# Bouncing Jacobi Fields and the Allen-Cahn equation on surfaces

Yong Liu  Department of Mathematics, University of Science and Technology of
China, Hefei, China<EMAIL_ADDRESS>,  Frank Pacard  Centre de
Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau, France.
<EMAIL_ADDRESS>and  Juncheng Wei  Department of Mathematics,
University of British Columbia, Vancouver, B.C. Canada, V6T 1Z2.
<EMAIL_ADDRESS>
###### Abstract.

The Allen-Cahn functional is a well studied variational problem which appears
in the modeling of phase transition phenomenon. This functional depends on a
parameter $\varepsilon>0$ and is intimately related to the area functional as
the parameter $\varepsilon$ tends to $0$. In the case where the ambient
manifold is a compact surface, we give sufficient assumptions which guarantee
the existence of countable families of critical points of the Allen-Cahn
functional whose nodal sets converge with multiplicity $2$ to a given embedded
geodesic, while their energies and Morse indices stays uniformly bounded, as
the parameter $\varepsilon$ tends to $0$. This result is specific to two
dimensional surfaces and, for generic metric, it does not occur in higher
dimension.

##  1\. Introduction

Given a compact $m$-dimensional Riemannian manifold $(M,g)$ without boundary
and a positive parameter $\varepsilon>0$, the Allen-Cahn equation arises as
the Euler-Lagrange equation of the energy functional

(1.1) $\mathcal{E}_{AC}(u):=\frac{\varepsilon}{2}\int_{M}|\nabla
u|_{g}^{2}\,\text{dvol}_{g}+\frac{1}{4\varepsilon}\,\int_{M}(1-u^{2})^{2}\,\text{dvol}_{g},$

which models a phase transition phenomenon. A classical result of L. Modica
[17] asserts that, under natural assumptions, the Allen-Cahn functional
$\Gamma$-converges to the volume functional. This seminal result has triggered
many studies to unravel the deep connections between the Allen-Cahn equation

(1.2) $\varepsilon^{2}\Delta_{g}u+u-u^{3}=0,$

which is the Euler-Lagrange equation associated to the functional (1.1) and
minimal surfaces which are critical points of the volume functional.

The zero set of solutions of (1.2) correspond to the _transition layers_ in
the phase transition phenomenon and the $\Gamma$-convergence theory tells us
that (under some natural assumption on their energy and Morse indices) the
zero sets of sequences of solutions of the Allen-Cahn equation converge to
minimal surfaces (eventually with multiplicity).

In the past decade, we have seen important advances in the min-max theory of
minimal surfaces in the varifold setting by C. Marques and A. Neves and their
collaborators [12, 14, 15, 16]. Given the strong links between the theory of
minimal surfaces and the Allen-Cahn equation, these results have in turn
triggered some work concerning the existence and behavior of sequences of
critical points of the Allen-Cahn functional.

The min-max theory for the Allen-Cahn equation which has been developed in [7,
9, 10] ensures the existence of min-max critical points of the Allen-Cahn
functional, and, eventually, this provides a powerful tool to prove the
existence of smooth minimal surfaces provided one is able to study the
convergence of transition layers of the critical points as the parameter
$\varepsilon$ tends to $0$. In this context, a delicate question arises since
the convergence of the transition layers to a minimal submanifold can be quite
complicated. For example, transition layers could possibly converge to a
minimal surface with multiplicity higher than $1$ or it might happen that the
transition layers converge to some immersed minimal surface or even the union
of minimal surfaces which intersect.

In the case where the dimension of the ambient manifold is equal to $3$, O.
Chodosh and C. Mantoulidis [2] have proven that, for generic metrics on the
manifold, uniform bounds on the Morse index and the energy of solutions of
(1.2) prevent the convergence of the transition layers with multiplicity to a
minimal surface.

Recall that the Jacobi operator about a given minimal surface $N$ is defined
by

$\mathfrak{J}_{N}:=-\left(\Delta_{N}+|A_{N}|^{2}+\text{Ric}(\nu,\nu)\right),$

where $|A_{N}|^{2}$ is the square of the norm of the second fundamental form
about $N$ and $\nu$ a unit normal vector field along $N$. The following result
is proven in [2]:

######  Theorem 1.1.

[2] Let $N$ denote a minimal surface which is embedded in a $3$-dimensional
manifold $M$ and is constructed as the limit, as $\varepsilon$ tends to $0$,
of zero sets of solutions to the Allen–Cahn equation with uniformly bounded
indices and energies. If the convergence of the transition layers towards $N$
occurs with multiplicity or is two-sided, then $N$ is stable and carries a
positive Jacobi field (on its two-sided double cover, in the second case).

Recall that, for bumpy metrics [23, 24], the Jacobi operator does not carry
any Jacobi field. This implies that, for generic metrics on the ambient
$3$-manifold, if an embedded minimal surface $N$ is obtained as in Theorem
1.1, then it is two-sided and the convergence of the nodal sets of the
solutions of the Allen-Cahn equation towards $N$, does not occur with
multiplicity larger than one.

One of the motivation of the present paper is to understand the set of
solutions of the Allen-Cahn equation which have energy and indices uniformly
bounded as the parameter $\varepsilon$ tends to $0$. The _Morse index_ of a
solution $u$ of (1.2) is defined to be the dimension of the space of (smooth)
functions over which the quadratic form

${\mathcal{Q}}_{u}(w):=\varepsilon\,\int_{M}|\nabla
w|_{g}^{2}\,\text{dvol}_{g}+\frac{1}{\varepsilon}\,\int_{M}\,(3u^{2}-1)\,w^{2}\,\text{dvol}_{g},$

is negative definite.

In view of all these developments, it is natural to ask whether Theorem 1.1
also holds when the ambient manifold is a $2$-dimensional surface. In other
words, is it possible to construct sequences of solutions of the Allen-Cahn
equation which are defined on a compact surface without boundary, whose
energies and Morse indices are uniformly bounded and whose transition layers
converge with multiplicity larger than one, to a given embedded geodesics as
the parameter $\varepsilon$ tends to $0$?

This question will be answered affirmatively in the present paper.

To better understand our result, let us recall a couple of constructions which
are relevant to the present paper. Given an embedded minimal hypersurface $N$
in $M$ a $m$-dimensional compact manifold without boundary, solutions whose
zero set is a (small) normal graph over $N$ have been constructed using a
Lyapunov-Schmidt reduction argument in [18, 19], provided the Jacobi operator
about $N$ does not carry any Jacobi field. This approach has been further
developed in [3] to construct counter-examples to the De Giorgi conjecture for
the Allen-Cahn equation in $\mathbf{R}^{9}$ and entire solutions of the Allen-
Cahn equation whose zero set is close to a given embedded finite total
curvature minimal surface in $\mathbf{R}^{3}$ [4]. In all these papers, the
transition layer (i.e. the zero set) of the solutions is a (small) normal
graphs over a minimal hypersurface $N$. In particular, they are diffeomorphic
to $N$ and hence this corresponds to the case where the zero set of the
solutions to the Allen-Cahn equation converges to a minimal hypersurface with
multiplicity one. The existence of solutions to the Allen-Cahn equation whose
transition layers consist of several disjoint (small) graphs over a given
minimal surface has been proven in [6], provided the potential
$|A_{N}|^{2}+\text{Ric}(\nu,\nu)$ which appears in the Jacobi operator, is
positive along $N$. In this case, the position of the transition layers of the
solutions of the Allen-Cahn equation is governed by solutions of the the so-
called _Jacobi-Toda system_ which will also play an important role in the
present paper. This last result provides examples of solutions to the Allen-
Cahn equation whose transition layers converge to a given embedded minimal
surface with multiplicity larger than one, as $\varepsilon$ tends to $0$.
However, these solutions do not satisfy all assumptions of the Theorem of O.
Chodosh and C. Mantoulidis, since their Morse indices are of the order
$\left|\ln\varepsilon\right|$, and hence tend to infinity as $\varepsilon$
tends to $0$.

In the present paper, we concentrate our attention to the case where the
ambient manifold $(M,g)$ is a compact surface without boundary and $\gamma$ is
a closed geodesic embedded in $M$. In this case, the Jacobi operator about
$\gamma$ reduces to

$\mathfrak{J}_{\gamma}:=-(\partial_{s}^{2}+K_{g}),$

where $s$ is the arc-length along $\gamma$ and where $K_{g}$ denotes the Gauss
curvature of $(M,g)$ along $\gamma$. Let us recall the well known:

######  Definition 1.2.

A closed geodesic $\gamma$ is said to be non-degenerate if the Jacobi operator
$\mathfrak{J}_{\gamma}$ does not have any nonzero Jacobi field, i.e. it has
trivial kernel.

We also define $\text{Ind}\,(\gamma)$, the _Morse index_ of a closed geodesic
$\gamma$, to be equal to the maximal dimension of the space of (smooth)
functions defined on $\gamma$, over which the quadratic form

$Q_{\gamma}(v):=\int_{\gamma}\left(|\partial_{s}v|^{2}-K_{g}\,v^{2}\right)\,ds,$

is negative definite.

To state our result, we need to introduce a new notion which we have
christened _Bouncing Jacobi Field_. It will be convenient to assume that the
geodesic $\gamma$ we are given, is parameterized by arc-length and, to allay
notations, we will identify $\gamma$ with $\mathbf{R}/\sim$ where $s\sim
s^{\prime}$ if and only if $s^{\prime}-s\in|\gamma|\,{\bf Z}$, where
$|\gamma|$ is the length of $\gamma$. Therefore, $s$ will denote both a point
in $\gamma$ and a point in $\mathbf{R}/\sim$.

Finally, if $s_{1},\ldots,s_{n}$ are point of $\gamma$, we will assume that
they (or their representatives) are arranged so that $0\leq
s_{1}<\ldots<s_{n}<|\gamma|$ and we will agree that $s_{0}:=s_{n}-|\gamma|$
and $s_{n+1}=s_{1}+|\gamma|$.

######  Definition 1.3 (Bouncing Jacobi Field).

We will say that a real valued function $\Phi:\gamma\to\mathbf{R}$ is a
_Bouncing Jacobi Field_ with $n\geq 1$ minimums if there exist $n$ distinct
points $s_{1},\ldots,s_{n}\in\gamma$, such that $\Phi$ is continuous on
$\gamma$, smooth on $\gamma-\left\\{s_{1},\ldots,s_{n}\right\\}$ and if it
satisfies

$\left\\{\begin{array}[]{rlllllll}\mathfrak{J}_{\gamma}\Phi=0&\text{and}&\Phi>1\quad\text{on}\quad\gamma-\\{s_{1},\ldots,s_{n}\\},\\\\[8.53581pt]
\Phi(s_{j})=1&\text{and}&\partial_{s}\Phi(s_{j}^{+})=-\partial_{s}\Phi(s_{j}^{-})>0\quad\text{for
all}\quad j=1,\ldots,n.\end{array}\right.$

In this definition as well as in the rest of the paper, if
$f:\gamma-\\{s_{1},\ldots,s_{n}\\}\to\mathbf{R}$ is a function, then
$f(s^{+}_{j})$ and $f(s^{-}_{j})$ denote respectively the left and right
limits of $f$ at $s_{j}$, assuming these limits are defined.

We will prove that the points $s_{1},\ldots,s_{n}$ where a given Bouncing
Jacobi Field is minimal are critical points of the function

$\mathcal{H}_{n}(s_{1},\ldots,s_{n}):=\sum_{j=1}^{n}\left(\inf_{\phi-1\in
H_{0}^{1}\left(s_{i},s_{i+1}\right)}\int_{(s_{i},s_{i+1})}\left(|\partial_{s}\phi|^{2}-K_{g}\,\phi^{2}\right)\,ds\right).$

The index of a Bouncing Jacobi Field $\Phi$ will be denoted by
$\text{Ind}\,(\Phi)$. It is by definition equal to the number of negative
eigenvalues of the symmetric matrix associated to the Hessian of
$\mathcal{H}_{n}$ at $(s_{1},\ldots,s_{n})$ where $\Phi$ is minimal. The
nullity of the Bouncing Jacobi Field $\Phi$ is by definition equal to the
dimension of the null-space of the matrix associated to the Hessian of
$\mathcal{H}_{n}$ at $(s_{1},\ldots,s_{n})$. A Bouncing Jacobi Field is said
to be non-degenerate if its nullity is $0$.

The existence of Bouncing Jacobi Fields is guaranteed by the following result:

######  Theorem 1.4.

Let $\gamma$ be an embedded geodesic in $M$ and $n\geq 1$. Assume that the
Gauss curvature $K_{g}$ is positive along $\gamma$ and

$|\gamma|^{2}\,\sup_{\gamma}K_{g}<\pi^{2}\,n^{2}.$

Then, there exists a _Bouncing Jacobi Field_ which has $n$ minimums and for
which, the sum of the nullity and index is equal to $n$.

We will also prove that, if $K_{g}$ is positive and $n\geq 1$, then

$|\gamma|^{2}\,\inf_{\gamma}K_{g}\leq\pi^{2}\,n^{2},$

is a necessary condition for the existence of a Bouncing-Jacobi Field with $n$
minimum along $\gamma$. As we will see (Corollary 5.6),

$2n\geq\text{Ind}(\gamma).$

is also a necessary condition for a Bouncing Jacobi Field $\Phi$ with $n$
minimums to exist.

Bouncing Jacobi Fields are at the heart of our construction. Starting from a
Bouncing Jacobi Field with suitable non-degeneracy condition, we can construct
solutions of Allen-Cahn equation with bounded energy and indices.

Let ${\bf G}^{1}$ and ${\bf G}^{2}$ be two (possibly empty) finite collections
of closed, embedded, non-degenerate geodesics on $(M,g)$. We assume that
geodesics in ${\bf G}^{1}\cup\bf G^{2}$ are disjoint and we further assume
that, for all $\gamma\in{\bf G}^{2}$, the Jacobi operator
$\mathfrak{J}_{\gamma}$ carries a non-degenerate _Bouncing Jacobi Field_
$\Phi=\Phi_{\gamma}$ with $n=n_{\gamma}\geq 1$ minimums. Finally, assume that
${\bf G}^{1}$ is the boundary of a smooth compact domain(This is to guarantee
that an approximate solution can be defined on the manifold) and that each
geodesic in ${\bf G}^{2}$ is two-sided (namely, that its normal bundle is
orientable).

The main result of the paper is the following:

######  Theorem 1.5.

Under the above assumptions, for all $\varepsilon$ small enough, the Allen-
Cahn equation has a solution $u_{\varepsilon}$ whose Morse index is equal to

$\sum_{\gamma\in{\bf G}^{1}}\text{\rm Ind}\,(\gamma)+\sum_{\gamma\in{\bf
G}^{2}}(\text{\rm Ind}\,(\gamma)+n_{\gamma}+\text{\rm Ind}(\Phi_{\gamma})),$

and whose energy converges to

$\frac{2\sqrt{2}}{3}\,\left(\sum_{\gamma\in{\bf
G}^{1}}|\gamma|+2\sum_{\gamma\in{\bf G}^{2}}|\gamma|\right),$

as $\varepsilon$ tends to $0$. Moreover, the transition layers of
$u_{\varepsilon}$ converge to geodesics in ${\bf G}^{2}$ with multiplicity two
and converge to geodesics in ${\bf G}^{1}$ with multiplicity one.

The constant $\frac{2\sqrt{2}}{3}$ which appears in the statement of the
theorem is nothing but the energy of the one dimensional heteroclinic solution

$x\mapsto\tanh\left(\frac{x}{\sqrt{2}}\right).$

Let us explain how this result relates to former results on the construction
of solutions of the Allen-Cahn equation.

First, the above result is a combination of the results which will be proved
in the present paper and the result of F. Pacard and M. Ritoré [18, 19] which
addresses the case where ${\bf G}^{2}$ is empty.

Our result is, in some sense, a generalization of earlier results obtained in
[5] and [11]. In these two papers, the Allen-Cahn equation is considered over
a smooth bounded domain in the Euclidean plane and solutions are assumed to
have $0$ Neumann condition on the boundary of the domain. In this context, a
geodesic is a line segment $\gamma$ meeting the boundary of the domain
orthogonally. The result in [11] corresponds to the case where the transition
layer of the solutions converges to $\gamma$ with multiplicity $1$. The result
in [5] corresponds exactly to solutions of the Allen-Cahn equation whose
transition layers converge to $\gamma$ with multiplicity $2$ and in fact, the
solutions in this paper are constructed starting from a Bouncing Jacobi Field,
which in this case is continuous, piecewise affine function of the form
$s\mapsto 1+a\,|s-\bar{s}|$, satisfying some boundary condition.

Observe that, in Theorem 1.5, we do not assume that the Gauss curvature is
everywhere positive along each geodesic of ${\bf G}^{2}$. If one assumes that
the Gauss curvature is positive along a closed embedded geodesic, then the
result in [6] already provides solutions of the Allen-Cahn equation whose
transition layers converges to $\gamma$ with multiplicity $2$ as the parameter
$\varepsilon$ tends to $0$. But as already explained, the Morse indices of
these solutions tends to $+\infty$ as $\varepsilon$ tends to $0$, while the
solutions provided by Theorem 1.5 have Morse indices which does not depend on
$\varepsilon$.

######  Remark 1.6.

In Theorem 1.5, we ask that the normal bundle over each geodesic of ${\bf
G}^{2}$ is two-sided. This assumption can be relaxed if one requires that both
the geodesic and the Bouncing Jacobi Fields, extended to the two-sided double-
cover, are non-degenerate.

Our result requires the geodesics and the Bouncing Jacobi Fields to be non-
degenerate. These two non-degeneracy conditions do hold for a generic choice
of the ambient metric on $M$. A metric $g$ on a surface $(M,g)$ is said to be
a _bumpy metric_ if there is no immersed closed geodesic carrying a non-
trivial Jacobi field. Recall that B. White [23, 24] has proven that bumpy
metrics are generic in the sense of Baire category. The existence of non-
degenerate geodesics carrying non-degenerate Bouncing Jacobi Fields is ensured
by the Theorem 1.4 and the:

######  Theorem 1.7.

For a generic choice of a $\mathcal{C}^{k}$, with $k\geq 3$, metric $g$ on $M$
(in the sense of Baire Category), embedded, closed geodesic $\gamma$ in
$(M,g)$ and all _Bouncing Jacobi Field_ they carry are non-degenerate.

Collecting Theorem 1.7, Theorem 1.5 and Theorem 1.4, we get the:

######  Corollary 1.8.

For generic metric on $M$, if $\gamma$ is an embedded closed geodesic such
that $K_{g}>0$ along $\gamma$, and if $n\in\mathbf{N}$ satisfies

$n^{2}\,\pi^{2}>|\gamma|^{2}\,\sup_{\gamma}K_{g},$

then, for all $\varepsilon>0$ small enough, there exist solutions of the
Allen-Cahn equation whose Morse index is given by $\text{Ind}\,(\gamma)+2n$
and whose transition layers converge to $\gamma$ with multiplicity $2$.

Given a closed embedded geodesic in $M$ and $n\geq 1$, uniqueness of the
Bouncing Jacobi Fields having $n$ minimums is not true in general. For
example, when $n=1$ and under suitable assumption on the Gauss curvature, we
will see that there are at least $2$ distinct Bouncing Jacobi Fields with one
minimum, giving rise to two distinct solutions of the Allen-Cahn equation, one
of which having a Morse index strictly smaller than the one described in
Theorem 1.4. Regarding other construction of solutions with bounded Morse
indices, it is also worth mentioning that existence and symmetry of ground
states for the Allen-Cahn equation on the round sphere have been studied in
[1].

The tools developed to prove Theorem 1.5 can very likely be generalized to
produce solutions of the Allen-Cahn equations whose transition layers converge
to a given geodesic with multiplicity $k\geq 3$, while their Morse indices
stays constant and their energy remains uniformly bounded as $\varepsilon$
tends to $0$.

As we have discussed above, Bouncing Jacobi Fields do exist in abundance, at
least when the Gauss curvature along a geodesic is positive, while, when the
Gauss curvature along the geodesic is negative, Bouncing Jacobi Fields do not
exist and we expect that, solutions of the Allen-Cahn equation whose
transition layers converge to a given geodesic $\gamma$ with multiplicity
larger than $1$ do not exist when the Gauss curvature along $\gamma$ is
negative.

To complete the picture, let us mention that, in a forthcoming paper [13], we
will consider the case where the transition layers of the solutions of the
Allen-Cahn equation converges to the union of of geodesics that are not
necessarily disjoint. In which case, we obtain the existence of solutions to
the Allen-Cahn equation whose Morse index is equal to the sum of the Morse
indices of the geodesics and the number of intersection points and self
intersections of the union of the geodesics. These results are first steps
toward a classification of critical points of the Allen-Cahn functional with
controlled energy and Morse index.

