}}$. In this case, the
information about the approximate Morse index of $\\{u_{n}\\}\subset S_{\mu}$,
in Theorem 1 (iv), can be used to infer information about the Morse index of
this critical point.

We recall, see for example [1, Definition 2.5], that

######  Definition 5.

If $u\in S_{\mu}$ is a critical point of $\Phi_{\rho}|_{S_{\mu}}$ with
Lagrange parameter $\lambda\in\mathbb{R}$.

  * 1)

The Morse index $m(u)\in\mathbb{N}\cup\\{0,\infty\\}$ of $u$ with respect to
$\Phi_{\rho}|_{S_{\mu}}$ is defined as

$m(u)=\sup\left\\{\dim\,L\left|\begin{array}[]{l}\ L\text{ is a subspace of
$T_{u}S_{\mu}$ such that:~{}$\forall\varphi\in L\backslash\\{0\\},\,$
}D^{2}\Phi_{\rho}(u)[\varphi,\varphi]<0\end{array}\right.\right\\}.$

  * 2)

The free Morse index $m_{f}(u)\in\mathbb{N}\cup\\{0,\infty\\}$ of $u$ is
defined as

$m_{f}(u)=\sup\left\\{\dim\,L\left|\begin{array}[]{l}\ L\text{ is a subspace
of $E$ such that:~{}$\forall\varphi\in L\backslash\\{0\\},\,$
}D^{2}\Phi_{\rho}(u)[\varphi,\varphi]<0\end{array}\right.\right\\}.$

######  Remark 3.1.

Recall that, for critical points $u\in S_{\mu}$ of $\Phi_{\rho}|_{S_{\mu}}$,
the restriction of $D^{2}\Phi_{\rho}(u)$ to $T_{u}S_{\mu}$ coincides with the
Hessian of $\Phi_{\rho}|_{S_{\mu}}$ at $u$. See Remark 2.3.

######  Theorem 3.

In the setting of Theorem 1, we assume that the sequence $\\{u_{n}\\}\subset
S_{\mu}$ converges to some $u\in S_{\mu}$. Then

  * 1)

Setting

$\lambda_{\rho}:=\frac{1}{\mu}\Phi^{\prime}_{\rho}(u)\cdot
u=\lim_{n\to\infty}\frac{1}{\mu}\Phi^{\prime}_{\rho}(u_{n})\cdot u_{n},$

we have

$\Phi^{\prime}_{\rho}(u)-\lambda_{\rho}\langle Gu,\cdot\rangle=0,\quad\mbox{in
}E^{\prime}.$ (3.1)

Here $G:E\rightarrow E$ is the injective linear map defined in (2.10).
Equation (3.1) indicates that $u\in S_{\mu}$ is a constrained critical point
of $\Phi_{\rho}$ with Lagrange parameter $\lambda_{\rho}\in\mathbb{R}$.
Alternatively, $u\in S_{\mu}$ can be viewed as a free critical point of the
functional $\Phi_{\rho}-\lambda_{\rho}Gu$ defined on $E$.

  * 2)

The Morse index $m(u)$ of $u$ is at most $1$.

  * 3)

The free Morse index $m_{f}(u)$ of $u$ is at most $2$.

###### Proof.

Using Equation (1.6) in Remark 1.3, we immediately deduce that 1) holds. To
show that $m(u)\leq 1$ we assume by contradiction that there exists a
$W_{0}\subset T_{u}S_{\mu}$ with $\dim W_{0}=2$ such that

$D^{2}\Phi_{\rho}(u)[w,w]<0,\quad\mbox{for all }w\in W_{0}\backslash\\{0\\}.$

Since $W_{0}$ is of finite dimension, by compactness and homogeneity, there
exists a $\beta_{0}>0$ such that

$D^{2}\Phi_{\rho}(u)[w,w]<-\beta_{0}||w||^{2},\quad\mbox{for all }w\in W_{0}.$

Now, from Corollary 1 we deduce, for $\delta_{1}>0$ given by Equation (2.8),
that for any $v\in B(S_{\mu};u,\delta_{1})$,

$ D^{2}\Phi_{\rho}(v)[w,w]<-\frac{3\beta_{0}}{4}||w||^{2},\quad\mbox{for all
}w\in W_{0}.$ (3.2)

Since $\\{u_{n}\\}\subset S_{\mu}$ converges to $u$ we have that $u_{n}\in
B(S_{\mu};u,\delta_{1})$ for $n\in\mathbb{N}$ large enough. Then since $\dim
W_{0}>1$, (3.2) provides a contradiction with Theorem 1 (iv) where we recall
that $\zeta_{n}\to 0^{+}$. This proves 2). Finally, recording that $S_{\mu}$
is of codimension 1 in $E$ we immediately obtain that $m_{f}(u)\leq 2$. ∎

######  Remark 3.2.

If in Theorem 1 the conclusion only holds for almost every $\rho\in I$ this is
due to the fact that it is not known if, for a given $\rho\in I$, the
functional $\Phi_{\rho}$ admits a sequence of paths
$\\{\gamma_{n}\\}\subset\Gamma$ as in Step 1 in the proof of Theorem 1. For a
functional for which this is known a priori Theorem 2 directly implies the
existence of sequences $\\{u_{n}\\}\subset S_{\mu}$ and $\zeta_{n}\to 0^{+}$
such that, as $n\to+\infty$, the properties (i)-(iv) in Theorem 1 hold.

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