The paper is organized in the following way. In Section 2, we discuss the
existence and non existence of Bouncing Jacobi Fields which have been defined
in Definition 1.3 and, in Section 3, we study the Morse Index of a given
Bouncing Jacobi Field. Bouncing Jacobi Fields are at the heart of our
construction since they are an essential element which will allow us to prove,
in Section 4, the existence of special solutions $\Psi:\gamma\to\mathbf{R}$ of
the Jacobi-Toda equation

$\varepsilon^{2}\,\mathfrak{J}_{\gamma}\Psi+\bar{c}^{2}\,e^{-2\Psi}=0,$

provided $\varepsilon>0$ is small enough, where $\bar{c}>0$ is a universal
constant. These special solutions are obtained by smoothing the singularities
(i.e. the points where the function is continuous and not smooth) of a given
Bouncing Jacobi Field $\Phi$ using suitably scaled copies of solutions
$T:\mathbf{R}\to\mathbf{R}$ of the Toda equation

$\partial_{s}^{2}T-e^{-2T}=0.$

Section 5 will be devoted to the study of the Morse index of the solutions
$\Psi$ constructed in the previous section. We will prove that the Morse index
of $\Psi$ is the sum of the Morse index of $\gamma$ and the number of minimums
of $\Phi$. In Section 6, starting from $\Psi$, a special solutions of the
Jacobi-Toda equation, we will construct a solution of the Allen-Cahn equation
whose transition layers are normal graphs over $\gamma$ for the functions
$\pm\varepsilon\,\Psi$. This construction is by now standard and starts with
the construction of an approximate solution which is then perturbed into a
genuine solution of the Allen-Cahn equation. Once the existence of the desired
solution of the Allen-Cahn equation is complete, we will study their Morse
index in Section 7. Therefore, in some sense, we reduce the existence of
solutions to the Allen-Cahn equation whose transition layers converge to a
given geodesic with multiplicity $2$ to the existence of Bouncing Jacobi
Fields. Finally, Section 8 is devoted to the proof of the fact that, for
generic metrics on $M$, Bouncing Jacobi Fields are non-degenerate.

Acknowledgement Y. Liu is supported by the National Key R&D Program of China
2022YFA1005400 and NSFC 11971026, NSFC 12141105. J. Wei is partially supported
by NSERC of Canada.

##  2\. Bouncing Jacobi Fields

In the case where the Gauss curvature is constant (positive) along the
embedded geodesic $\gamma$, the existence of Bouncing Jacobi Fields with $n$
minimums is straightforward. Indeed, we have the:

######  Lemma 2.1.

When $K_{g}$ is constant and positive along $\gamma$, Bouncing Jacobi Fields
with $n$ minimums do exist if and only if
$\pi^{2}\,n^{2}>|\gamma|^{2}\,K_{g}$, where $|\gamma|$ denotes the length of
$\gamma$.

###### Proof.

We set $s_{n}:=\frac{|\gamma|}{2n}$, the Bouncing Jacobi field (unique up to
translation) is given by function

$\Phi(s)=\frac{\cos\left(\sqrt{K_{g}}s\right)\,}{\cos\left(\sqrt{K_{g}}s_{n}\right)},$

on $\left(-s_{n},s_{n}\right)$ and extended to be $2s_{n}$-periodic. ∎

In the general case, we have the following theorem, where the existence part
is a direct application of Theorem 1.2 in [20] (the equation being known as
Hill’s equation with obstacle).

######  Theorem 2.2.

Assume that $K_{g}\geq 0$ along $\gamma$ and let $n\geq 1$ be given. Then,
$\mathfrak{J}_{\gamma}$ has at least one Bouncing Jacobi Field with $n$ local
minimums provided

(2.1) $\pi^{2}\,n^{2}>|\gamma|^{2}\,\sup K_{g},$

and $\mathfrak{J}_{\gamma}$ does not have any Bouncing Jacobi Field with $n$
local minimums when

(2.2) $\pi^{2}\,n^{2}<|\gamma|^{2}\,\inf K_{g}.$

The method used in [20] is the Poincaré-Birkhoff Theorem about the existence
of fixed points for area preserving maps in dimension two. We also refer to
Theorem 8.4 and Corollary 8.6 in [8], for a detailed introduction to the
Poincaré-Birkhoff Theorem.

We point out that the proof in [20] does not provide any information about the
Morse indices of the Bouncing Jacobi Fields. For this reason, we give here a
(rather independent) proof of Theorem 2.2. As we will see, our proof captures
more information about the Morse indices of the solutions.

Our proof is based on the existence of a max-min solution for some energy we
now describe. For $s_{1}<s_{2}$, we define

$H\left(s_{1},s_{2}\right):=\inf_{\phi-1\in
H_{0}^{1}\left(s_{1},s_{2}\right)}\int_{s_{1}}^{s_{2}}\left(|\partial_{s}\phi|^{2}-K_{g}\,\phi^{2}\right)\,ds.$

Observe that the infimum may well be equal to $-\infty$. On the other hand, if
$H\left(s_{1},s_{2}\right)>-\infty,$ then the infimum is achieved for some
function $\phi$ solution of

$\mathfrak{J}_{\gamma}\phi=0,$

in $(s_{1},s_{2})$ with boundary data $\phi(s_{1})=\phi(s_{2})=1$. Standard
arguments imply that $\phi>0$ in $[s_{1},s_{2}]$ (since $|\phi|$ is also a
minimizer). The maximum principle, together with the fact that we have assumed
that $K_{g}$ is positive along $\gamma$, then implies that

$\phi\left(s\right)>1,$

in $(s_{1},s_{2})$. Finally, using an integration by parts, we conclude that

$H\left(s_{1},s_{2}\right)=\partial_{s}\phi\left(s_{2}\right)-\partial_{s}\phi\left(s_{1}\right)\leq
0.$

This being understood, we consider the set

$\Omega_{n}:=\left\\{\left(s_{1},s_{2},\ldots,s_{n}\right)\,:\,0\leq
s_{1}<\ldots<s_{n}<|\gamma|\right\\},$

and, for all $\left(s_{1},\ldots,s_{n}\right)\in\Omega_{n}$, we define

$\mathcal{H}_{n}\left(s_{1},\ldots,s_{n}\right):=\sum_{i=1}^{n}H\left(s_{i},s_{i+1}\right).$

where we agree that $s_{n+1}:=s_{1}+|\gamma|$.

###### Proof of Theorem 2.2.

The proof is decomposed into three parts. In the first part, we give a
sufficient condition for the existence of a critical point of
$\mathcal{H}_{n}$. Then, we derive the first variation of $\mathcal{H}_{n}$
and prove that a critical point of this functional is associated to a Bouncing
Jacobi Field with $n$ minimums. Finally, we give some necessary conditions for
the existence of Bouncing Jacobi Field with $n$ minimums.

######  Lemma 2.3.

Assume that $K_{g}>0$ and

(2.3) $\pi^{2}\,n^{2}>|\gamma|^{2}\displaystyle\sup_{\gamma}K_{g}.$

Then, there exists a critical point of $\mathcal{H}_{n}$ and in fact this
point can be chosen to be a point where $\mathcal{H}_{n}$ is maximal.

###### Proof.

As we have mentioned above, since it is assumed that $K_{g}>0$ along $\gamma$,
there holds

$H\left(s_{i},s_{i+1}\right)\leq 0,$

and hence $\mathcal{H}_{n}\leq 0$. To produce critical points of
$\mathcal{H}_{n}$, it is therefore reasonable to look for a point of
$\Omega_{n}$ where $\mathcal{H}_{n}$ achieves a local maximum.

Let us prove that (2.3) ensures that
$\displaystyle\sup_{\Omega_{n}}\mathcal{H}_{n}>-\infty$. Indeed, the $n$-tuple
$\left(\bar{s}_{1},\ldots,\bar{s}_{n}\right)$ where
$\bar{s}_{j}:=\frac{|\gamma|}{n}\,j$ belongs to $\Omega_{n}$ and, using
Poincaré inequality, we estimate

$\displaystyle H\left(\bar{s}_{i},\bar{s}_{i+1}\right)$
$\displaystyle=\inf_{\phi\in
H_{0}^{1}\left(\bar{s}_{i},\bar{s}_{i+1}\right)}\int_{\bar{s}_{i}}^{\bar{s}_{i+1}}\left(|\partial_{s}\phi|^{2}-K_{g}\,\left(\phi+1\right)^{2}\right)ds$
$\displaystyle\geq\inf_{\phi\in
H_{0}^{1}\left(\bar{s}_{i},\bar{s}_{i+1}\right)}\int_{\bar{s}_{i}}^{\bar{s}_{i+1}}\left(\left(\frac{n^{2}\pi^{2}}{|\gamma|^{2}}-K_{g}\right)\phi^{2}-2K_{g}\phi-
K_{g}\right)ds$ $\displaystyle>-\infty.$

Therefore,

$\mathcal{M}_{n}:=\sup_{{\bf s}\in\Omega_{n}}\mathcal{H}_{n}\left({\bf
s}\right)>-\infty.$

We claim that $\mathcal{M}_{n}$ is achieved in $\Omega_{n}$. The _key
observation_ is the fact that, if $s<t$ satisfy $H\left(s,t\right)>-\infty,$
then

$H\left(s,t\right)<H\left(s,\frac{s+t}{2}\right)+H\left(\frac{s+t}{2},t\right).$

Therefore, introducing an extra minimum increases the energy.

Now, let ${\bf s}_{i}\in\Omega_{n}$ be a maximizing sequence for
$\mathcal{H}_{n}$ over $\Omega_{n}$. Then, by definition,

$-\infty<\mathcal{M}_{n}=\lim_{i\to+\infty}\mathcal{H}_{n}({\bf s}_{i}).$

and, up to a subsequence, we can assume that $({\bf s}_{i})_{i\in\bf N}$
converges to ${\bf s}_{\infty}\in\overline{\Omega}_{n}$. We now argue by
contradiction and assume that ${\bf s}_{\infty}\notin\Omega_{n}$ and hence
${\bf s}_{\infty}\in\Omega_{k}\cap\partial\Omega_{n}$, for some $k<n$. It is
easy to see that

$-\infty<\mathcal{M}_{n}=\lim_{i\to+\infty}\mathcal{H}_{n}({\bf
z}_{i})=\mathcal{H}_{k}({\bf s}_{\infty}).$

But, using the above key observation $n-k$ times, if this were the case, we
would conclude that

$\mathcal{H}_{k}({\bf z}_{\infty})<\mathcal{M}_{n},$

which is a contradiction and this proves the claim. To conclude, we have
obtained a critical point of $\mathcal{H}_{n}$ and, in fact, this critical
point maximises $\mathcal{H}_{n}$ over $\Omega_{n}$. ∎

######  Remark 2.4.

In the case where $n=1,$ the analysis simplifies since we just consider the
function

$\mathcal{H}_{1}\left(s\right):=H\left(s,s+|\gamma|\right),$

for $s\in\mathbf{R}$. This is a continuous and periodic function (of period
$|\gamma|$), and if we assume that $|\gamma|^{2}\,\sup_{\gamma}K_{g}<\pi^{2}$,
the function $\mathcal{H}_{1}$ is bounded from below. It follows that
$\mathcal{H}_{1}$ has at least two critical points, corresponding to a global
maximum and minimum of $\mathcal{H}_{1}$ on $\gamma$. If the maximum equals
minimum, then $\mathcal{H}_{1}$ is constant and hence has infinitely many
critical points (this also implies that $K_{g}$ is constant along $\gamma$).

We will now prove that the function $\Phi$ associated to a critical point of
$\mathcal{H}_{n}$ is a Bouncing Jacobi Fields with $n$ minimums. This will be
a consequence of the computation of the first variation of $\mathcal{H}_{n}$
at a point $\left(s_{1},\ldots,s_{n}\right)\in\Omega_{n}$.

######  Proposition 2.5.

Assume that $K_{g}>0$ along $\gamma$ and that $(s_{1},\ldots,s_{n})$ is a
critical point of $\mathcal{H}_{n}$, then the associated function $\Phi$ is a
Bouncing Jacobi Field with $n$ minimums.

###### Proof.

Let us assume that $(s_{1},\ldots,s_{n})$, with $s_{1}<\ldots<s_{n}$ is a
critical point of $\mathcal{H}_{n}$. We agree that $s_{n+1}:=s_{1}+|\gamma|$
and $s_{0}:=s_{n}-|\gamma|$. We denote by $\Phi$ the function associated to
this critical point and denote the restriction of $\Phi$ to $(s_{j},s_{j+1})$
by $\phi_{j}$. Therefore, for all $j=1,\ldots,n$, the function $\phi_{j}$ is a
solution of $\mathfrak{J}_{\gamma}\phi_{j}=0$ in $(s_{j},s_{j+1})$ with
$\phi_{j}(s_{j})=\phi_{j}(s_{j+1})=1$, and $\phi_{j}>1$ on $(s_{j},s_{j+1})$
(observe that, since we have assumed that $K_{g}>0$, $\phi_{j}$ can’t have a
positive local minimum in $(s_{j},s_{j+1})$).

For all $s^{*}_{j}$ close to $s_{j}$ (for $j=1,\ldots,n$), we would like to
understand the solution of $\mathfrak{J}_{\gamma}\tilde{\phi}_{j}=0$ on
$(s_{j}^{*},s_{j+1}^{*})$ which satisfies
$\tilde{\phi}_{j}(s_{j})=\tilde{\phi}_{j}(s_{j+1})=1$ and $\tilde{\phi}_{j}>0$
in $(s^{*}_{j},s^{*}_{j+1})$ and which is close to $\phi_{j}$.

Given $\delta_{j}^{-},\delta_{j}^{+}\in\mathbf{R}$, for $j=1,\ldots,n$, we can
solve

$\mathfrak{J}_{\gamma}\eta_{j}=0,$

in $(s_{j},s_{j+1})$ with boundary data $\eta_{j}(s_{j})=\delta_{j}^{+}$ and
$\eta_{j}(s_{j+1})=\delta_{j+1}^{-}$. The existence of $\eta_{j}$ is
guarantied by the fact that $\phi_{j}>0$ over $[s_{j},s_{j+1}]$ and hence the
equation $\mathfrak{J}_{\gamma}\zeta=0$ on $(s_{j},s_{j+1})$ with boundary
data $\zeta(s_{j})=\zeta(s_{j+1})=0$ does not have any nontrivial solution.

Since we are dealing with a linear second order ordinary differential equation
with bounded coefficients, the functions $s\mapsto(\phi_{j}+\eta_{j})(s)$
which are smooth on $[s_{j},s_{j+1}]$ can be extended as _smooth_ functions
$\psi_{j}$, solutions of $\mathfrak{J}_{\gamma}\psi_{j}=0$ in an open interval
containing $[s_{j},s_{j+1}]$. Using the fact that that $\phi_{j}(s_{j})=1$ and
also that

$\partial_{s}\psi_{j}(s_{j})=\partial_{s}(\phi_{j}+\eta_{j})(s_{j})\qquad\text{and}\qquad\partial_{s}\psi_{j}(s_{j+1})=\partial_{s}(\phi_{j}+\eta_{j})(s_{j+1}),$

we get the Taylor expansion of $\psi_{j}$ at $s_{j}$

$\begin{array}[]{rllll}\psi_{j}(s_{j}+s)&=&\displaystyle
1+\delta^{+}_{j}+\partial_{s}(\phi_{j}+\eta_{j})(s_{j})\,s-K_{g}(s_{j})(1+\delta^{+}_{j})\,\frac{s^{2}}{2}\\\\[8.53581pt]
&-&\displaystyle\left(\partial_{s}K_{g}(s_{j})\,(1+\delta^{+}_{j})+K_{g}(s_{j})\,\partial_{s}(\phi_{j}+\eta_{j})(s_{j})\right)\,\frac{s^{3}}{6}+\mathcal{O}(s^{4}),\end{array}$

and the Taylor expansion of $\psi_{j-1}$ at $s_{j}$

$\begin{array}[]{rllll}\psi_{j-1}(s_{j}+s)&=&\displaystyle
1+\delta^{-}_{j}+\partial_{s}(\phi_{j-1}+\eta_{j-1})(s_{j})\,s-K_{g}(s_{j})(1+\delta^{-}_{j})\,\frac{s^{2}}{2}\\\\[8.53581pt]
&-&\displaystyle\left(\partial_{s}K_{g}(s_{j})(1+\delta^{-}_{j})+K_{g}(s_{j})\,\partial_{s}(\phi_{j-1}+\eta_{j-1})(s_{j})\right)\,\frac{s^{3}}{6}+\mathcal{O}(s^{4}).\end{array}$

Now, $s_{j}^{*}$ being chosen close enough to $s_{j}$, for all $j=1,\ldots,n$,
we can determine $\delta_{j}^{\pm}$ (close to $0$) so that

$\psi_{j-1}(s_{j}^{*})=\psi_{j}(s_{j}^{*})=1.$

It is easy to check that the $\delta^{\pm}_{j}$ depend smoothly on the
$s^{*}_{j}$ and we have the expansions

(2.4)
$\delta_{j}^{+}-\delta_{j}^{-}=-N_{j}\,\dot{s}_{j}+\mathcal{O}(\dot{s}^{2}),$

and

(2.5)
$\begin{array}[]{rllll}\delta^{+}_{j}+\delta^{-}_{j}&=&\displaystyle-\partial_{s}(\phi_{j}+\phi_{j-1})(s_{j})\,\dot{s}_{j}-(\partial_{s}\eta_{j}+\partial_{s}\eta_{j-1})(s_{j})\,\dot{s}_{j}\\\\[8.53581pt]
&+&\displaystyle
K_{g}(s_{j})\,\dot{s}_{j}^{2}+\mathcal{O}(\dot{s}^{3}),\end{array}$

where we have defined

$\dot{s}_{j}:=s_{j}^{*}-s_{j},$

and

$N_{j}:=-\partial_{s}(\phi_{j}-\phi_{j-1})(s_{j}).$

It is important to keep in mind that

$\delta_{j}^{\pm}=\mathcal{O}(\dot{s}).$

We also have the expansions

$\begin{array}[]{rllll}\partial_{s}\psi_{j}(s_{j}^{*})&=&\partial_{s}\psi_{j}(s_{j})-K_{g}(s_{j})\,(1+\delta^{+}_{j})\,\dot{s}_{j}-\partial_{s}K_{g}(s_{j})(1+\delta^{+}_{j})\\\\[5.69054pt]
&+&\displaystyle
K_{g}(s_{j})\,\partial_{s}\phi_{j}(s_{j})\,\frac{\dot{s}_{j}^{2}}{2}+\mathcal{O}(\dot{s}^{3}),\end{array}$

and

$\begin{array}[]{rllll}\partial_{s}\psi_{j}(s_{j+1}^{*})&=&\partial_{s}\psi_{j}(s_{j+1})-K_{g}(s_{j+1})\,(1+\delta^{-}_{j+1})\,\dot{s}_{j+1}-\partial_{s}K_{g}(s_{j+1})\,(1+\delta^{-}_{j+1})\\\\[5.69054pt]
&+&\displaystyle
K_{g}(s_{j})\,\partial_{s}\phi_{j}(s_{j+1})\,\frac{\dot{s}_{j+1}^{2}}{2}+\mathcal{O}(\dot{s}^{3}).\end{array}$

Since the functions $\psi_{j-1}$ and $\psi_{j}$ take the same value (equal to
$1$) at $s_{j}^{*}$, the restrictions of $\psi_{j}$ to
$[s_{j}^{*},s_{j+1}^{*}$] can be patched together to obtain a function which
is continuous on $\gamma$, smooth away from the $s_{j}^{*}$ and which solves
$\mathfrak{J}_{\gamma}\psi_{j}=0$ on $(s^{*}_{j},s_{j+1}^{*})$ with boundary
data $\psi_{j}(s_{j}^{*})=\psi_{j}(s_{j+1}^{*})=1$.

We are now in a position to compute the expansion of the functional
$\mathcal{H}_{n}$ in powers of $\dot{s}_{j}$. We start with an integration by
parts which leads to

$\begin{array}[]{rlll}\mathcal{H}_{n}(s_{1}^{*},\ldots,s_{n}^{*})&=&\displaystyle\sum_{j=1}^{n}\int_{s_{j}^{*}}^{s_{j+1}^{*}}\left((\partial_{s}\psi_{j})^{2}-K_{g}\,\psi_{j}^{2}\right)\,ds\\\\[8.53581pt]
&=&\displaystyle\sum_{j=1}^{n}\left(\partial_{s}\psi_{j}(s_{j}^{*})-\partial_{s}\psi_{j}(s_{j+1}^{*})\right).\end{array}$

Using the above expansions, we get

$\begin{array}[]{rllll}\mathcal{H}_{n}(s_{1}^{*},\ldots,s_{n}^{*})&=&\mathcal{H}_{n}(s_{1},\ldots,s_{n})-\displaystyle\sum_{j=1}^{n}\left(\partial_{s}\eta_{j}(s_{j+1})-\partial_{s}\eta_{j}(s_{j})\right)\\\\[8.53581pt]
&+&\displaystyle\sum_{j=1}^{n}K_{g}(s_{j+1})\,\delta_{j+1}^{-}\,\dot{s}_{j+1}-\sum_{j=1}^{n}K_{g}(s_{j})\,\delta_{j}^{+}\,\dot{s}_{j}\\\\[8.53581pt]
&+&\displaystyle\sum_{j=1}^{n}K_{g}(s_{j+1})\,\partial_{s}\phi_{j}(s_{j+1})\frac{\dot{s}_{j+1}^{2}}{2}\\\\[8.53581pt]
&-&\displaystyle\sum_{j=1}^{n}K_{g}(s_{j})\,\partial_{s}\phi_{j}(s_{j})\frac{\dot{s}_{j}^{2}}{2}+\mathcal{O}(\dot{s}^{3})\\\\[8.53581pt]
&=&\mathcal{H}_{n}(s_{1},\ldots,s_{n})-\displaystyle\sum_{j=1}^{n}\left(\partial_{s}\eta_{j}(s_{j+1})-\partial_{s}\eta_{j}(s_{j})\right)\\\\[8.53581pt]
&-&\displaystyle\sum_{j=1}^{n}K_{g}(s_{j})\,(\delta_{j}^{+}-\delta_{j}^{-})\,\dot{s}_{j}\\\\[8.53581pt]
&-&\displaystyle\sum_{j=1}^{n}K_{g}(s_{j})\,\partial_{s}(\phi_{j}(s_{j})-\phi_{j-1})(s_{j})\,\frac{\dot{s}_{j}^{2}}{2}+\mathcal{O}(\dot{s}^{3}).\end{array}$

Observe that, since $\phi_{j}$ and $\eta_{j}$ are both solutions of the same
second order linear ordinary differential equation, we have

$0=\int_{s_{j}}^{s_{j+1}}\left(\eta_{j}\mathfrak{J}_{\gamma}\phi_{j}-\phi_{j}\mathfrak{J}_{\gamma}\eta_{j}\right)\,ds=(\eta_{j}\partial_{s}\phi_{j}-\phi_{j}\partial_{s}\eta_{j})(s_{j})-(\eta_{j}\partial_{s}\phi_{j}-\phi_{j}\partial_{s}\eta_{j})(s_{j+1}).$

Hence

$\partial_{s}\eta_{j}(s_{j+1})-\partial_{s}\eta_{j}(s_{j})=\delta_{j+1}^{-}\,\partial_{s}\phi_{j}(s_{j+1})-\delta^{+}_{j}\,\partial_{s}\phi_{j}(s_{j}).$

Using this, we get with little work

$\begin{array}[]{rllll}\mathcal{H}(s^{*}_{1},\ldots,s_{n}^{*})&=&\mathcal{H}(s_{1},\ldots,s_{n})\displaystyle+\frac{1}{2}\sum_{j=1}^{n}\partial_{s}\left(\phi_{j}+\phi_{j-1}\right)(s_{j})\,(\delta^{+}_{j}-\delta_{j}^{-})\\\\[8.53581pt]
&+&\displaystyle\frac{1}{2}\sum_{j=1}^{n}\partial_{s}\left(\phi_{j}-\phi_{j-1}\right)(s_{j})\,(\delta^{+}_{j}+\delta_{j}^{-})-\displaystyle\sum_{j=1}^{n}K_{g}(s_{j})\,(\delta_{j}^{+}-\delta_{j}^{-})\,\dot{s}_{j}\\\\[8.53581pt]
&-&\displaystyle\sum_{j=1}^{n}K_{g}(s_{j})\,\partial_{s}(\phi_{j}-\phi_{j-1})(s_{j})\,\frac{\dot{s}_{j}^{2}}{2}+\mathcal{O}(\dot{s}^{3}).\end{array}$

Finally, we use (2.4) and (2.5) to conclude that

(2.6)
$\begin{array}[]{llll}\mathcal{H}_{n}(s^{*}_{1},\ldots,s^{*}_{n})&=&\displaystyle\mathcal{H}_{n}(s_{1},\ldots,s_{n})-\sum_{j=1}^{n}((\partial_{s}\phi_{j})^{2}-(\partial_{s}\phi_{j-1})^{2})(s_{j})\,\dot{s}_{j}\\\\[8.53581pt]
&-&\displaystyle\sum_{j=1}^{n}\partial_{s}(\phi_{j}-\phi_{j-1})(s_{j})\,\partial_{s}(\eta_{j}+\eta_{j-1})(s_{j})\,\frac{\dot{s}_{j}^{2}}{2}\\\\[8.53581pt]
&+&\displaystyle\sum_{j=1}^{n}K_{g}(s_{j})\,\partial_{s}(\phi_{j}-\phi_{j-1})(s_{j})\,\frac{\dot{s}_{j}^{2}}{2}\\\\[8.53581pt]
&-&\displaystyle\sum_{j=1}^{n}((\partial_{s}\phi_{j})^{2}-(\partial_{s}\phi_{j-1})^{2})(s_{j})\,\mathcal{O}(\dot{s}^{2})+\mathcal{O}(\dot{s}^{3}).\end{array}$

Since we have assumed that $(s_{1},\ldots,s_{n})$ is a critical point of
$\mathcal{H}_{n}$, the linear term in $\dot{s}_{j}$ has to vanish, and hence
we get

$(\partial_{s}\phi_{j})^{2}(s_{j})=(\partial_{s}\phi_{j-1})^{2}(s_{j}).$

But, $\phi_{j-1}$ and $\phi_{j}$ are minimal at $s_{j}$, therefore,
$\partial_{s}\phi_{j}(s_{j})\geq 0$ and $\partial_{s}\phi_{j-1}(s_{j})\leq 0$
and this implies that

$\partial_{s}(\phi_{j}+\phi_{j-1})(s_{j})=0,$

for all $j=1,\ldots,n$. Since $K_{g}>0$ along $\gamma$, it is not possible for
$\partial_{s}\phi_{j}(s_{j})$ to be equal to $0$ since if this were the case
then the functions $\psi_{j}$ on $(s_{j-1},s_{j}]$ and $\psi_{j+1}$ on
$[s_{j},s_{j+1})$ can be glued together to provide a solution of
$\mathfrak{J}_{\gamma}\psi=0$ on $(s_{j-1},s_{j+1}$ which has a positive local
minimum at $s_{j}$, which is a contradiction. This proves that $\Phi$ is a
Bouncing Jacobi Field. ∎

To proceed, let us give a necessary condition for the existence of a Bouncing
Jacobi Field.

######  Lemma 2.6.

Assume that

$|\gamma|^{2}\,\inf_{\gamma}K_{g}>\pi^{2}\,n^{2}.$

Then $\gamma$ does not carry any Bouncing Jacobi Field with $n$ minimums.

###### Proof.

We argue by contradiction and assume that there was a Bouncing Jacobi Field
$\Phi$ with $n$ consecutive local minimums at $s_{1}<\ldots<s_{n}$. We can
find an index $j$ such that

$n\,\sqrt{\inf K_{g}}\,|s_{j+1}-s_{j}|\geq|\gamma|\sqrt{\inf K_{g}}>n\,\pi.$

It follows that the equation

$\left(\partial_{s}^{2}+\inf K_{g}\right)\,\eta=0,$

has a nontrivial solution $\eta$ which vanishes at two distinct points
$\overline{s}<\underline{s}$ in the interval $(s_{j},s_{j+1})$. Now,

$\mathfrak{J}_{\gamma}\Phi=0,$

in $(s_{j},s_{j+1})$, with $\Phi(s_{j})=\Phi(s_{j+1})=1$. Since $\Phi>1$ in
$(\underline{s},\overline{s})$, one can choose a constant $c>0$ such that
$\Phi-c\,\eta$ has a local minimum in $(\underline{s},\overline{s})$. This
implies that $\Phi-c\,\eta\equiv 0$ and this is a contradiction since this
function is not zero at $\overline{s}$ and $\underline{s}$. ∎

The proof of Theorem 2.2 is now complete. This also completes the proof of
Theorem 1.4 mentioned in the first section. ∎

######  Remark 2.7.

In the special case where $n=1$ and where (2.3) holds, there exist at least
two distinct Bouncing Jacobi Fields with one local minimum since, in this
case, $s\mapsto\mathcal{H}_{1}(s)$ is a smooth periodic function which has at
least one minimum and one maximum.

##  3\. The index of Bouncing Jacobi Fields

Observe that, in the case where the Gauss curvature is constant along $\gamma$
and if $\gamma$ is assumed to be non-degenerate, then inequality
$\left(\ref{th:1.5}\right)$ tells us that $n$, the number of minimum of a
Bouncing Jacobi Field, satisfies

$n\geq\text{Ind}\left(\gamma\right),$

where we recall that $\text{Ind}\left(\gamma\right)$ denotes the Morse index
of the geodesic $\gamma$.

In the case where the Gauss curvature is not constant, the relation between
$n$ and the index of the geodesic is not clear. In this section, we will
analyze the properties of the index of the Bouncing Jacobi Field. These
properties will be useful in our later analysis of the Jacobi-Toda system. As
a by product, in Corollary 5.6, we will also prove a weaker lower bound for
$n$ in terms of the index of $\gamma$, which has independent interest.

To proceed, we will now compute the the second variation of $\mathcal{H}_{n}$
at a critical point.

######  Proposition 3.1.

Assume that $\Phi$ is Bouncing Jacobi Field with $n$ minimums at $0\leq
s_{1}<\ldots<s_{n}<|\gamma|$. As usual, we agree that
$s_{n+1}:=s_{1}+|\gamma|$ and $s_{0}=s_{n}-|\gamma|$ and we define

$N_{j}:=\partial_{s}\Phi(s_{j}^{+})-\partial_{s}\Phi(s_{j}^{-}).$

Then, the following expansion holds

$\begin{array}[]{lllll}\displaystyle{\mathcal{H}}_{n}\left(s_{1}+\frac{\sigma_{1}}{N_{1}},\ldots,s_{n}+\frac{\sigma_{n}}{N_{n}}\right)={\mathcal{H}}_{n}(s_{1},\ldots,s_{n})\\\\[8.53581pt]
\hskip
71.13188pt\displaystyle-\frac{1}{4}\sum_{j=1}^{n}{\bigg{\\{}}\left(\partial_{s}\eta(s_{j}^{-})+\partial_{s}\eta(s_{j}^{+})\right)\,\sigma_{j}+\frac{4\,K_{g}(s_{j})}{N_{j}}\,\sigma_{j}^{2}{\bigg{\\}}}+\mathcal{O}(\sigma_{j}^{3}),\end{array}$

or equivalently

$\begin{array}[]{lllll}\displaystyle\mathcal{H}_{n}\left(s_{1}+\frac{\sigma_{1}}{N_{1}},\ldots,s_{n}+\frac{\sigma_{n}}{N_{n}}\right)={\mathcal{H}}_{n}(s_{1},\ldots,s_{n})\\\\[8.53581pt]
\hskip
56.9055pt\displaystyle+\frac{1}{4}\sum_{j=1}^{n}{\bigg{\\{}}\int_{s_{j}}^{s_{j+1}}\left((\partial_{s}\eta)^{2}-K_{g}\,\eta^{2}\right)\,ds-\frac{4\,K_{g}(s_{j})}{N_{j}}\,\sigma_{j}^{2}{\bigg{\\}}}+\displaystyle\sum_{j=1}^{n}\mathcal{O}(\sigma_{j}^{3}),\end{array}$

where the function $\eta:\gamma-\\{s_{1},\ldots,s_{n}\\}\to\mathbf{R}$ is the
(unique) solution of

$\left\\{\begin{array}[]{rlllll}\mathfrak{J}_{\gamma}\eta&=&0\qquad\text{on}\qquad\gamma-\\{s_{1},\ldots,s_{n}\\},\\\\[8.53581pt]
\eta(s_{j}^{+})=-\eta(s_{j}^{-})&=&\sigma_{j}\qquad\text{for all}\quad
j=1,\ldots,n.\end{array}\right.$

###### Proof.

We keep the notations of the proof of Proposition 2.5 and start with (2.6)
which we specialize to the case where $\Phi$ is Bouncing Jacobi Field. In this
case, the formula simplifies and we get

$\begin{array}[]{llll}\mathcal{H}_{n}(s^{*}_{1},\ldots,s^{*}_{n})&=&\displaystyle\mathcal{H}_{n}(s_{1},\ldots,s_{n})\\\\[8.53581pt]
&-&\displaystyle\sum_{j=1}^{n}\partial_{s}(\phi_{j}-\phi_{j-1})(s_{j})\,\partial_{s}(\eta_{j}+\eta_{j-1})(s_{j})\,\frac{\dot{s}_{j}^{2}}{2}\\\\[8.53581pt]
&+&\displaystyle\sum_{j=1}^{n}K_{g}(s_{j})\,\partial_{s}(\phi_{j}-\phi_{j-1})(s_{j})\,\frac{\dot{s}_{j}^{2}}{2}+\mathcal{O}(\dot{s}^{3}).\end{array}$

where we recall that $\dot{s}_{j}:=s^{*}_{j}-s_{j}$. Using the fact that

$\delta_{j}^{\pm}=\pm\frac{N_{j}}{2}\dot{s}_{j}+\mathcal{O}(\dot{s}^{2}),$

where we recall that

$N_{j}:=\partial_{s}(\phi_{j}-\phi_{j-1})(s_{j}),$

this can be rewritten as

$\begin{array}[]{lllll}\mathcal{H}(s^{*}_{1},\ldots,s_{n}^{*})&=&\mathcal{H}(s_{1},\ldots,s_{n})\\\\[8.53581pt]
&-&\displaystyle\frac{1}{4}\,\sum_{j=1}^{n}N_{j}^{2}\,\bigg{\\{}\partial_{s}(\dot{\eta}_{j}+\dot{\eta}_{j-1})(s_{j})\,\dot{s}_{j}+\frac{4\,K_{g}(s_{j})}{N_{j}}\dot{s}_{j}^{2}\bigg{\\}}+\sum_{j=1}^{n}\mathcal{O}(\dot{s}_{j}^{3}),\end{array}$

where $\dot{\eta}_{j}$ is the solution of
$\mathfrak{J}_{\gamma}\dot{\eta}_{j}=0$ on $(s_{j},s_{j+1})$, with boundary
data $\dot{s}_{j}$ at $s_{j}$ and $\dot{s}_{j+1}$ at $s_{j+1}$.

Since we also have

$\int_{s_{j}}^{s_{j+1}}\left((\partial_{s}\dot{\eta}_{j})^{2}-K_{g}\,\dot{\eta}_{j}^{2}\right)\,ds=-\left(\partial_{s}\eta_{j}(s_{j+1})\,\dot{s}_{j+1}+\partial_{s}\eta_{j}(s_{j})\,\dot{s}_{j}\right),$

we get the alternative formula

$\begin{array}[]{llll}\mathcal{H}(s_{1}^{*},\ldots,s_{n}^{*})=\mathcal{H}(s_{1},\ldots,s_{n})\\\\[8.53581pt]
\hskip
59.75078pt\,+\displaystyle\frac{1}{4}\sum_{j=1}^{n}N_{j}^{2}\,\bigg{\\{}\int_{s_{j}}^{s_{j+1}}\left((\partial_{s}\dot{\eta}_{j})^{2}-K_{g}(\dot{\eta}_{j})^{2}\right)\,ds-\frac{4\,K_{g}(s_{j})}{N_{j}}\dot{s}_{j}^{2}\bigg{\\}}+\mathcal{O}(\dot{s}_{j}^{3}).\end{array}$

This completes the proof. ∎

We now introduce the definition of a non-degenerate Bouncing Jacobi Field.

######  Definition 3.2.

A Bouncing Jacobi Field $\Phi$ with $n$ minimums at the points
$s_{1}<\ldots<s_{n}$ (being understood that $s_{n+1}:=s_{1}+|\gamma|$ and
$s_{0}:=s_{n}-|\gamma|$) is said to be non-degenerate, if the following system
does not have any non-trivial solution:

$\left\\{\begin{array}[]{rllll}\mathfrak{J}_{\gamma}\eta&=&0\quad\text{in}\quad\gamma-\\{s_{1},\ldots,s_{n}\\},\\\\[8.53581pt]
\eta(s^{+}_{j})+\eta(s^{-}_{j})&=&0\quad\text{for all}\quad
j=1,\ldots,n,\\\\[8.53581pt]
\displaystyle\partial_{s}\eta(s_{j}^{+})+\partial_{s}\eta(s_{j}^{-})+\displaystyle
2K_{g}\,\frac{\eta(s_{j}^{+})-\eta(s_{j}^{-})}{N_{j}}&=&0\quad\text{for
all}\quad j=1,\ldots,n,\end{array}\right.$

where $N_{j}:=\partial_{s}\Phi(s_{j}^{+})-\partial_{s}\Phi(s_{j}^{-})$.

Observe that the operator which appears in this definition also appears in the
second variation of the functional $\mathcal{H}_{n}$ which is given in
Proposition 3.1.

The non-degeneracy condition will now be used to prove that we can embed the
Bouncing Jacobi Field $\Phi$ as a $2n$ dimensional family of continuous,
piecewise $\mathcal{C}^{1}$ Jacobi Fields whose $n$ local minimum have a
prescribed value close to $1$ and have the property that the sum of the right
and left derivatives at these points can also be prescribed to some value
close to $0$. More precisely, we have the:

######  Proposition 3.3.

Assume that $\Phi$ is a non-degenerate Bouncing Jacobi Field with $n$ minimums
at the points $s_{1},\ldots,s_{n}\in\gamma$. Then, for all
$\delta_{1},\ldots,\delta_{n}\in\mathbf{R}$ and
$\theta_{1},\ldots,\theta_{n}\in\mathbf{R}$ close enough to $0$, there exists
$\tilde{s}_{1},\ldots,\tilde{s}_{n}$ in $\gamma$ and $\tilde{\Phi}$ solution
of:

(3.1)
$\left\\{\begin{array}[]{rllll}\mathfrak{J}_{\gamma}\tilde{\Phi}&=&0\quad\text{on}\quad\gamma-\\{\tilde{s}_{1},\ldots,\tilde{s}_{n}\\},\\\\[8.53581pt]
\tilde{\Phi}(\tilde{s}_{j})&=&1+\delta_{j}\quad\text{for}\quad
j=1,\ldots,n,\\\\[8.53581pt]
\partial_{s}\tilde{\Phi}(\tilde{s}_{j}^{+})+\partial_{s}\tilde{\Phi}(\tilde{s}_{j}^{-})&=&\theta_{j}\quad\text{for}\quad
j=1,\ldots,n.\end{array}\right.$

Moreover, $\tilde{\Phi}$ depends smoothly on the $\delta_{j}$ and the
$\theta_{j}$, and $\tilde{\Phi}=\Phi$ when $\delta_{j}=\theta_{j}=0$ for
$j=1,\ldots,n$. Finally, the points $\tilde{s}_{1},\ldots,\tilde{s}_{n}$ and
the function $\tilde{\Phi}$ depend smoothly on the parameters
$\delta_{1},\ldots,\delta_{n}\in\mathbf{R}$ and
$\theta_{1},\ldots,\theta_{n}\in\mathbf{R}$ (provided these are close enough
to $0$).

###### Proof.

Keeping the notation of the proof of Proposition 2.5, we can write

$\psi_{j}(s_{j}+s)=1+\delta_{j}^{+}+\partial_{s}(\phi_{j}+\eta_{j})(s_{j})\,s-K_{g}(s_{j})(1+\delta_{j}^{+})\frac{s^{2}}{2}+\mathcal{O}(s^{3}),$

and

$\psi_{j-1}(s_{j}+s)=1+\delta_{j}^{-}+\partial_{s}(\phi_{j-1}+\eta_{j-1})(s_{j})\,s-K_{g}(s_{j})(1+\delta_{j}^{-})\frac{s^{2}}{2}+\mathcal{O}(s^{3}),$

Given the $\theta_{j}$ and the $\delta_{j}$, we determine $\tilde{s}_{j}$
close to $s_{j}$, and $\delta_{j}^{\pm}$ close to $0$ such that

$\psi_{j-1}(\tilde{s}_{j})=\psi_{j-1}(\tilde{s}_{j})=1+\delta_{j},$

and

$\partial_{s}(\psi_{j}+\psi_{j-1})(\tilde{s}_{j})=\theta_{j}.$

With little work, we get

$\psi_{j}(\tilde{s}_{j})=\psi_{j-1}(\tilde{s}_{j})=1+\frac{\delta_{j}^{+}+\delta_{j}^{-}}{2}+\sum_{i}\mathcal{O}(\delta_{i}^{2}+\theta_{i}^{2}),$
$\partial_{s}(\phi_{j}-\phi_{j-1})(s_{j})\,(\tilde{s}_{j}-s_{j})+\delta_{j}^{+}-\delta_{j}^{-}=\sum_{i}\mathcal{O}(\delta_{i}^{2}+\theta_{i}^{2}),$

and

$\partial_{s}\left(\psi_{j}+\psi_{j-1}\right)(\tilde{s}_{j})=\displaystyle\partial_{s}(\eta_{j}+\eta_{j-1})(s_{j})-2K_{g}(s_{j})\,(\tilde{s}_{j}-s_{j})\displaystyle+\sum_{i}\mathcal{O}(\delta_{i}^{2}+\theta_{i}^{2}).$

So our equations become

$\delta_{j}^{+}+\delta_{j}^{-}=2\,\delta_{j}+\sum_{i}\mathcal{O}(\delta_{i}^{2}+\theta_{i}^{2}),$

and

$\displaystyle\partial_{s}(\eta_{j}+\eta_{j-1})(s_{j})+2\,\frac{K_{g}(s_{j})}{\partial_{s}(\phi_{j}-\phi_{j-1})(s_{j})}(\delta_{j}^{+}-\delta_{j}^{-})=\theta_{j}+\displaystyle\sum_{i}\mathcal{O}(\delta_{i}^{2}+\theta_{i}^{2}).$

The non-degeneracy of the Bouncing Jacobi Field ensures that the linear map

$(\delta^{-}_{j},\delta^{+}_{j})_{j=1,\ldots,n}\mapsto\left(\delta^{+}_{j}+\delta^{-}_{j},\partial_{s}(\eta_{j}+\eta_{j-1})(s_{j})+K_{g}(s_{j})\,\displaystyle\frac{\delta^{+}_{j}-\delta^{-}_{j}}{N_{j}}\right)_{j=1,\ldots,n},$

defined from $(\mathbf{R}^{2})^{n}$ into $(\mathbf{R}^{2})^{n}$, is one-to-one
and hence, it is invertible. Recall that the functions $\eta_{j}$ solve
$\mathfrak{J}_{\gamma}\eta_{j}=0$ in $(s_{j},s_{j+1})$ with
$\eta_{j}(s_{j})=\delta_{j}^{+}$ and $\eta_{j}(s_{j+1})=\delta^{-}_{j+1}$ and
hence they depend linearly on the $\delta^{\pm}_{j}$.

The inverse function Theorem can be applied to determine the
$\delta^{\pm}_{j}$ as functions of the $\delta_{i}$ and $\theta_{i}$ provided
these are close to $0$. This completes the proof of the existence of
$\tilde{\Phi}$ solution of (3.1). ∎

There is a functional which is closely related to the second variation of
$\mathcal{H}_{n}$ and which will be very useful when we will need to determine
the Morse index of the solutions of the Jacobi-Toda system and then of the
solutions of the Allen-Cahn equation.

Given a Bouncing Jacobi Field $\Phi$ with $n$ minimums at
$s_{1}<\ldots<s_{n}$, we define

(3.2)
$\begin{array}[]{lllll}\mathcal{Q}(\eta):=\displaystyle\sum_{j=1}^{n}{\bigg{\\{}}\int_{s_{j}}^{s_{j+1}}\left((\partial_{s}\eta)^{2}-K_{g}\,\eta^{2}\right)\,ds-\frac{K_{g}(s_{j})}{N_{j}}\,\left(\eta(s_{j}^{+})-\eta(s_{j}^{-})\right)^{2}{\bigg{\\}}},\end{array}$

for all functions $\eta:\gamma-\\{s_{1},\ldots,s_{n}\\}\to\mathbf{R}$ such
that $\eta_{|(s_{j},s_{j+1})}\in H^{1}(s_{j},s_{j+1})$ and
$\eta(s_{j}^{+})+\eta_{j}(s_{j}^{-})=0$ for all $j=1,\ldots,n$. As usual,
$N_{j}:=\partial_{s}\Phi(s_{j}^{+})-\partial_{s}\Phi(s_{j}^{-})$. We now prove
that the index of $\mathcal{Q}$ at $\Phi$ (which is the maximal dimension of
the space of functions over which $\mathcal{Q}$ is negative definite), is
equal to the index of the Bouncing Jacobi Field $\Phi$ as defined in the
introduction.

######  Proposition 3.4.

Given a Bouncing Jacobi Field $\Phi$ with $n$ minimums at
$s_{1}<\ldots<s_{n}$, its index is equal to the index of $\mathcal{Q}$ at
$\Phi$

###### Proof.

Let $E$ be the vector space spanned by the functions
$\xi^{i}:\gamma-\\{s_{1},\ldots,s_{n}\\}\to\mathbf{R}$, for $i=1,\ldots,n$,
where $\xi^{i}$ is a solutions of the Jacobi equation
$\mathfrak{J}_{\gamma}\xi^{i}=0$ on each $(s_{j},s_{j+1})$ with boundary data

$\xi^{i}(s_{j}^{-})=\xi_{j}(s_{j}^{+})=0\quad\text{ for all }\quad j\neq
i\quad\text{ and }\quad\xi^{i}(s_{i}^{+})=-\xi^{i}(s_{i}^{-})=1.$

The existence of $\xi^{i}$ follows from the fact that, since we have a
Bouncing Jacobi Field, the operator $\mathfrak{J}_{\gamma}$, restricted to
$H^{1}_{0}(s_{j},s_{j+1})$, is non-degenerate. It follows from the alternative
formula in Proposition 3.1, that the index of the quadratic form
$\mathcal{Q}$, restricted to $E$, is equal to the index of $\mathcal{H}_{n}$
at the point $(s_{1},\ldots,s_{n})$.

Next, we define $F$ to be the vector space of functions
$\psi:\gamma-\\{s_{1},\ldots,s_{n}\\}\to\mathbf{R}$ such that
$\psi_{|(s_{j},s_{j+1})}\in H^{1}_{0}(s_{j},s_{j+1})$, for all $j=1,\ldots,n$.
The existence of the Bouncing Jacobi Field $\Phi$ implies that the operator
$\mathfrak{J}_{\gamma}$ has index $0$ and nullity $0$ on each
$(s_{j},s_{j+1})$), therefore, the restriction of the quadratic form
$\mathcal{Q}$ to $F$ is positive definite.

Now, observe that any $\eta:\gamma-\\{s_{1},\ldots,s_{n}\\}\to\mathbf{R}$ with
$\eta_{|(s_{j},s_{j+1})}\in H^{1}(s_{j},s_{j+1})$ and
$\eta(s_{j}^{+})+\eta(s_{j}^{-})=0$, for $j=1,\ldots,n$, can be uniquely
decomposed as

$\eta=\xi+\psi,$

where $\xi\in E$ and $\psi\in F$. Moreover, a simple integration by parts
shows that

$\mathcal{Q}(\eta)=\mathcal{Q}(\xi)+\mathcal{Q}(\psi).$

The index of $\mathcal{Q}$ restricted to $E$ being equal to the index of
$\mathcal{H}_{n}$ and $\mathcal{Q}$ being positive definite on $F$, we
conclude that the index of $\mathcal{Q}$ is equal to the index of
$\mathcal{H}_{n}$. ∎

##  4\. The Jacobi-Toda equation

With some existence results for Bouncing Jacobi Fields at hand, we prove in
this section the existence of solutions $\Psi:\gamma\to\mathbf{R}$ to the so-
called Jacobi-Toda equation:

(4.1) $\varepsilon^{2}\,\mathfrak{J}_{\gamma}\Psi+\bar{c}^{2}\,e^{-2\Psi}=0,$

where $\Psi:\gamma\to\mathbf{R}$ and $\bar{c}>0$ is a constant which will be
defined later on.

Actually we are interested in solutions of this equation whose Morse index is
bounded uniformly in $\varepsilon$ since, as we will see later on, these
solutions will hopefully lead to the construction of solutions of the Allen-
Cahn equation which have $2$ transition layers converging to $\gamma$ and also
have uniformly bounded Morse indices. We will show that solutions of the
Jacobi-Toda equation can be obtained by desingularizing a Bouncing Jacobi
Fields using solutions of a Toda equation.

For all $\varepsilon>0$, it will be convenient to define
$\alpha_{\varepsilon}>0$ to be the unique solution of

(4.2) $\alpha_{\varepsilon}\,e^{\alpha_{\varepsilon}}=\frac{1}{\varepsilon}.$

It is an easy exercise to check that, as a function of $\varepsilon$,
$\alpha_{\varepsilon}$ can be expanded as

$\alpha_{\varepsilon}=-\ln\varepsilon-\ln\,(-\ln\varepsilon)+\mathcal{O}\left(\frac{\ln\,(-\ln\varepsilon)}{\ln\varepsilon}\right),$

as $\varepsilon$ tends to $0$.

In the following, we agree that $C$ are positive constants, which may depend
from one inequality to the next, but which can be fixed large enough
independently of $\varepsilon$. These constants only depend on the geometry of
our problem, namely the geodesic, curvature of the surface along the geodesic
and the Bouncing Jacobi Field we consider to start the construction.
Similarly, we agree that $c$ are positive constants which may also depend from
one inequality to the next, but can be fixed small enough independently of
$\varepsilon$. Again, the constants $c$ only depend on the geometry of our
problem and and the Bouncing Jacobi Field we consider.

Given a non-degenerate Bouncing Jacobi Field $\Phi$ with $n$ minimums at
$s_{1}<\ldots<s_{n}$, we first explain how to perturb
$\alpha_{\varepsilon}\,\Phi$ into a $2n$-dimensional family of solutions of
(4.1) which are defined away from small intervals centered at the $s_{j}$’s.

Assume we are given $\delta_{j}\in\mathbf{R}$ and $\theta_{j}\in\mathbf{R}$
such that

$|\delta_{j}|+\alpha_{\varepsilon}\,|\theta_{j}|\leq\frac{C}{\alpha_{\varepsilon}},$

where $C>0$ is a constant fixed large enough independently of $\varepsilon$.

Proposition 3.3 provides a continuous function
$\tilde{\Phi}=\tilde{\Phi}(\delta_{1},\ldots,\delta_{n},\theta_{1},\ldots,\theta_{n})$
which has $n$ minimums at points $\tilde{s}_{1}<\ldots<\tilde{s}_{n}$ (each of
which is a function of the $\delta_{j}$’s and the $\theta_{j}$’s), and which
solves the equation $\mathfrak{J}_{\gamma}\,\tilde{\Phi}=0$ in each
$(\tilde{s}_{j},\tilde{s}_{j+1})$ with initial data given by

$\tilde{\Phi}(\tilde{s}_{j})=1+\delta_{j}\qquad\text{and}\qquad\partial_{s}\tilde{\Phi}(\tilde{s}_{j}^{+})+\partial_{s}\tilde{\Phi}(\tilde{s}_{j}^{-})=\theta_{j}.$

Observe that the function $\tilde{\Phi}$ and the $\tilde{s}_{j}$’s depend
(smoothly) on the parameters $\delta_{j}$ and $\theta_{j}$, for $j=1,\ldots,n$
and we have the estimate

$\sup_{j}|\tilde{s}_{j}-s_{j}|\leq\frac{C}{\alpha_{\varepsilon}}.$

We are now going to perturb the function $\alpha_{\varepsilon}\,\tilde{\Phi}$
into a function $\Psi$ which solves the Jacobi-Toda equation (4.1), away from
some small intervals of size $2C/\alpha_{\varepsilon}$, centered at the
$\tilde{s}_{j}$’s. We are solving a second order ordinary differential
equation and since we do not prescribe the boundary data of $\Psi$ at the ends
of the intervals, we need to address the issue of the non-uniqueness of the
solution. This is the reason why, for each $j=1,\ldots,n$, we ask that

$\Psi(\hat{s}_{j})=\alpha_{\varepsilon}\,\tilde{\Phi}(\hat{s}_{j})\qquad\text{and}\qquad\partial_{s}\Psi(\hat{s}_{j})=\alpha_{\varepsilon}\partial_{s}\tilde{\Phi}(\hat{s}_{j}),$

where $\hat{s}_{j}$ is the middle of the interval
$(\tilde{s}_{j},\tilde{s}_{j+1})$. This choice might seem quite arbitrary and
the sceptic reader might ask if this is a reasonable choice. Let us just
recall that we are solving a second order ordinary differential equation, so
the space of solutions is $2$-dimensional. But in our construction we have two
parameters $\delta_{j}$ and $\theta_{j}$ which are free and can be used to
parameterize, close to $\alpha_{\varepsilon}\,\Phi$, the set of all solutions
to the equation.

The existence of a solution $\Psi$ close to
$\alpha_{\varepsilon}\,\tilde{\Phi}$ follows from a perturbation argument, the
rational being that, away from small intervals centered at the
$\tilde{s}_{j}$’s, the function $\alpha_{\varepsilon}\,\tilde{\Phi}$ is much
larger than $\alpha_{\varepsilon}$ and hence the term
$\bar{c}^{2}\,e^{-2\alpha_{\varepsilon}\,\tilde{\Phi}}$ in the equation is
small enough to ensure the perturbation argument to work.

To be more precise, we have the:

######  Proposition 4.1.

There exist constants $C>0$ and, for all $\varepsilon>0$ small enough, there
exists a function
$\psi=\psi(\varepsilon,\delta_{1},\ldots,\delta_{n},\theta_{1},\ldots\theta_{n})$
such that

$\Psi:=\alpha_{\varepsilon}\,(\tilde{\Phi}+\psi),$

solves $\mathfrak{J}_{\gamma}\,\Psi=0$ in
$(\tilde{s}_{j}+\frac{C}{\alpha_{\varepsilon}},\tilde{s}_{j+1}-\frac{C}{\alpha_{\varepsilon}})$
with

$\Psi=\alpha_{\varepsilon}\tilde{\Phi}\qquad\text{and}\qquad\partial_{s}\Psi=\alpha_{\varepsilon}\partial_{s}\tilde{\Phi},$

at $\hat{s}_{j}$, for all $j=1,\ldots,n$. Moreover, the function $\psi$
depends smoothly on the $\delta_{j}$’s and $\theta_{j}$’s and satisfies the
estimates

$|\psi|+|\partial_{\delta_{j}}\psi|+|\partial_{\theta_{j}}\psi|\leq
C\,e^{-\frac{\alpha_{\varepsilon}}{C}\,|s-\tilde{s}_{j}|},$

and

$|\partial_{s}\psi|+|\partial_{\delta_{j}}\partial_{s}\psi|+|\partial_{\theta_{j}}\partial_{s}\psi|\leq
C\,\alpha_{\varepsilon}\,e^{-\frac{\alpha_{\varepsilon}}{C}\,|s-\tilde{s}_{j}|}.$

###### Proof.

If $f:\gamma-\\{\tilde{s}_{1},\ldots,\tilde{s}_{n}\\}\to\mathbf{R}$, we define
the operator $\mathcal{S}$ by

$\mathcal{S}(f)(s):=-\tilde{\Phi}(s)\int_{\hat{s}_{j}}^{s}\frac{1}{(\tilde{\Phi}(t))^{2}}\left(\int_{\hat{s}_{j}}^{t}\tilde{\Phi}(t^{\prime})\,f(t^{\prime})\,dt^{\prime}\,\right)\,dt,$

for all $s\in(\tilde{s}_{j},\tilde{s}_{j+1})$. Since $\tilde{\Phi}$ is a
Jacobi field which does not vanish on $(\tilde{s}_{j},\tilde{s}_{j+1})$, the
operator ${\mathcal{S}}$ is well defined and we have

$-(\partial_{s}^{2}+K_{g})\,{\mathcal{S}}(f)=f,$

on $\gamma-\\{\tilde{s}_{1},\ldots,\tilde{s}_{n}\\}$ with
${\mathcal{S}}(f)(\hat{s}_{j})=\partial_{s}{\mathcal{S}}(f)(\hat{s}_{j})=0$.

We can then rephrase the problem as finding $\psi$, solution of

$\psi=-\frac{1}{\alpha_{\varepsilon}}\mathcal{S}\left(\frac{\bar{c}^{2}}{\varepsilon^{2}}\,e^{-2\alpha_{\varepsilon}\,(\tilde{\Phi}+\psi)}\right).$

It is a simple (but tedious) exercise to check that, there exists $C>0$ large
enough, independent of $\varepsilon$, such that this nonlinear equation has a
solution well defined on each
$(\tilde{s}_{j}+\frac{C}{\alpha_{\varepsilon}},\tilde{s}_{j+1}-\frac{C}{\alpha_{\varepsilon}})$.
The solution can be obtained applying a fixed point argument for contraction
mappings. Moreover, using (4.2), we get the estimates

$\alpha_{\varepsilon}\,|\psi(s)|+|\partial_{s}\psi(s)|\leq
C\,\alpha_{\varepsilon}\,e^{-\frac{\alpha_{\varepsilon}}{C}\,|s-\tilde{s}_{j}|}.$

Similar estimates can be derived for the partial derivative of $\psi$ with
respect to the parameters $\delta_{j}$ and $\theta_{j}$. ∎

We now turn to the second ingredient in our construction. To start with,
observe that the Toda equation

(4.3) $\varepsilon^{2}\,\partial_{s}^{2}T+\bar{c}^{2}\,e^{-2T}=0,$

where $T:\mathbf{R}\to\mathbf{R}$, can be explicitly integrated and all
solutions are given by

$s\mapsto\ln\left(\cosh\left(\frac{\kappa}{\varepsilon}\,(s-\bar{s})s\right)\right)-\ln\kappa+\ln\bar{c},$

for some $\kappa>0$ and some $\bar{s}\in\mathbf{R}$. Our next result states
that one can perturb this function into a solution of

(4.4) $\varepsilon^{2}\,\mathfrak{J}_{\gamma}T+\bar{c}^{2}\,e^{-2T}=0$

on some small interval centered at $\bar{s}$, with initial conditions given by

$T(\bar{s})=\partial_{s}T(\bar{s})=0.$

The rational being that in the regime where the term $\varepsilon^{2}\,K_{g}T$
is small compared to the other terms, a perturbation argument will lead to a
solution $v$ of (4.4) close to $T_{0}$. Again, the initial conditions we
impose might seem arbitrary but the interested reader will check that playing
with the parameters $\kappa$ and translation $\bar{s}$, one can prescribe any
small value for the function $T$ and its first derivative at a fixed point
$\bar{s}$.

We define

$T_{0}(s):=\ln\left(\cosh\left(\frac{\kappa}{\varepsilon}\,(s-\bar{s})s\right)\right)-\ln\kappa+\ln\bar{c}+K_{g}(\bar{s})\,\ln\kappa\,\frac{(s-\bar{s})^{2}}{2}.$

With this definition at hand, we have the:

######  Proposition 4.2.

For all $\varepsilon>0$ small enough, all $\bar{s}\in\mathbf{R}$ and all
$\kappa>0$ satisfying

$\frac{1}{C}\,\alpha_{\varepsilon}\leq\frac{\kappa}{\varepsilon}\leq
C\,\alpha_{\varepsilon},$

there exists $c>0$ and a unique $T=T(\varepsilon,\kappa,\bar{s})$ solution of
(4.4) on $[\bar{s}-c,\bar{s}+c]$ which depends smoothly on $\varepsilon$,
$\bar{s}$ and $\kappa$. Moreover

$|(T-T_{0})(s)|+|\kappa\,\partial_{\kappa}(T-T_{0})(s)|+|\partial_{\bar{s}}(T-T_{0})(s)|\leq
C\,\left(\alpha_{\varepsilon}\,|s-\bar{s}|^{3}+\frac{|s-\bar{s}|}{\alpha_{\varepsilon}}\right),$

and

$|\partial_{s}(T-T_{0})(s)|+|\kappa\,\partial_{\kappa}\partial_{s}(T-T_{0})(s)|+|\partial_{\bar{s}}\partial_{s}(T-T_{0})(s)|\leq
C\,\left(\alpha_{\varepsilon}\,|s-\bar{s}|^{2}+\frac{1}{\alpha_{\varepsilon}}\right),$

for all $\displaystyle|s-\bar{s}|<c$.

###### Proof.

We write

$T:=T_{0}+S,$

where we hope that $S$ will be small. Inserting this into (4.4) and using the
fact that

$\bar{c}^{2}\,e^{-2T_{0}}=\frac{\kappa^{2}}{(\cosh(\kappa
s/\varepsilon))^{2}},$

we can rewrite the equation we want to solve as

$\partial_{s}^{2}S=-K_{g}\,(T_{0}+S)-\frac{(\kappa/\varepsilon)^{2}}{(\cosh(\kappa\cdot/\varepsilon))^{2}}\,(1-e^{-2S}).$

with $S(\bar{s})=\partial_{s}S(\bar{s})=0$. Hence

(4.5)
$S=\int_{\bar{s}}^{s}\left(\int_{\bar{s}}^{t}\left(-K_{g}\,(T_{0}+S)-\frac{(\kappa/\varepsilon)^{2}}{(\cosh(\kappa\cdot/\varepsilon))^{2}}\,(1-e^{-2S})\right)\,dt\right)\,ds.$

Observe that, on the right-hand-side, the term which does not depend on $S$
can be estimated by

$\left|\int_{\bar{s}}^{s}\left(\int_{\bar{s}}^{t}K_{g}\,T_{0}\,dt\right)\,ds+K_{g}(\bar{s})\,\ln\kappa\,\frac{(s-\bar{s})^{2}}{2}\right|\leq
C\,\left(\alpha_{\varepsilon}\,|s-\bar{s}|^{3}+\frac{|s-\bar{s}|}{\alpha_{\varepsilon}}\right),$

and its derivative is given by

$\left|\int_{\bar{s}}^{s}K_{g}\,T_{0}\,dt+K_{g}(\bar{s})\,\ln\kappa\,\frac{(s-\bar{s})^{2}}{2}\right|\leq
C\,\left(\alpha_{\varepsilon}\,|s-\bar{s}|^{2}+\frac{1}{\alpha_{\varepsilon}}\right).$

Once these estimates have been derived, it is easy exercise the existence of
$S$ using a fixed-point argument building on the fact that, in (4.5), the
linear terms in $S$

$\int_{\bar{s}}^{s}\left(\int_{0}^{t}-K_{g}\,S\,dt\right)\,ds,$

and the nonlinear terms

$\int_{\bar{s}}^{s}\left(\int_{0}^{t}\frac{(\kappa/\varepsilon)^{2}}{(\cosh(\kappa\cdot/\varepsilon))^{2}}\,(e^{-2S}-1)\,dt\right)\,ds,$

can be considered as perturbations provided $|s-\bar{s}|<c$ where $c>0$ is
fixed small enough, independently of $\varepsilon$. We leave the details to
the interested reader. ∎

As alluded above, the idea for constructing solutions of the Jacobi-Toda
equation (4.1) is start with a non-degenerate Bouncing Jacobi Field $\Phi$
with minimums at points of coordinates $s_{1}<\ldots<s_{n}$. We then perturb
$\alpha_{\varepsilon}\,\Phi$ as we have done in Proposition 4.1. This yields a
solution $\Psi$ of (4.1) which is well defined away from small intervals of
size $C/\alpha_{\varepsilon}$ centered at the $s_{j}$’s and which depends on
the $2n$ parameters $\delta_{j}$ and $\theta_{j}$, for $j=1,\ldots,n$. Next we
apply the last Proposition using the parameters $\kappa=\kappa_{j}$ and
$\bar{s}=\bar{s}_{j}$ to define functions $T_{j}$ which solve (4.1) on some
small interval centered at the $s_{j}$’s. Our aim is to prove that one can
choose the parameters $\delta_{j}$, $\theta_{j}$, $\kappa_{j}$ and
$\bar{s}_{j}$ in such a way that $\Psi$ and $v_{j}$ coincide on the intervals
where they are both defined. Since all functions are solution of the same
second order differential equation, it is enough to check that one can choose
the $4n$ parameters so that the value and derivatives of the different
functions agree at one point of the intervals where they are both defined and
this will produce the solution of the Jacobi-Toda equation we are looking for.

To be more precise, we have the:

######  Proposition 4.3.

Suppose $\Phi$ is a non-degenerate Bouncing Jacobi Field with $n$ minimums at
$s_{1},\ldots,s_{n}$. Then, there exists a constant $C>0$ and, for all
$\varepsilon>0$ small enough, there exists $\Psi_{\varepsilon}$ a solution of
(4.1) obtained by patching together the solutions
$T(\varepsilon,\kappa_{j},\bar{s}_{j})$ given by Proposition 4.2 centered at
points close to the $s_{j}$’s and the solution
$\Psi(\varepsilon,\delta_{j},\theta_{j})$ given by Proposition 4.1 for
parameters satisfying

$\left|\frac{\kappa_{j}}{\varepsilon}-\alpha_{\varepsilon}\,\frac{N_{j}}{2}\right|+\alpha_{\varepsilon}^{2}\,\left|\delta_{j}+\frac{\ln(N_{j}/\bar{c})}{\alpha_{\varepsilon}}\right|+\alpha_{\varepsilon}^{2}\,|\theta_{j}|+\alpha_{\varepsilon}\,|\bar{s}_{j}-s_{j}|\leq
C.$

###### Proof.

For all $j=1,\ldots,n$, we define $T_{j}$ to be the solution of (4.1) given by
Proposition 4.2 when $\kappa=\kappa_{j}$ and $\bar{s}=\bar{s}_{j}$. This
function can be expanded as

$T_{j}(s)=\frac{\kappa_{j}}{\varepsilon}\,|s-\bar{s}_{j}|-\ln\left(\frac{2\,\kappa_{j}}{\bar{c}}\right)+K_{g}(\bar{s}_{j})\,\ln\kappa_{j}\,\frac{(s-\bar{s}_{j})^{2}}{2}+\mathcal{O}\left(\alpha_{\varepsilon}\,(s-\bar{s}_{j})^{3}\right),$

for $C/\alpha_{\varepsilon}\leq|s-\bar{s}_{j}|\leq c$.

Next, we expand the solution $\Psi$ obtained in Proposition 4.1 as

$\Psi(s)=\displaystyle\alpha_{\varepsilon}\,\left(1+\delta_{j}+\partial_{+}\tilde{\Phi}(\tilde{s}_{j})\,|s-\tilde{s}_{j}|-K_{g}(\tilde{s}_{j})\,\frac{(s-\tilde{s}_{j})^{2}}{2}\right)+\mathcal{O}(e^{-\alpha_{\varepsilon}|s-\tilde{s}_{j}|/C}),$

for $|s-\tilde{s}_{j}|\geq C/\alpha_{\varepsilon}$.

We look for the parameters so that the following system of equation is
fulfilled

(4.6)
$\Psi(s)=T_{j}(s)\qquad\text{and}\qquad\partial_{s}\Psi(s)=\partial_{s}T_{j}(s),$

for all $s$ satisfying

$\frac{C}{\alpha_{\varepsilon}}\leq|s-s_{j}|\leq c.$

Using the estimates we have obtained in the previous two Propositions, one can
apply some fixed point argument (or the implicit function Theorem) to get the
existence of parameters $\kappa_{j}$, $\bar{s}_{j}$, $\delta_{j}$ and
$\theta_{j}$ for which (4.6) holds. The estimates in the statement follow from
these equations evaluated at $s-s_{j}=\alpha_{\varepsilon}^{-2/3}$. ∎

######  Remark 4.4.

In [6], the Jacobi-Toda equation

$\varepsilon^{2}\,{\mathfrak{J}}_{\gamma}\Psi+\bar{c}^{2}\,e^{-2\Psi}=0$

also plays a very important role as it does in the present paper. To explain
the difference between the two approaches, let us consider the case where the
Gauss curvature is constant positive.

In this case it is clear that there exists a constant solution of the Jacobi-
Toda equation which can be expanded as

$\Psi=-\ln\varepsilon-\frac{1}{2}\,\ln\left(-\ln\varepsilon\right)+\mathcal{O}(1),$

In the case where the Gauss curvature is not constant, one can get an
approximate solution of the Jacobi-Toda equation by using a fixed point
iteration scheme

$\Psi_{k+1}=\frac{1}{K_{g}}\left(\frac{\bar{c}^{2}}{\varepsilon^{2}}\,e^{-2\Psi_{k}}-\partial_{s}^{2}\Psi_{k}\right).$

which, for $k$ large enough, can be expanded as

$\Psi_{k}=-\ln\varepsilon-\frac{1}{2}\,\ln\left(-\ln\varepsilon\right)+\mathcal{O}(1),$

where, this time, $\mathcal{O}(1)$ depends on $s$.

In contract, our solution of the Jacobi Toda equation is close to
$\alpha_{\varepsilon}\,\Phi$ where $\Phi$ is a Bouncing Jacobi Field and where
$\alpha_{\varepsilon}$ is determined by (4.2).

##  5\. The Morse index of solutions to the Jacobi-Toda equation

In the previous section, starting from a non-degenerate Bouncing Jacobi Field
$\Phi$, we have constructed a solution $\Psi_{\varepsilon}$ of the Jacobi-Toda
equation (4.1), for all $\varepsilon>0$ small enough.

Our aim is now to analyse the mapping properties and the spectrum of
$\mathcal{L}_{\varepsilon}$, the linearized Jacobi-Toda equation about
$\Psi_{\varepsilon}$

$\mathcal{L}_{\varepsilon}:=\varepsilon^{2}\,\mathfrak{J}_{\gamma}-2\,\bar{c}^{2}\,e^{-2\Psi_{\varepsilon}}.$

In particular, we are interested in the Morse index of the solution
$\Psi_{\varepsilon}$.

Since $\bar{c}^{2}e^{-2\Psi_{\varepsilon}}>0,$ we know that the Morse index of
the operator $\mathcal{L}_{\varepsilon}$ is bounded from below by the Morse
index of the geodesic $\gamma$ which, by definition, is the Morse index of the
Jacobi operator $\mathfrak{J}_{\gamma}$ also defined on
$H^{1}\left(\gamma\right)$. Therefore, we have

$\text{Ind}(\Psi_{\varepsilon})\geq\text{Ind}(\gamma).$

We will prove that, for all $\varepsilon>0$ close enough to $0$, the operator
$\mathcal{L}_{\varepsilon}$ is injective and that it has Morse index equal to
$n+\text{Ind}\,(\Phi)$, where $n$ is the number of mimimums of the Bouncing
Jacobi Field $\Phi$ we have used to construct $\Psi_{\varepsilon}$ and
$\text{Ind}(\Phi)$ is the Morse index of $\Phi$ as defined in the introduction
which agrees with the Morse index of the Hessian of $\mathcal{H}_{n}$. The
rational behind this result is that the negative eigenvalues of
$\mathcal{L}_{\varepsilon}$ arise either from negative directions of
deformation of the critical point with respect to the functional
$\mathcal{H}_{n}$, which in turn correspond to negative eigenvalues of the
quadratic form $\mathcal{Q}$ which has been defined in (3.2), or they arise
from the fact that, close to the points where $\Phi$ is minimal,
$\Psi_{\varepsilon}$ is close to a solution of the Toda equation whose
linearized operator has one negative eigenvalue and, since we have used $n$
copies of a solution of the Toda equation to construct $\Psi_{\varepsilon}$,
this contributes to $n$ additional negative eigenvalues of
$\mathcal{L}_{\varepsilon}$.

As a warm up, let us analyse the spectrum of the operator

$L_{0}:=-\left(\partial_{s}^{2}+\frac{2}{(\cosh s)^{2}}\right).$

which arises as the linearized operator of the Toda equation

$\partial_{s}^{2}T-e^{-2T}=0,$

at

$T(s):=\ln\,(\cosh s).$

It can be checked through direct computation that $-1$ is an eigenvalue
associated to the eigenfunction

$s\mapsto\frac{1}{\cosh s}.$

In fact it is known that this is the only negative eigenvalue and hence
$L_{0}$ has index one. The continuous spectrum of $L_{0}$ is given by
$[0,+\infty)$ and the null space of $L_{0}$ is spanned by the functions

$\varphi_{0}^{-}(s):=\tanh s\qquad\text{and}\qquad\varphi_{0}^{+}(s):=s\,\tanh
s-1.$

These functions reflect the fact that the Toda equation is invariant under
translation and scaling, namely, if $T$ is a solution of the Toda equation,
then $s\mapsto T(\kappa\,s+\bar{s})-\ln\kappa$ is also a solution for all
$\kappa>0$ and $\bar{s}\in\mathbf{R}$.

The main result of this section reads:

######  Proposition 5.1.

For all $\varepsilon>0$ close enough to $0$, the Morse index of the operator
$\mathcal{L}_{\varepsilon}$ is equal to

$n+\text{\rm Ind}(\Phi),$

where $n$ is the number of minimums of the non-degenerate Bouncing Jacobi
Field $\Phi$ we have used in Proposition 4.3 to construct
$\Psi_{\varepsilon}$.

The proof of this Proposition is decomposed into two parts. First, we prove
that, provided $\varepsilon$ is small enough, the index of
$\mathcal{L}_{\varepsilon}$ is _at least_ equal to $n+\text{\rm Ind}(\Phi)$
and then, we prove that it is at most equal to this value and that
$\mathcal{L}_{\varepsilon}$ does not have any kernel.

We start with the:

######  Lemma 5.2.

For all $\varepsilon$ small enough, the index of $\mathcal{L}_{\varepsilon}$
is _at least_ equal to $n+\text{\rm Ind}(\Phi)$.

###### Proof.

The construction in Proposition 4.3 tells that $\Psi_{\varepsilon}$ has local
minimums at points $\bar{s}_{j}$ which depend on $\varepsilon$ but are close
to the points $s_{j}$ where the Bouncing Jacobi Field $\Phi$ is minimal. It
also follows from Proposition 4.3 and Proposition 4.1, that, in a fixed small
neighborhood of $\bar{s}_{j}$ (which contains $s_{j}$), the function
$\Psi_{\varepsilon}$ can be expanded as

$\begin{array}[]{rllll}\Psi_{\varepsilon}(s)&=&\displaystyle\ln\left(\cosh\left(\frac{\kappa_{j}}{\varepsilon}(s-\bar{s}_{j})\right)\right)-\ln\kappa_{j}+\ln\bar{c}+K_{g}(\bar{s})\,\ln\kappa_{j}\,\frac{(s-\bar{s}_{j})^{2}}{2}\\\\[8.53581pt]
&+&\displaystyle\mathcal{O}\left(\alpha_{\varepsilon}\,|s-\bar{s}_{j}|^{3}+\frac{|s-\bar{s}_{j}|}{\alpha_{\varepsilon}}\right),\end{array}$

where $\kappa_{j}$ and $\bar{s}_{j}$ depend on $\varepsilon$.

For all nonzero function $v\in H^{1}(\gamma)$, we define

$\mathcal{Q}(v):=\displaystyle\int_{\gamma}\left(\varepsilon^{2}\,(\partial_{s}v)^{2}-\left(\varepsilon^{2}\,K_{g}+2\,\bar{c}^{2}\,e^{-2\Psi_{\varepsilon}}\right)\,v^{2}\right)\,ds.$

Given $j=1,\ldots,n$, we define the function $e_{j}:\gamma\to\mathbf{R}$ by

$e_{j}(s):=\frac{\chi(s-\bar{s}_{j})}{\displaystyle\cosh\left(\frac{\kappa_{j}}{\varepsilon}(s-\bar{s}_{j})\right)},$

where $\chi$ is a cutoff function identically equal to $1$ in some fixed small
interval $(-c,c)$ and identically equal to $0$ outside $(-2c,2c)$. Using the
fact that

$\left(\varepsilon^{2}\,\partial_{s}^{2}+\frac{2\,\kappa_{j}^{2}}{\displaystyle\cosh^{2}\left(\frac{\kappa_{j}}{\varepsilon}(\cdot-\bar{s}_{j})\right)}\right)\,\tilde{e}_{j}=\kappa_{j}^{2}\,\tilde{e}_{j},$

in $(-c,c)$, together with the estimates in Proposition 4.1, one can check
that

$\left|\mathcal{Q}(e_{j})+\kappa_{j}^{2}\,\int_{\gamma}e_{j}^{2}\,ds\right|\leq\frac{C}{\alpha_{\varepsilon}^{2}}.$

This estimate proves that, for $\varepsilon$ small enough, the vector space of
functions over which $\mathcal{Q}$ is negative definite is at least $n$
dimensional.

Now, assume that we are given a nonzero function
$\eta:\gamma-\\{s_{1},\ldots,s_{n}\\}\to\mathbf{R}$ solution of

(5.1)
$\left\\{\begin{array}[]{rlllll}\mathfrak{J}_{\gamma}\eta&=&\lambda\,\eta\quad\text{on}\quad\gamma-\\{s_{1},\ldots,s_{n}\\}\\\\[8.53581pt]
\eta(s_{j}^{+})+\eta(s_{j}^{-})&=&0\quad\text{for all}\quad
j=1,\ldots,n\\\\[8.53581pt]
\displaystyle\partial_{s}\eta(s_{j}^{+})+\partial_{-}\eta(s_{j}^{-})+\displaystyle
2K_{g}(s_{j})\,\frac{\eta(s_{j}^{+})-\eta(s_{j}^{-})}{N_{j}}&=&0\quad\text{for
all}\quad j=1,\ldots,n.\end{array}\right.$

with $\lambda\leq 0$, where we recall that
$N_{j}:=\partial_{s}\Phi(s_{j}^{+})-\partial_{s}\Phi(s_{j}^{-})$. Observe that
the function $\eta$ is not necessarily continuous at $s_{j}$.

We set

$\varphi_{j}(s):=\partial_{s}\Psi_{\varepsilon}(s),$

and

$\psi_{j}(s):=\partial_{\kappa}T_{\kappa}\,{}_{|\kappa=\kappa_{j}}(s).$

where $T_{\kappa}=T(\varepsilon,\kappa,\bar{s}_{j})$ is the solution given in
Proposition 4.2 for $\kappa$ close to $\kappa_{j}$ and $\bar{s}=\bar{s}_{j}$
and which agrees with $\Psi_{\varepsilon}$ when $\kappa=\kappa_{j}$. We have

$\left(\varepsilon^{2}\,(\partial_{s}^{2}+K_{g})+2\,\bar{c}^{2}\,e^{-2\Psi_{\varepsilon}}\right)\,\varphi_{j}=\varepsilon^{2}\,\partial_{s}K_{g}\,\Psi_{\varepsilon},$

and, since $\kappa\mapsto T_{\kappa}$ is differentiable, we also have

$\left(\varepsilon^{2}\,(\partial_{s}^{2}+K_{g})+2\,\bar{c}^{2}\,e^{-2\Psi_{\varepsilon}}\right)\,\psi_{j}=0,$

in some small neighborhood of fixed size, centered at $\bar{s}_{j}$. These
functions can be used to desingularize $\eta$, which as mentioned above, might
not even be continuous at the $s_{j}$’s. More precisely, since the function
$\varphi_{j}$ can be expanded as

$\varphi_{j}(s)=\displaystyle\frac{\kappa_{j}}{\varepsilon}\,\tanh\left(\frac{\kappa_{j}}{\varepsilon}(s-\bar{s}_{j})\right)+K_{g}(\bar{s})\,\ln\kappa_{j}\,(s-\bar{s}_{j})+\displaystyle\mathcal{O}\left(\alpha_{\varepsilon}\,(s-\bar{s}_{j})^{2}+\frac{1}{\alpha_{\varepsilon}}\right),$

and since the function $\psi_{j}$ can be expanded as

$\psi_{j}(s)=\displaystyle\frac{s-\bar{s}_{j}}{\varepsilon}\,\tanh\left(\frac{\kappa_{j}}{\varepsilon}(s-\bar{s}_{j})\right)\displaystyle+\mathcal{O}\left(\frac{1}{\kappa_{j}}+\frac{|s-\bar{s}_{j}|^{3}}{\varepsilon}+\frac{|s-\bar{s}_{j}|}{\kappa_{j}\alpha_{\varepsilon}}\right),$

We can desingularize $\eta$ in a neighborhood of $s_{j}$ by defining

$\begin{array}[]{rllll}\tilde{\eta}(s)&:=&\displaystyle(1-\chi(s-\bar{s}_{j}))\,\eta(s)+\chi(s-\bar{s}_{j})\,\frac{\varepsilon}{\kappa_{j}}\frac{\eta(s_{j}^{+})-\eta(s_{j}^{-})}{2}\,\varphi_{j}(s)\\\\[8.53581pt]
&+&\displaystyle\chi(s-\bar{s}_{j})\,\,\varepsilon\,\frac{\partial_{s}\eta(s_{j}^{+})-\partial_{s}\eta(s_{j}^{-})}{2}\,\psi_{j}(s),\end{array}$

where $\chi$ is a cutoff function identically equal to $1$ in
$(s_{j}-1/\alpha_{\varepsilon},s_{j}+1/\alpha_{\varepsilon})$ and identically
equal to $0$ outside
$(s_{j}-2/\alpha_{\varepsilon},s_{j}+2/\alpha_{\varepsilon})$.

Using the fact that

$\varepsilon\frac{\ln\kappa_{j}}{\kappa_{j}}=-\frac{2}{N_{j}}+\mathcal{O}\left(\frac{1}{\alpha_{\varepsilon}}\right),$

we conclude with some work that

$\left|\mathcal{Q}(\tilde{\eta})-\varepsilon^{2}\,\lambda\,\int_{\gamma}\tilde{\eta}^{2}\,ds\right|\leq
C\,\frac{\varepsilon^{2}}{\alpha_{\varepsilon}}.$

This estimate implies that, for $\varepsilon$ small enough, $\mathcal{D}$ is
negative definite over the vector space of functions obtained by
desingularizing the solutions of (5.1) for $\lambda<0$. We then conclude that
the vector space of functions over which $\mathcal{Q}$ is negative definite is
at least $n+\text{Ind}(\Phi)$ dimensional. ∎

We now turn to the second part of the proof. This is the content of the:

######  Lemma 5.3.

For all $\varepsilon$ small enough, the index of $\mathcal{L}_{\varepsilon}$
is at most equal to $n+\text{\rm Ind}(\Phi)$ and $\mathcal{L}_{\varepsilon}$
is injective.

###### Proof.

Let us assume that we have a sequence $(\varepsilon_{k})_{k=0}^{\infty}$
tending to $0$ and $\Psi_{\varepsilon_{k}}$ the corresponding solutions of the
Jacobi-Toda equation given by Proposition 4.3 and $w_{k}$ a non-zero
eigenfunction of $\mathcal{L}_{\varepsilon_{k}}$, associated to a non-positive
eigenvalue $\lambda_{k}\leq 0$, namely

$\mathcal{L}_{\varepsilon_{k}}\,w_{k}=\lambda_{k}\,w_{k}.$

Without loss of generality, we can normalize $w_{k}$ so that
$\|w_{k}\|_{L^{\infty}}=1$ and choose a point $t_{k}$ in $\gamma$ where
$w_{k}(t_{k})=1$.

We now distinguish two cases.

Case 1. Assume that, up to a subsequence, $(t_{k})_{k\geq 0}$ converges to a
point $t_{\infty}\notin\\{s_{1},\ldots,s_{n}\\}$, or that it converges
_slowly_ to some $s_{j}$ in the sense that

$A:=\lim_{k\to+\infty}\frac{1}{\varepsilon_{k}^{2}}e^{-2\Psi_{\varepsilon_{k}}(t_{k})}<+\infty.$

In this case, it follows from the maximum principle that

$-\sup_{\gamma}K_{g}-2\,\bar{c}^{2}\,A\leq\limsup_{k\to+\infty}\frac{\lambda_{k}}{\varepsilon^{2}_{k}}\leq
0,$

and, extracting a subsequence if necessary, we can assume that
$(\varepsilon^{-2}_{k}\,\lambda_{k})_{k\geq 0}$ converges to some
$\lambda_{\infty}\leq 0$ and we can also assume that $w_{k}$ converges to
$w_{\infty}$ which is smooth away from the points $s_{1},\ldots,s_{n}$.
Finally, $w_{\infty}$ is a solution of

$\mathfrak{J}_{\gamma}w_{\infty}=\lambda_{\infty}w_{\infty}.$

The convergence of $(w_{k})_{k=0}^{\infty}$ to $w_{\infty}$ is uniform on
compacts of $\gamma-\\{s_{1},\ldots,s_{n}\\}$ and, a priori, the function
$w_{\infty}$ (which is a solution of a second order linear ordinary
differential equation) has both right and left limits and right and left
derivatives at $s_{1},\ldots,s_{n}$.

Observe that $w_{\infty}$ is not identically equal to $0$. This is clear when
$(t_{k})_{k\geq 0}$ converges to $t_{\infty}\notin\\{s_{1},\ldots,s_{n}\\}$.
If $(t_{k})_{k\geq 0}$ converges slowly to $s_{j}$ as explained above, one can
use the fact that $w_{k}(t_{k})=1$ and $w_{k}$ solves a second order ordinary
differential equation with coefficients uniformly bounded to check that
$w_{k}$ remains larger than $1/2$ on some interval $[t_{k},t_{k}+c)$, if
$t_{k}>s_{j}$ (or $(t_{k}-c,t_{k}]$, if $t_{k}<s_{j}$). This again implies
that $w_{\infty}$ is not identically equal to $0$ either on the interval
$(s_{j},s_{j+1})$ (or on the interval $(s_{j-1},s_{j})$).

Now, for each $j=1,\ldots,n$, we fix $\dot{s}>0$ small and compute

(5.2)
$\begin{array}[]{rllll}\displaystyle\int_{s_{j}-\dot{s}}^{s_{j}+\dot{s}}\left(\psi\,\partial_{s}^{2}w_{k}-w_{k}\,\partial_{s}^{2}\psi\right)ds&=&\left(\psi\,\partial_{s}w_{k}-w_{k}\,\partial_{s}\psi\right)(s_{j}+\dot{s})\\\\[8.53581pt]
&-&\left(\psi\,\partial_{s}w_{k}-w_{k}\,\partial_{s}\psi\right)(s_{j}-\dot{s}),\end{array}$

where

$\psi:=\partial_{s}\Psi_{\varepsilon_{k}}.$

We have

$\mathcal{L}_{\varepsilon_{k}}\psi=\varepsilon_{k}^{2}\,\partial_{s}K_{g}\,\Psi_{\varepsilon_{k}},$

hence

$\int_{s_{j}-\dot{s}}^{s_{j}+\dot{s}}\left(\psi\,\partial_{s}^{2}w_{k}-w_{k}\,\partial_{s}^{2}\psi\right)ds=\int_{s_{j}-\dot{s}}^{s_{j}+\dot{s}}\left(\partial_{s}K_{g}\,\Psi_{\varepsilon_{k}}\,w_{k}-\frac{\lambda_{k}}{\varepsilon_{k}^{2}}w_{k}\,\psi\right)\,ds.$

Since $\Psi_{\varepsilon_{k}}+|\partial_{s}\Psi_{\varepsilon_{k}}|\leq
C\,\alpha_{\varepsilon_{k}}$ for some constant $C>0$ independent of $k$, we
get

$\left|\int_{s_{j}-\dot{s}}^{s_{j}+\dot{s}}\left(\partial_{s}K_{g}\,\Psi_{\varepsilon_{k}}\,w_{k}-\frac{\lambda_{k}}{\varepsilon_{k}^{2}}w_{k}\,\psi\right)\,ds\right|\leq
C\,\dot{s}\,\alpha_{\varepsilon_{k}}.$

It follows from the construction of $\Psi_{\varepsilon_{k}}$ that, away from
$s_{j}$, the function $\Psi_{\varepsilon_{k}}$ is a small perturbation of
$\alpha_{\varepsilon_{k}}\,\Phi$. Hence,

$\lim_{k\to+\infty}\frac{\psi(s_{j}\pm\dot{s})}{\alpha_{\varepsilon_{k}}}=\partial_{s}\Phi(s_{j}+\dot{s}),$

and

$\lim_{k\to+\infty}\frac{\partial_{s}\psi(s_{j}\pm\dot{s})}{\alpha_{\varepsilon_{k}}}=\partial^{2}_{s}\Phi(s_{j}\pm\dot{s})=K_{g}(s_{j}\pm\dot{s})\,\Phi(s_{j}\pm\dot{s}).$

Letting $k$ tend to infinity in (5.2), we conclude that

$\left|(\partial_{s}\Phi\,\partial_{s}w_{\infty}-K_{g}w_{\infty})(s_{j}+\dot{s})-(\partial_{s}\Phi\,\partial_{s}w_{\infty}-K_{g}w_{\infty})(s_{j}-\dot{s})\right|\leq
C\,\dot{s}.$

Letting $\dot{s}$ tend to $0$, we conclude that

(5.3)
$(\partial_{+}w_{\infty}+\partial_{-}w_{\infty})(s_{j})+\frac{2K_{g}(s_{j})}{N_{j}}\,(w_{\infty}(s_{j}^{+})-w_{\infty}(s_{j}^{-}))=0.$

Performing the same analysis with

$\psi(s):=\partial_{\kappa}T_{\kappa}|_{\kappa=\kappa_{j}}(s),$

we obtain

$w_{\infty}(s_{j}^{+})=-w_{\infty}(s_{j}^{-}).$

which together with (5.3) implies that $w_{\infty}$ is continuous on $\gamma$
away from $s_{j}$, solves
$\mathfrak{J}_{\gamma}w_{\infty}=-\lambda_{\infty}w_{\infty}(s^{+}_{j})$ on
$(s_{j},s_{j+1})$ and

$\partial_{s}w_{\infty}(s_{j}^{+})+\partial_{s}w_{\infty}(s_{j}^{-})+\frac{4K_{g}}{N_{j}}\,w_{\infty}=0,$

at $s_{j}$, for all $j=1,\ldots,n$. Since we have assumed the Bouncing Jacobi
Field to be non-degenerate, we also conclude that $\lambda_{\infty}\neq 0$.

Case 2. Assume that (up to a subsequence), the sequence
$(t_{k})_{k=0}^{\infty}$ converges _fast_ to some $s_{j}$, in the sense that

$\lim_{k\to+\infty}\frac{1}{\varepsilon_{k}^{2}}\,e^{-2\Psi_{\varepsilon_{k}}(t_{k})}=+\infty$

In this case, we define

$\hat{w}_{k}(s):=w_{k}\left(s_{j}+\frac{\varepsilon_{k}}{\kappa_{k,j}}s\right),$

which is a solution of

$\left(\partial^{2}_{s}+\left(\frac{\varepsilon_{k}}{\kappa_{k,j}}\right)^{2}\,\hat{K}_{g}\right)\,\hat{w}_{k}+2\,\left(\frac{\bar{c}}{\kappa_{k,j}}\right)^{2}\,e^{-2\hat{\Psi}_{k}}\hat{w}_{k}=-\frac{\lambda_{k}}{\kappa_{k,j}^{2}}\,\hat{w}_{k},$

where

$\hat{K}_{g}(s):=K_{g}\,\left(s_{j}+\frac{\varepsilon_{k}}{\kappa_{k,j}}s\right)\qquad\text{and}\qquad\hat{\Psi}_{k}(s):=\Psi_{\varepsilon_{k}}\left(s_{j}+\frac{\varepsilon_{k}}{\kappa_{k,j}}s\right).$

Since we have assumed that the $L^{\infty}$ norm of $w_{k}$ tends to zero on
any compact interval away from the local minimums, there exists a constant
$c<0$ independent of $\varepsilon$, such that

$\frac{\lambda_{k}}{\kappa^{2}_{k,j}}<c.$

The maximum principle also tells that for some constant $C>0$ independent of
$\varepsilon$,

$\frac{\lambda_{k}}{\kappa^{2}_{k,j}}>-C.$

Therefore we can assume that up to a subsequence,

$\lim_{k\to\infty}\frac{\lambda_{k}}{\kappa^{2}_{k,j}}=\lambda_{\infty}.$

Then the sequence of functions $\hat{w}_{k}$ will converge to $w_{\infty}$
which satisfies

$-(\partial^{2}_{s}+2\bar{c}^{2}e^{-2u})\,w_{\infty}=\lambda_{\infty}w_{\infty}.$

We claim that $w_{\infty}$ is not identically equal to $0$. Assuming the claim
is proven, this implies that that $w_{\infty}$ is an eigenfunction of the
linearized Toda equation with negative eigenvalue, which is unique as we
mentioned before. In particular, this implies that $\lambda_{\infty}=-1$.

It remains to prove the claim. Obviously, the claim holds if (for a
subsequence),

$\lim_{k\to+\infty}\frac{\kappa_{k,j}}{\varepsilon_{k}}\,(t_{k}-s_{j})\in\mathbf{R}.$

Now, if, for example,

$\lim_{k\to+\infty}\frac{\kappa_{k,j}}{\varepsilon_{k}}\,(t_{k}-s_{j})=+\infty,$

we can write

$\partial_{s}^{2}\hat{w}_{k}\geq-\left(\frac{\varepsilon_{k}}{\kappa_{k,j}}\right)^{2}\,\hat{K}_{g}-2\,\left(\frac{\bar{c}}{\kappa_{k,j}}\right)^{2}\,e^{-2\hat{\Psi}_{k}}=:f_{k}.$

since $\lambda_{k}\leq 0$ and $\hat{w}_{k}\leq 1$. We set

$\hat{t}_{k}:=\frac{\kappa_{k,j}}{\varepsilon_{k}}\,(t_{k}-s_{j}).$

Using the fact that $\hat{w}_{k}(\hat{t}_{k})=1$,
$\partial_{s}\hat{w}_{k}(\hat{t}_{k})=0$, we get

$\begin{array}[]{rllll}\hat{w}_{k}(s)&\geq&\displaystyle
1-\int_{s}^{\hat{t}_{k}}\int_{x}^{\hat{t}_{k}}f_{k}(y)\,dy\,dx\\\\[8.53581pt]
&\geq&\displaystyle
1-C\,\left(\left(\frac{\varepsilon_{k}}{\kappa_{k,j}}\right)^{2}\,(\hat{t}_{k}-s)^{2}-e^{-2\,(\hat{t}_{k}-s)}\right),\end{array}$

for all $s\in(0,\hat{t}_{k})$. Observe that

$\lim_{k\to+\infty}\frac{\varepsilon_{k}}{\kappa_{k,j}}\,\hat{t}_{k}=0,$

hence, we conclude that, if $\hat{s}>0$ is fixed large enough, then
$\hat{w}_{k}(\hat{s})\geq 1/2$ for all $k$ large enough. This implies that
$w_{\infty}$ is not identically equal to $0$ and the proof of the claim is
complete.

Combining the discussion for all cases, we see that an eigenfunction of
$\mathcal{L}_{\varepsilon}$ either converges to an eigenfunction of the
linearized problem associated to the quadratic form $\mathcal{Q}$ (after a
suitable rescaling), or to an eigenfunction of the linearized Toda equation.
The conclusion of the lemma then follows from Proposition 3.4 and Lemma 5.2. ∎

###### Proof of Proposition 5.1.

Collecting the results of Lemma 5.2 and Lemma 5.3, we see that the proof of
Proposition 5.1 is now complete. ∎

Observe that, as a by product of the proof of Lemma 5.3, we have the:

######  Proposition 5.4.

There exists a constant $c>0$ such that, for all $\varepsilon$ small enough,
there are no eigenvalues of $\mathcal{L}_{\varepsilon}$ in
$[-c\,\varepsilon^{2},c\,\varepsilon^{2}]$.

###### Proof.

The proof is by contradiction. We assume that there was a sequence of
eigenvalues $\lambda_{k}$ of $\mathcal{L}_{\varepsilon_{k}}$ such that

$\lim_{k\to+\infty}\frac{\lambda_{k}}{\varepsilon_{k}^{2}}=0.$

and $\lim_{k\to+\infty}\varepsilon_{k}=0$. Then, the proof of Lemma 5.3
applies verbatim to get a contradiction. ∎

######  Remark 5.5.

We have explained in Remark 4.4 that the construction in [6] was making use of
an approximate solution of the Jacobi-Toda equation which is close to a
constant which can be expanded as

$\psi=-\ln\varepsilon-\frac{1}{2}\,\ln\left(-\ln\varepsilon\right)+\mathcal{O}(1),$

The linearized Jacobi-Toda equation about this solution reads

$-\varepsilon^{2}\,\left(\partial_{s}^{2}+K_{g}\right)+2\,\bar{c}^{2}\,e^{-2\psi},$

but

$2\,\bar{c}^{2}\,e^{-2\psi}=-\varepsilon^{2}\,\left(K_{g}\ln\varepsilon+\mathcal{O}\left(\ln\,(-\ln\varepsilon)\right)\right),$

and this implies that the Morse index of this linearized equation is of order
$-\ln\varepsilon$ and hence tends to infinity as $\varepsilon$ tends to $0$.
This in turn implies that the solutions of the Allen-Cahn equation constructed
in [6] have Morse indices which tend to infinity as $\varepsilon$ tends to
$0$.

Since the index of $\mathcal{L}_{\varepsilon}$ is bounded from below by the
index of geodesic $\gamma,$ we have the following:

######  Corollary 5.6.

Suppose there is a non-degenerate Bouncing Jacobi Field $\Phi$ with $n$
minimums for the geodesic $\gamma$. Then $2n\geq\text{\rm
Ind}\left(\gamma\right).$

###### Proof.

We can construct a solution $\Psi_{\varepsilon}$ of the Jacobi-Toda equation
using the Bouncing Jacobi Field $\Phi$. Now we know that the corresponding
index of $\mathcal{L}_{\varepsilon}$ is equal to $n+\text{\rm Ind}(\Phi)$.
From this, we deduce that

$n+\text{\rm Ind}(\Phi)\geq\text{\rm Ind}(\gamma).$

Since $\text{\rm Ind}(\Phi)\leq n$, we conclude that

$2n\geq\text{\rm Ind}(\gamma),$

which is the required inequality. ∎

This last result provides a necessary condition for the existence of Bouncing
Jacobi Fields with prescribed number of minimums.

##  6\. Local geometry about a closed geodesic and the existence of solutions
to the Allen-Cahn equation

In this section, we explain how the previous analysis leads to solutions of
the Allen-Cahn equation. Observe that, away from the transition layers, the
solutions of the Allen-Cahn equation converge exponentially fast to either $1$
or $-1$ (in fact the convergence is like
$\exp(-\text{dist}(\cdot,\gamma)/(\sqrt{2}\,\varepsilon))$. This implies that
the construction near a given geodesic is hardly perturbed by the construction
near a different geodesic. The construction of solutions whose transition
layer converges with multiplicity one to a geodesic is the same as the one
done in [18, 19]). Hence, to simplify the exposition, we will assume ${\bf
G}^{1}$ is empty. The construction of solutions whose transition layer
converges with multiplicity two to a geodesic follows the same lines as in [6]
with some differences we will explain. To simplify the exposition, we will
assume that ${\bf G}^{2}$ is reduced to a single geodesic. Thanks to the
exponential convergence away from the transition layers, the construction in
the general case follows the same line with more involved notations.

Assume we are given a two-sided, closed embedded geodesic $\gamma$ in $(M,g)$.
We choose a unit normal vector field $\nu$ along $\gamma$ and we define

$Z(s,t):=\exp_{s}\left(t\,\nu(s)\right),$

where $s\in\gamma$ and $t\in\mathbf{R}$ is close to $0$. It is well known that
$Z$ is a diffeomorphism from $\gamma\times(-\tau,\tau)$ into a tubular
neighborhood of $\gamma$ in $M$, provided $\tau>0$ is fixed small enough. The
coordinates $(s,t)\in\gamma\times(-\tau,\tau)$ are called _Fermi coordinates_
relative to the geodesic $\gamma$. Observe that $t$ is _signed distance_ to
the geodesic $\gamma$.

In these coordinates, the metric on $M$ can be expanded as

$g=\left(1-K_{g}(s)\,t^{2}+\mathcal{O}(t^{3})\right)\,ds^{2}+dt^{2},$

where $s\mapsto K_{g}(s)$ is the Gauss curvature along $\gamma$ [18, 19]. The
fact that, in this expansion, the coefficient of $ds^{2}$ does not involve any
linear term in $t$ is a consequence of the fact that $\gamma$ is a geodesic,
and the fact that there are no terms in $ds\,dt$ in the expression of $g$ in
these coordinates is a consequence of Gauss’s Lemma.

In these coordinates, the Laplace-Beltrami operator on $M$ is given by

$\Delta_{g}=\displaystyle\frac{1}{\sqrt{|g|}}\partial_{s}\left(\sqrt{|g|}\,g^{ss}\,\partial_{s}\right)+\frac{1}{\sqrt{|g|}}\partial_{t}\left(\sqrt{|g|}\,g^{tt}\,\partial_{t}\right)$

where $|g|$ is the coefficient of $ds^{2}$ in the expression of $g$ and
$g^{ss}$ is the inverse of this coefficient. Moreover, $g^{tt}=1$. One can
check that $\Delta_{g}$ has the following expansion close to $\gamma$

(6.1)
$\Delta_{g}=(1+\mathcal{O}(t^{2}))\,\partial_{s}^{2}+{\mathcal{O}}(t^{2})\,\partial_{s}+\partial_{t}^{2}-\left(K_{g}(s)+\mathcal{O}(t)\right)\,t\,\partial_{t}.$

Recall that the _heteroclinic solution_ of the Allen-Cahn equation is defined
to be the unique entire solution of

$W^{\prime\prime}+W-W^{3}=0$

satisfying $\displaystyle\lim_{x\to\pm\infty}W(x)=\pm 1$ and $W(0)=0$. It is
explicitly given by

$W(x):=\tanh\left(\frac{x}{\sqrt{2}}\right).$

To build a solution of the Allen-Cahn equation on $M$ whose transition layer
converges, with multiplicity $1$, to a given closed embedded geodesic, it is
reasonable to look for a (small) perturbation of the function explicitly given
in Fermi coordinates by

$U_{\varepsilon}(s,t):=W\left(t/\varepsilon\right).$

To allay notations, we set

$\mathcal{N}_{\varepsilon}(U):=\varepsilon^{2}\,\Delta_{g}U+U-U^{3}.$

Using the expansion of the Laplace-Beltrami operator in Fermi coordinates, we
get the expression of the error

$\displaystyle{\mathcal{N}}_{\varepsilon}(U_{\varepsilon})(s,t)=-\varepsilon^{2}\,K_{g}(s)\,\frac{t}{\varepsilon}\,W^{\prime}\left(t/\varepsilon\right)\,(1+\mathcal{O}(t)).$

Observe that, the function on the right-hand-side is bounded by a constant
$\varepsilon^{2}$ and decays exponentially fast as one moves away from
$\gamma$.

In the normal direction to the geodesic, the approximate solution is, up to a
scaling factor, close to a translated copy of the one dimensional heteroclinic
solution $W$. Since the one dimensional Allen-Cahn equation is translation
invariant, the linearized Allen-Cahn operator about $W$ has a kernel spanned
by

$W^{\prime}(x)=\displaystyle\frac{1}{\sqrt{2}}\,\frac{1}{\cosh^{2}\left(x/\sqrt{2}\right)}.$

That is

$\left(W^{\prime}\right)^{\prime\prime}+(1-3W^{2})\,W^{\prime}=0.$

This implies that

$\varepsilon^{2}\,\Delta_{g}+1-3\,U_{\varepsilon}^{2},$

the linearized Allen-Cahn operator about $U_{\varepsilon}$ might not be
invertible but even if it were, it would have a right inverse whose norm will
blow up as the parameter $\varepsilon$ tends to $0$. To overcome this issue,
the classical strategy [18, 19] is to work orthogonally to an approximate
kernel which turns out to be the space of functions which can be written as

$(s,t)\mapsto\varphi(s)\,W^{\prime}(t/\varepsilon),$

for any $\varphi\in H^{1}(\gamma)$ in some tubular neighbourhood of $\gamma$.

The price to pay is that we have to modify the approximate solution by
introducing perturbations of its zero set. Given a (small) function
$f:\gamma\to\mathbf{R}$ at least $\mathcal{C}^{2}$, we are led to alter the
previous approximate solution and consider instead the function

$U_{\varepsilon}(s,t):=W\left(\frac{t-\varepsilon\,f(s)}{\varepsilon}\right),$

as a reasonable approximate solution. Indeed, using (6.1), we get the
structure of the error

(6.2)
$\begin{array}[]{rlll}\displaystyle{\mathcal{N}}_{\varepsilon}(U_{\varepsilon})(s,t)&=&\displaystyle\varepsilon^{2}\,\left({\mathfrak{J}}_{\gamma}f(s)-K_{g}(s)\,\left(\frac{t-\varepsilon\,f(s)}{\varepsilon}\right)\right)\,W^{\prime}\left(\frac{t-\varepsilon\,f(s)}{\varepsilon}\right)\\\\[8.53581pt]
&+&\displaystyle\varepsilon^{2}\,(\partial_{s}f)^{2}\,W^{\prime\prime}\left(\frac{t-\varepsilon\,f(s)}{\varepsilon}\right)\\\\[8.53581pt]
&+&\displaystyle\varepsilon^{2}\,\left(\mathcal{O}\left(\frac{t^{2}}{\varepsilon}\right)+\mathcal{O}(t^{2})\,\partial_{s}f+\mathcal{O}(t^{2})\,\partial^{2}_{s}f\right)W^{\prime}\left(\frac{t-\varepsilon\,f(s)}{\varepsilon}\right)\\\\[8.53581pt]
&+&\displaystyle\varepsilon^{2}\,\mathcal{O}(t^{2})\,(\partial_{s}f)^{2}\,W^{\prime\prime}\left(\frac{t-\varepsilon\,f(s)}{\varepsilon}\right).\end{array}$

Observe that if we fix $c>0$ small enough and if we require that
$\mathcal{N}_{\varepsilon}(U_{\varepsilon})$ is $L^{2}(-c,c)$ orthogonal to

$t\mapsto W^{\prime}\left(\frac{t-\varepsilon\,f(s)}{\varepsilon}\right),$

we find that $f$ has to solve an equation of the form

${\mathfrak{J}}_{\gamma}f=\mathcal{O}(\varepsilon)+\varepsilon\,L(f)+\varepsilon\,Q(f),$

where $L$ is a linear second order differential operator and $Q$ a quadratic
second order differential operator in the sense that its Taylor expansion in
$f$ vanishes at order $2$. To derive this equation it is important to use the
fact that $x\mapsto x\,W^{\prime}(x)$ and $x\mapsto W^{\prime\prime}(x)$ are
both $L^{2}(\mathbf{R})$ orthogonal to $x\mapsto W^{\prime}(x)$. The fact that
we work on some interval $(-c,c)$ instead of $\mathbf{R}$ is not a real issue
since the function $W^{\prime}$ tends exponentially fast to $0$ at infinity.

Therefore, assuming that ${\mathfrak{J}}_{\gamma}$ is invertible, one can
improve the approximate solution by choosing $f$ to be a solution of the above
equation (which can then be solved using a fixed point argument) and this
explains why we ask the geodesic $\gamma$ to be non-degenerate. The existence
of solutions of the Allen-Cahn equation having a transition layer which
converges with multiplicity one is well known [19], therefore, we will not
copy this construction here. However, it is interesting to compare this
construction with the construction of the present paper. In particular, the
above formal computation (which can be made rigorous) governs the size of the
function $f$ and the solution of the Allen-Cahn equation can be decomposed in
a tubular neighborhood of $\gamma$ as

$u_{\varepsilon}=W\left(\frac{t-\varepsilon\,f}{\varepsilon}\right)+V,$

where $\|V\|_{L^{\infty}}\leq C\,\varepsilon^{2}$ and $\|f\|_{L^{\infty}}\leq
C\,\varepsilon$. As we will see shortly the situation is totally different in
the construction of solutions whose transition layers converges with
multiplicity $2$ to a given geodesic.

Let us now explain how to build solutions of the Allen-Cahn equation whose
transition layers converge to a given geodesic with multiplicity $2$. We have
in mind to look for solutions whose zero set consists of two components (we
assume here that the geodesic is two-sided), each one being a normal graph
over $\gamma$ for some small function. This time, a reasonable approximate
solution $U_{\varepsilon}$ has the form

$U_{\varepsilon}(s,t):=W\left(\frac{t-\varepsilon\,f_{2}(s)}{\varepsilon}\right)-W\left(\frac{t-\varepsilon\,f_{1}(s)}{\varepsilon}\right)+1,$

where $f_{1}<0<f_{2}$. More precisely, we will prove that $\sqrt{2}\,f_{2}$ is
close to $\Psi_{\varepsilon}$ and $\sqrt{2}\,f_{1}$ is close to
$-\Psi_{\varepsilon}$ where $\Psi_{\varepsilon}$ is a solution of the Jacobi-
Toda equation. Hence, the distance between the graph of $\varepsilon\,f_{2}$
and the graph of $\varepsilon\,f_{1}$ is of order
$\varepsilon\,\alpha_{\varepsilon}$ and hence is much larger than
$\varepsilon$, which is the characteristic size of the transition layers of
the solutions of the Allen-Cahn equation. Even though the distance between the
two transition layers is much larger than $\varepsilon$, we will see that
there will be some interaction between the two transition layers at points
where $\Psi_{\varepsilon}$, the solution of the Jacobi-Toda equation, is
minimal.

To simplify the notation, let us set

$W_{j}(s,t):=W\left(\frac{t-\varepsilon\,f_{j}(s)}{\varepsilon}\right)\qquad\text{and}\qquad
W_{j}^{\prime}(s,t):=W^{\prime}\left(\frac{t-\varepsilon\,f_{j}(s)}{\varepsilon}\right)$

with similar notations for higher derivatives.

Given a small function $V$, we compute

(6.3)
$\begin{array}[]{rlllll}\mathcal{N}_{\varepsilon}(U_{\varepsilon}+V)&=&\mathcal{N}_{\varepsilon}(W_{2})-\mathcal{N}_{\varepsilon}(W_{1})\\\\[8.53581pt]
&+&3\,(1-W_{2}^{2})\,(1-W_{1})-3\,(W_{2}+1)\,(1-W_{1})^{2}\\\\[8.53581pt]
&+&\varepsilon^{2}\,\Delta_{g}V+V-3\,U^{2}_{\varepsilon}\,V\\\\[8.53581pt]
&-&3\,U_{\varepsilon}\,V^{2}-V^{3}.\end{array}$

The expression of $\mathcal{N}_{\varepsilon}(W_{2})$ and
$\mathcal{N}_{\varepsilon}(W_{1})$ are given by (6.2) with $f$ replaced
respectively by $f_{2}$ or $f_{1}$.

As in the case of a single transition layer, the projection of the error
$\mathcal{N}_{\varepsilon}(U_{\varepsilon})$ over the approximate kernel plays
an important role in our analysis. This time, the approximate kernel is
spanned by the functions

$(s,t)\mapsto\varphi_{j}(s)\,W^{\prime}\left(\frac{t-\varepsilon\,f_{j}(s)}{\varepsilon}\right)$

for $j=1,2$, where the functions $\varphi_{j}\in H^{1}(\gamma)$.

For example, the projection of $\mathcal{N}_{\varepsilon}$ over
$W^{\prime}_{2}$ reads

(6.4)
$\begin{array}[]{rllll}\displaystyle\int_{[-c,c]}\mathcal{N}_{\varepsilon}(U_{\varepsilon})\,W_{2}^{\prime}\,dt&=&\displaystyle\varepsilon\,\left(c_{0}\,\varepsilon^{2}\,\mathfrak{J}_{\gamma}f_{2}+c_{1}\,e^{-\sqrt{2}(f_{2}-f_{1})}\right)\\\\[8.53581pt]
&+&\varepsilon^{4}\,\mathcal{O}(1)+\varepsilon^{3}\,e^{-\sqrt{2}(f_{2}-f_{1})}\,\mathcal{O}(1)+e^{-2\sqrt{2}(f_{2}-f_{1})}\,\mathcal{O}(1)\\\\[8.53581pt]
&+&\varepsilon^{4}\,L(f_{2})+\varepsilon^{3}\,e^{-\sqrt{2}(f_{2}-f_{1})}\,L(f_{1})+\varepsilon^{4}\,Q(f_{1},f_{2}),\end{array}$

where $L$ are second order differential operators and $Q$ a quadratic second
order differential operator and where the constants $c_{0}$ and $c_{1}$ are
explicitly given by

(6.5) $c_{0}:=\int_{{\mathbf{R}}}(W^{\prime}(x))^{2}\,dx\qquad\text{and}\qquad
c_{1}:=6\int_{{\mathbf{R}}}e^{-\sqrt{2}x}\,\left(1-W^{2}(x)\right)W^{\prime}(x)\,dx.$

The term $\varepsilon^{2}\,c_{0}\,\,\mathfrak{J}_{\gamma}f$ comes from the
first term in (6.2) while the term $c_{1}\,e^{-\sqrt{2}(f_{2}-f_{1})}$ comes
from the $L^{2}$ projection of $3\,(1-W_{2}^{2})\,(1-W_{1})$ over
$W_{2}^{\prime}$ using the fact that

$1-W(x)=2\,e^{-\sqrt{2}x}+\mathcal{O}\left(e^{-2\sqrt{2}x}\right)$

for $x>0$. A similar analysis can be performed for the projection of
$\mathcal{N}_{\varepsilon}$ over $W^{\prime}_{1}$. The result is similar with
a change of sign due to the fact that $3\,(1-W_{2}^{2})\,(1-W_{1})$ is now
replaced by $-3\,(1-W_{1}^{2})\,(1+W_{2})$.

Neglecting the higher order terms, we conclude that $U_{\varepsilon}$ could be
a reasonable approximate solution if we are able to find solutions of the
system

$\varepsilon^{2}\,c_{0}\,\mathfrak{J}_{\gamma}\,f_{2}+c_{1}\,e^{-\sqrt{2}(f_{2}-f_{1})}=0\qquad\text{and}\qquad\varepsilon^{2}\,c_{0}\,\mathfrak{J}_{\gamma}\,f_{1}-c_{1}\,e^{-\sqrt{2}\,(f_{2}-f_{1})}=0.$

If we set

$\phi_{j}:=\sqrt{2}\,f_{j},$

for $j=1,2$ and $\bar{c}^{2}:=\frac{\sqrt{2}c_{1}}{c_{0}}$, the above system
becomes

$\left\\{\begin{array}[]{rllll}\varepsilon^{2}\,\mathfrak{J}_{\gamma}\,\phi_{2}+\bar{c}^{2}\,e^{-(\phi_{2}-\phi_{1})}&=&0\\\\[8.53581pt]
\varepsilon^{2}\,\mathfrak{J}_{\gamma}\,\phi_{1}-\bar{c}^{2}\,e^{-(\phi_{2}-\phi_{1})}&=&0,\end{array}\right.$

Summing up the two equations, we obtain

$\mathfrak{J}_{\gamma}\left(\phi_{1}+\phi_{2}\right)=0,$

and, if the geodesic $\gamma$ is assumed to be non-degenerate, we conclude
that $\phi_{2}=-\phi_{1}$. Therefore, our problem reduces to the Jacobi-Toda
equation (4.1) which has been studied in Section 4.

We now come to the main result of this section. Starting from a non-degenerate
embedded, two-sided geodesic $\gamma$ and a non-degenerate Bouncing Jacobi
Field $\Phi$ along $\gamma$, we prove the existence of solutions to the Allen-
Cahn equation whose nodal set is close to the normal graph over $\gamma$ for
the functions $s\mapsto\varepsilon\,\Psi_{\varepsilon}(s)$, where
$\Psi_{\varepsilon}$ is the solution of the Jacobi-Toda equation associated to
$\Phi$, given by Proposition 4.3.

We keep the notations already used and set

$\sqrt{2}\,f_{2}:=\Psi_{\varepsilon}+h_{2}\qquad\text{and}\qquad\sqrt{2}\,f_{1}:=-\Psi_{\varepsilon}+h_{1},$

where $\Psi_{\varepsilon}$ is the solution of the Jacobi-Toda equation
provided by Proposition 4.3 starting with the Bouncing Jacobi Field $\Phi$ and
where $h_{2}$ and $h_{1}$ are small functions to be determined.

We define the approximate solution as

$U_{\varepsilon}(s,t)=\chi(t)\,\left(W\left(\frac{t-\varepsilon\,f_{2}(s)}{\varepsilon}\right)-W\left(\frac{t-\varepsilon\,f_{1}(s)}{\varepsilon}\right)\right)+1,$

where $\chi$ is a smooth cutoff function identically equal to $1$ in a tubular
neighbourhood of width $c$ around $\gamma$ and identically equal to $0$
outside a tubular neighbourhood of width $2\,c$ around $\gamma$, for some
$c>0$ fixed small enough but independent of $\varepsilon$.

We look for a solution of the Allen-Cahn equation of the form
$U_{\varepsilon}+V$ where $V$ is a solution of

(6.6)
$\left(\varepsilon^{2}\,\Delta_{g}+1-3U_{\varepsilon}^{2}\right)\,V+\mathcal{N}_{\varepsilon}(U_{\varepsilon})-3\,U_{\varepsilon}\,V^{2}-V^{3}=0,$

which satisfies the orthogonality conditions

(6.7)
$\int_{(-c,c)}V\,W_{1}^{\prime}\,dt=\int_{(-c,c)}V\,W_{2}^{\prime}\,dt=0,$

for all $s\in\gamma$.

The next statement involves some weighted norm on the space of
$\mathcal{C}^{2,\alpha}$ function which is defined by

$\|V\|_{\mathcal{C}^{k,\alpha}_{\varepsilon}}=\sup_{i}\left(\sum_{j=0}^{k}\varepsilon^{j}\,\sup_{\Omega_{i}}\,|\nabla^{j}V|+\varepsilon^{k+\alpha}\,\sup_{p\neq
q\in\Omega_{i}}\frac{|\nabla^{k}V(p)-\nabla^{k}V(q)|}{|p-q|^{\alpha}}\right),$

where $(\Omega_{i})_{i\in I}$ is a finite open cover of $M$. In other words,
with this definition, in a local chart, the $L^{\infty}$ norm of any $j$-th
partial derivatives of $f$ is controlled by
$\varepsilon^{-j}\,\|f\|_{\mathcal{C}^{k,\alpha}_{\varepsilon}}$.

The main result of this section is the:

######  Proposition 6.1.

Let $\gamma$ be a closed, embedded, non-degenerate two-sided geodesic in
$(M,g)$. Assume that $\Phi$ is a non-degenerate _Bouncing Jacobi Field_ with
$n\geq 1$ minimums. Then, for all $\varepsilon$ small enough, the Allen-Cahn
equation has a solution $u_{\varepsilon}$ whose transition layers converge to
$\gamma$ with multiplicity two. Moreover, $u_{\varepsilon}=U_{\varepsilon}+V$,
where

$\|V\|_{\mathcal{C}^{2,\alpha}_{\varepsilon}}\leq
C\,\varepsilon^{2}\,(\ln\varepsilon)^{2}$

and

$\|h_{1}\|_{\mathcal{C}^{2,\alpha}}+\|h_{2}\|_{\mathcal{C}^{2,\alpha}}\leq
C\,\varepsilon\,(-\ln\varepsilon)^{5}.$

###### Proof.

We need to find $V:M\to\mathbf{R}$ and the functions
$h_{1}:\gamma\in\mathbf{R}$ and $h_{2}:\gamma\to\mathbf{R}$ such that (6.6)
and (6.7) hold.

As explained above, the linear operator

$-\left(\varepsilon^{2}\,\Delta_{g}+1-3\,U_{\varepsilon}^{2}\right)$

can be inverted up to functions of the form
$(s,t)\mapsto\ell_{1}(s)\,\chi(t)\,W_{1}^{\prime}(t)$ and
$(s,t)\mapsto\ell_{2}(s)\,\chi(t)\,W_{2}^{\prime}(t)$ where $\chi$ is a cutoff
function identically equal to $1$ in a tubular neighborhood of width $c$ about
$\gamma$ and identically equal to $0$ outside a tubular neighborhood of width
$2c$ about $\gamma$. We refer to [5, 18, 19] and especially [4] for more
details.

This leads to a coupled system of equations

(6.8)
$\begin{array}[]{rlll}-\left(\varepsilon^{2}\,\Delta_{g}+1-3\,U_{\varepsilon}^{2}\right)V&=&\mathcal{N}_{\varepsilon}\left(W_{2}\right)-\mathcal{N}_{\varepsilon}\left(W_{1}\right)\\\\[8.53581pt]
&+&3\,\left(1-W_{2}^{2}\right)\left(1-W_{1}\right)-3\,\left(1+W_{2}\right)\left(1-W_{1}\right)^{2}\\\\[8.53581pt]
&-&3\,U_{\varepsilon}\,V^{2}-V^{3}+\chi\,\left(\ell_{2}\,W^{\prime}_{2}+\ell_{1}\,W^{\prime}_{1}\right),\end{array}$

and

(6.9)
$\left\\{\begin{array}[]{rllll}\varepsilon^{2}\,\mathfrak{J}_{\gamma}h_{2}-\bar{c}^{2}\,e^{-2\Psi_{\varepsilon}}\left(h_{2}-h_{1}\right)&=&\mathcal{O}(\varepsilon^{3}\,(\ln\varepsilon)^{4})\,+\varepsilon^{3}\,(\ln\varepsilon)^{4}\,L(h_{1},h_{2})\\\\[8.53581pt]
&+&\displaystyle\varepsilon\,L(V)+Q(h_{1},h_{2})+\frac{1}{\varepsilon}\,Q(V),\\\\[8.53581pt]
\varepsilon^{2}\,\mathfrak{J}_{\gamma}h_{1}+\bar{c}^{2}e^{-2\Psi_{\varepsilon}}\left(h_{2}-h_{1}\right)&=&\mathcal{O}(\varepsilon^{3}\,(\ln\varepsilon)^{4})+\varepsilon^{3}\,(\ln\varepsilon)^{4}\,L(h_{1},h_{2})\\\\[8.53581pt]
&+&\displaystyle\varepsilon\,L(V)+Q(h_{1},h_{2})+\frac{1}{\varepsilon}\,Q(V).\\\\[8.53581pt]
\end{array}\right.$

If we define $\xi:=h_{1}-h_{2}$ and $\zeta:=h_{1}+h_{2}$, we see that the
second system of equation translates into

$\left\\{\begin{array}[]{rllll}\mathcal{L}_{\varepsilon}\xi&=&\mathcal{O}(\varepsilon^{3}\,(\ln\varepsilon)^{4})+\varepsilon^{3}\,(\ln\varepsilon)^{4}\,L(\xi,\zeta)+Q(\xi,\zeta)\\\\[8.53581pt]
&+&\displaystyle\varepsilon\,L(V)+\frac{1}{\varepsilon}\,Q(V)\\\\[8.53581pt]
\mathfrak{J}_{\gamma}\zeta&=&\displaystyle\mathcal{O}(\varepsilon\,(\ln\varepsilon)^{4})+\varepsilon\,(\ln\varepsilon)^{4}\,L(\xi,\zeta)+\frac{1}{\varepsilon^{2}}\,Q(\xi,\zeta)\\\\[8.53581pt]
&+&\displaystyle\frac{1}{\varepsilon}\,L(V)+\frac{1}{\varepsilon^{3}}\,Q(V).\end{array}\right.$

Let us now point out the main ingredients in the estimate for $V$. We need to
estimate the projection of the right hand side of (6.8) on the orthogonal
complement of $\chi\,W^{\prime}_{1}$ and $\chi\,W^{\prime}_{2}$ when $V=0$.
Due to the exponential convergence of $W$ to $\pm 1$, it is clear that the
main contribution will come from the estimate of this quantity in a tubular
neighborhood about $\gamma$.

A close inspection of (6.2) shows that, in a tubular neighbourhood of the
curve determined by the graph of $\varepsilon\,\Psi_{\varepsilon}$, the terms
in
$\mathcal{N}_{\varepsilon}(W^{\prime}_{2})-\mathcal{N}_{\varepsilon}(W^{\prime}_{1})$
which govern the size of $V$ are

$\varepsilon^{2}\,K_{g}(s)\,\left(\frac{t-\varepsilon\,\Psi_{\varepsilon}(s)}{\varepsilon}\right)\,W_{2}^{\prime\prime}\left(\frac{t-\varepsilon\,\Psi_{\varepsilon}(s)}{\varepsilon}\right)+\varepsilon^{2}\,\left(\partial_{s}\Psi_{\varepsilon}(s)\right)^{2}\,W_{2}^{\prime\prime}\left(\frac{t-\varepsilon\,\Psi_{\varepsilon}(s)}{\varepsilon}\right).$

To evaluate their contribution to the size of $V$, we use Proposition 4.3
which provides estimates for $\partial_{s}\Psi$ and we get most
$C\,\varepsilon^{2}(\ln\varepsilon)^{2}$.

In
$3\left(1-W_{2}^{2}\right)\left(1-W_{1}\right)-3\left(1+W_{2}\right)\left(1-W_{1}\right)^{2}$,
the term which governs the size of $V$ is

$3\left(1-W_{2}^{2}\right)\left(1-W_{1}\right),$

whose contribution to the size of $V$ can be estimated by
$C\,e^{-2\alpha_{\varepsilon}}$ since $\sqrt{2}\,(f_{2}-f_{1})$ is bounded
from below by $2\,\alpha_{\varepsilon}-C$.

The existence of $V$, $h_{1}$ and $h_{2}$ follows from a fixed point argument.
Beside the analysis of which is given in [4], invertibility of
$\mathfrak{J}_{\gamma}$ is needed and we also need to use Proposition 5.4. ∎

In the next section, we will study the Morse index of the solution we have
just constructed. To this aim, we will need some better understanding of the
function $V_{\varepsilon}$. This understanding is based on the following
technical lemma part of which has been already used :

######  Lemma 6.2.

Let $X,Y$ and $Z$ be the solutions of

$\begin{array}[]{rllll}\left(\partial_{x}^{2}+1-3\,W^{2}\right)\,X&=&W^{\prime\prime}\\\\[8.53581pt]
\left(\partial_{x}^{2}+1-3\,W^{2}\right)\,Y&=&x\,W^{\prime}\\\\[8.53581pt]
\left(\partial_{x}^{2}+1-3\,W^{2}\right)\,Z&=&6\,(1-W^{2})\,\left(e^{-\sqrt{2}\,x}-\rho\right),\end{array}$

which are $L^{2}(\mathbf{R})$ orthogonal to $W^{\prime}$, where the constant
$\rho$ is determined so that the right-hand-side of the third equation is
$L^{2}(\mathbf{R})$ orthogonal to $W^{\prime}$. Then

$\begin{array}[]{rlll}\displaystyle
6\,\int_{\mathbf{R}}W\,(W^{\prime})^{2}\,X\,dx&=&\displaystyle\int_{\mathbf{R}}(W^{\prime\prime})^{2}\,dx\\\\[8.53581pt]
\displaystyle
6\,\int_{\mathbf{R}}W\,(W^{\prime})^{2}\,Y\,dx&=-&\displaystyle\frac{1}{2}\,\int_{\mathbf{R}}(W^{\prime})^{2}\,dx\\\\[8.53581pt]
\displaystyle
6\,\int_{\mathbf{R}}W\,(W^{\prime})^{2}\,Z\,dx&=&6\,\displaystyle\int_{\mathbf{R}}e^{-\sqrt{2}\,x}\,(1-W^{2})\,W^{\prime\prime}\,dx.\end{array}$

###### Proof.

Assume that $A:\mathbf{R}\to\mathbf{R}$ is a solution of

$\left(\partial_{x}^{2}+1-3\,W^{2}\right)\,A=B,$

where $B:\mathbf{R}\to\mathbf{R}$ is a smooth function. Taking the derivative
with respect to $x$, we get

$\left(\partial_{x}^{2}+1-3\,W^{2}\right)\,\partial_{x}A-6\,W\,W^{\prime}\,A=\partial_{x}B,$

which we multiply by $W^{\prime}$ and integrate by parts over $\mathbf{R}$ to
conclude that

$6\,\int_{\mathbf{R}}W\,(W^{\prime})^{2}\,A\,dx=\int_{\mathbf{R}}B\,W^{\prime\prime}\,dx.$

The result then follows easily. ∎

As a consequence we get the:

######  Corollary 6.3.

The following estimate holds

$\begin{array}[]{rllll}\displaystyle
6\,\int_{(-c,c)}W_{2}\,V\,(W_{2}^{\prime})^{2}\,dt&=&\displaystyle-\varepsilon^{3}\,(\partial_{s}f_{2})^{2}\,\int_{\mathbf{R}}(W^{\prime\prime})^{2}\,dx+\varepsilon^{3}\,K_{g}(s)\,\frac{1}{2}\,\int_{\mathbf{R}}(W^{\prime})^{2}\,dx\\\\[8.53581pt]
&&\displaystyle-6\,\varepsilon\,\,e^{-\sqrt{2}\,(f_{2}-f_{1})}\,\int_{\mathbf{R}}e^{-\sqrt{2}\,x}\,(1-W^{2})\,W^{\prime\prime}\,dx\\\\[8.53581pt]
&&+\mathcal{O}(\varepsilon^{5}\,(\ln\varepsilon)^{4}).\end{array}$

###### Proof.

The result follows directly from Lemma 6.2 and the fact that the remainder
terms are of the order $\varepsilon^{5}\,(\ln\varepsilon)^{4}$. ∎

##  7\. The index of the solutions of the Allen-Cahn equation

In this section, we study the Morse index of the solutions to the Allen-Cahn
equation we have constructed. We will also prove that our solutions are non-
degenerate, i.e. that they have nullity equal to $0$.

######  Proposition 7.1.

Assume that $\Phi$ is a non-degenerate Bouncing Jacobi Field with $n$
minimums. Then, for all $\varepsilon>0$ close enough to $0$, the solution of
the Allen-Cahn equation which has been constructed in Proposition 6.1 has
nullity $0$ and Morse Index equal to
$n+\text{Ind}\,(\Phi)+\text{Ind}\,(\gamma)$.

The proof of this Proposition is decomposed into two parts. First, we show
that the linearized operator

$\mathfrak{L}_{\varepsilon}:=-\left(\varepsilon^{2}\,\Delta_{g}+1-3\,u_{\varepsilon}^{2}\right),$

has Morse index at least equal to $n+\text{Ind}(\Phi)+\text{ind}(\gamma)$ and
then we prove that it has nullity $0$ and Morse index at most equal to
$n+\text{Ind}(\Phi)+\text{ind}(\gamma)$.

We define

$\mathfrak{Q}_{\varepsilon}(\phi):=\displaystyle\int_{M}\left(\varepsilon^{2}\,|\nabla\phi|_{g}^{2}-(1-3\,u_{\varepsilon}^{2})\,\phi^{2}\right)\,\text{dvol}_{g}.$

Observe that, close to $\gamma$, we have, in Fermi coordinates,

$|\nabla\phi|_{g}^{2}=(\partial_{t}\phi)^{2}+\left(1+K_{g}(s)\,t^{2}+\mathcal{O}(t^{3})\right)\,(\partial_{s}\phi)^{2},$

and

$\text{dvol}_{g}=\left(1-K_{g}(s)\,\frac{t^{2}}{2}+\mathcal{O}(t^{3})\right)\,ds\,dt.$

We start with the following technical Lemma:

######  Lemma 7.2.

Assume that $k:\gamma\to\mathbf{R}$ is a smooth function and

$\phi(s,t)=\chi(t)\,\left(W^{\prime}_{2}(s,t)\pm
W^{\prime}_{1}(s,t)\right)\,k(s),$

which is extended by $0$ away from the support of $\chi$. Then

$\begin{array}[]{rlll}\mathfrak{Q}_{\varepsilon}(\phi)&=&\displaystyle
2\,\varepsilon\,\left(\int_{\mathbf{R}}(W^{\prime})^{2}\,dx\right)\,\left(\int_{\gamma}\varepsilon^{2}\,\left((\partial_{s}k)^{2}-K_{g}\,k^{2}\right)\,ds\right)\\\\[8.53581pt]
&-&\displaystyle(1\mp
1)\,2\,\varepsilon\,\left(6\,\int_{\mathbf{R}}e^{-\sqrt{2}\,x}\,(1-W^{2})\,W^{\prime}\,dx\right)\,\left(\int_{\gamma}e^{-\sqrt{2}(f_{2}-f_{1})}\,k^{2}\,ds\right)\\\\[8.53581pt]
&+&\mathcal{O}(\varepsilon^{4}\,(-\ln\varepsilon)^{3}).\end{array}$

###### Proof.

The proof is a simple consequence of Lemma 7.4 and an integration by parts.
Observe that the error is not as good as in this Lemma because the term
$\mathcal{O}(t^{3})$ in the expansion of the volume form. ∎

Thanks to this technical result, we are able to prove the:

######  Lemma 7.3.

For all $\varepsilon$ small enough, $\mathfrak{L}_{\varepsilon}$ has Morse
index _at least_ equal to $n+\text{Ind}\,(\Phi)+\text{Ind}\,(\gamma)$.

###### Proof.

We consider $k$ to be one of the functions $e_{j}$ and $\tilde{\eta}$ which
have been constructed in the proof of Lemma 5.2. Then, the functions

$\phi(s,t):=\chi(t)\,\left(W^{\prime}_{2}(s,t)-W^{\prime}_{1}(s,t)\right)\,k(s).$

provide a $n+\text{Ind}(\Phi)$ dimensional space over which the quadratic form
$\mathfrak{Q}_{\varepsilon}$ is negative definite, provided $\varepsilon$ is
small enough.

Now, if we assume that $k$ is one of the nonzero eigenfunction of
$\mathfrak{J}_{\gamma}$ associated to a negative eigenvalue. We set

$\psi(s,t):=\chi(t)\,\left(W^{\prime}_{2}(s,t)+W^{\prime}_{1}(s,t)\right)\,k(s).$

As above, we get that the quadratic form $\mathfrak{Q}_{\varepsilon}$ is
negative on the vector space spanned by these functions and this provides a
$\text{Ind}(\gamma)$ dimensional space over which the quadratic form
$\mathfrak{Q}_{\varepsilon}$ is negative definite, provided $\varepsilon$ is
small enough.

Since the above functions are linearly independent and this implies that the
linearized operator $\mathfrak{L}_{\varepsilon}$ has Morse index at least
equal to $n+\text{Ind}(\Phi)+\text{Ind}(\gamma)$. ∎

We keep the notations we have used in the previous section and we consider an
eigenfunction $\phi$ of

$\mathfrak{L}_{\varepsilon}:=-\left(\varepsilon^{2}\,\Delta_{g}+1-3\,u_{\varepsilon}^{2}\right),$

the linearized operator at $u_{\varepsilon}$, associated to an eigenvalue
$\lambda\leq 0$. Hence

$\mathfrak{L}_{\varepsilon}\,\phi=\lambda\,\phi.$

We assume the $L^{\infty}$ norm of $\phi$ equals $1$. In a fixed, small
tubular neighborhood of size $2c$ about $\gamma$, we decompose the
eigenfunction $\phi$ as

(7.1)
$\phi(s,t)=k_{2}(s)\,W_{2}^{\prime}(s,t)+k_{1}(s)\,W_{1}^{\prime}(s,t)+\eta(s,t),$

with

$\int_{(-c,c)}\eta(s,t)\,W_{2}^{\prime}(s,t)\,dt=\int_{(-c,c)}\eta(s,t)\,W_{1}^{\prime}(s,t)\,dt,$

to address the uniqueness question of the decomposition.

Observe that,

$\varepsilon^{2}\,\Delta_{g}\phi-(3\,u_{\varepsilon}^{2}-1-\lambda)\,\phi=0.$

Since $\lambda\leq 0$ and since, for any given $\delta>0$,
$u_{\varepsilon}>1-2\delta/3$ away from the transition layers, which are close
to the graphs of $\varepsilon\,f_{2}$ and $\varepsilon\,f_{1}$, the potential
$3\,u_{\varepsilon}^{2}-1-\lambda\geq 2-2\delta$ we conclude that $\phi$ tends
to $0$ exponentially fast, as a function of the distance to the transition
layers, like

$(s,t)\mapsto\left(W^{\prime}_{2}(s,t)\right)^{\sqrt{1-\delta}}+\left(W^{\prime}_{1}(s,t)\right)^{\sqrt{1-\delta}}.$

To get more insight on the eigenfunction $\phi$, we project the equation
$\mathfrak{L}_{\varepsilon}\phi=-\lambda\phi$ over $W_{2}^{\prime}$ and
$W_{1}^{\prime}$ and over the orthogonal complement of these functions. We
start with the projection over $W_{2}^{\prime}$.

######  Lemma 7.4.

The following estimates hold:

$\begin{array}[]{rlllll}\displaystyle\int_{(-c,c)}\mathfrak{L}_{\varepsilon}(k_{2}\,W_{2}^{\prime})\,W_{2}^{\prime}\,dt=\varepsilon\,c_{0}\,\left(\varepsilon^{2}\,\mathfrak{J}_{\gamma}+\bar{c}^{2}\,e^{-2\Psi_{\varepsilon}}\right)\,k_{2}+\varepsilon^{5}\,(\ln\varepsilon)^{4}\,L(k_{2}),\end{array}$

and

$\begin{array}[]{rlllll}\displaystyle\int_{(-c,c)}\mathfrak{L}_{\varepsilon}(k_{1}\,W_{1}^{\prime})\,W_{1}^{\prime}\,dt=\varepsilon\,c_{0}\,\left(\varepsilon^{2}\,\mathfrak{J}_{\gamma}+\bar{c}^{2}\,e^{-2\Psi_{\varepsilon}}\right)\,k_{1}+\varepsilon^{5}\,(\ln\varepsilon)^{4}\,L(k_{2}),\end{array}$

where $L$ are second order linear operators with bounded coefficients and
where the constants $c_{0}$ and $c_{1}$ have been defined in (6.5).

###### Proof.

We compute, using the expansion of the Laplace-Beltrami in Fermi coordinates
(6.1), we get

$\begin{array}[]{rllll}\mathfrak{L}_{\varepsilon}(k_{2}W^{\prime}_{2})&=&-\varepsilon^{2}\,\left(\partial_{s}^{2}k_{2}\,W^{\prime}_{2}-(2\,\partial_{s}f_{2}\,\partial_{s}k_{2}\,-\partial_{s}^{2}f_{2}\,k_{2})\,W_{2}^{\prime\prime}+(\partial_{s}f_{2})^{2}\,k_{2}\,W_{2}^{\prime\prime\prime}\right)\\\\[8.53581pt]
&-&\varepsilon^{2}\,\mathcal{O}(t^{2})\left(\partial_{s}^{2}k_{2}\,W^{\prime}_{2}-(2\,\partial_{s}f_{2}\,\partial_{s}k_{2}\,-\partial_{s}^{2}f_{2}\,k_{2})\,W_{2}^{\prime\prime}+(\partial_{s}f_{2})^{2}\,k_{2}\,W_{2}^{\prime\prime\prime}\right)\\\\[8.53581pt]
&-&\varepsilon^{2}\,\mathcal{O}(t^{2})\,\left(\partial_{s}k_{2}\,W_{2}^{\prime}-\partial_{s}f_{2}\,k_{2}\,W_{2}^{\prime\prime}\right)\
+k_{2}\,W_{2}^{\prime\prime\prime}\\\\[8.53581pt]
&+&\varepsilon^{2}\,\left(K_{g}+\mathcal{O}(t)\right)\,k_{2}\,t\,W_{2}^{\prime\prime}+k_{2}W_{2}^{\prime}-3(W_{2}-(W_{1}-1)+V)^{2}\,k_{2}\,W_{2}^{\prime}\,.\end{array}$

Using the fact that
$(W^{\prime})^{\prime\prime}+W^{\prime}-3W^{2}\,W^{\prime}=0$, the above
expression simplifies slightly. Then, we can multiply the above expression by
$W^{\prime}_{2}$ and integrate (by parts) the result over $(-c,c)$. With
little work, we find

$\begin{array}[]{rllll}\displaystyle\int_{(-c,c)}\mathfrak{L}_{\varepsilon}(k_{2}W^{\prime}_{2})\,W_{2}^{\prime}\,dt&=&\displaystyle-\varepsilon^{3}\,\partial_{s}^{2}k_{2}\,\int_{\mathbf{R}}(W^{\prime})^{2}\,dx+\varepsilon^{3}\,(\partial_{s}f_{2})^{2}\,k_{2}\,\int_{\mathbf{R}}(W^{\prime\prime})^{2}\,dx\\\\[8.53581pt]
&-&\displaystyle\varepsilon^{3}\,K_{g}\,k_{2}\,\frac{1}{2}\int_{\mathbf{R}}(W^{\prime})^{2}\,dx+6\,k_{2}\,\int_{(-c,c)}W_{2}\,V\,(W_{2}^{\prime})^{2}\,dt\\\\[8.53581pt]
&+&\displaystyle
12\,\varepsilon\,e^{-\sqrt{2}\,(f_{2}-f_{1})}\,k_{2}\,\int_{\mathbf{R}}e^{-\sqrt{2}\,x}\,W\,(W^{\prime})^{2}\,dx\\\\[8.53581pt]
&+&\varepsilon^{5}\,(\ln\varepsilon)^{4}\,L(k_{2}),\end{array}$

where $L$ is a second order linear operator with bounded coefficients. To
obtain this estimate, we have implicitly used the fact that

$e^{-\sqrt{2}\,(f_{2}-f_{1})}=\mathcal{O}(\varepsilon^{2}\,(\ln\varepsilon)^{2})$

which follows from the construction of $f_{2}$ and $f_{1}$. Finally, one can
use Corollary 6.3 together with the fact that

$\displaystyle\int_{\mathbf{R}}e^{-\sqrt{2}\,x}\,(1-W^{2})\,W^{\prime\prime}\,dx\\\
=\displaystyle
2\,\int_{\mathbf{R}}e^{-\sqrt{2}\,x}\,W\,(W^{\prime})^{2}\,dx+\sqrt{2}\,\int_{\mathbf{R}}e^{-\sqrt{2}\,x}\,(1-W^{2})\,W^{\prime}\,dx,$

to conclude that

$\begin{array}[]{rllll}\displaystyle\int_{(-c,c)}\mathfrak{L}_{\varepsilon}(k_{2}W^{\prime}_{2})\,W_{2}^{\prime}\,dt&=&\displaystyle-\varepsilon^{3}\,\partial_{s}^{2}k_{2}\,\int_{\mathbf{R}}(W^{\prime})^{2}\,dx-\displaystyle\varepsilon^{3}\,K_{g}\,k_{2}\,\frac{1}{2}\int_{\mathbf{R}}(W^{\prime})^{2}\,dx\\\\[8.53581pt]
&+&\displaystyle
6\,\varepsilon\,e^{-\sqrt{2}\,(f_{2}-f_{1})}\,k_{2}\,\int_{\mathbf{R}}e^{-\sqrt{2}\,x}\,(1-W^{2})\,W^{\prime}\,dx\\\\[8.53581pt]
&+&\varepsilon^{5}\,(\ln\varepsilon)^{4}\,L(k_{2}),\end{array}$

The result follows from the definition of $\bar{c}^{2}$ and the fact that

$\sqrt{2}\,(f_{2}-f_{1})=2\,\Psi_{\varepsilon}+\mathcal{O}(\varepsilon^{2}\,(\ln\varepsilon)^{2}).$

This completes the proof of the Lemma. ∎

Our next task is to perform a similar analysis for the projection of
$\mathfrak{L}_{\varepsilon}\,(k_{1}\,W_{1}^{\prime})$ over $W^{\prime}_{2}$.

######  Lemma 7.5.

The following estimates hold:

$\displaystyle\int_{(-c,c)}\mathfrak{L}_{\varepsilon}(k_{1}\,W_{1}^{\prime})\,W_{2}^{\prime}\,dt=-\varepsilon\,c_{0}\,\bar{c}^{2}\,e^{-2\Psi_{\varepsilon}}\,k_{1}+\varepsilon^{5}\,(\ln\varepsilon)^{4}\,L(k_{1},k_{2}),$

and

$\displaystyle\int_{(-c,c)}\mathfrak{L}_{\varepsilon}(k_{2}\,W_{2}^{\prime})\,W_{1}^{\prime}\,dt=\varepsilon\,c_{0}\,\bar{c}^{2}\,e^{-2\Psi_{\varepsilon}}\,k_{2}+\varepsilon^{5}\,(\ln\varepsilon)^{4}\,L(k_{1},k_{2}),$

where $L$ are second order linear operators with bounded coefficients.

###### Proof.

The structure of $\mathfrak{L}_{\varepsilon}(k_{1}\,W_{1}^{\prime})$ is
similar to the structure we have obtained in the previous Lemma for
$\mathfrak{L}_{\varepsilon}(k_{2}\,W_{2}^{\prime})$, exchanging the indices
$1$ and $2$. As above we multiply the expression of
$\mathfrak{L}_{\varepsilon}(k_{1}\,W_{1}^{\prime})$ by $W^{\prime}_{2}$ and
integrate the result over $(-c,c)$. Using the fact that
$(W^{\prime\prime})^{\prime}+W^{\prime}-3\,W^{2}\,W^{\prime}=0$, we get

$\begin{array}[]{rllll}\displaystyle\int_{(-c,c)}\mathfrak{L}_{\varepsilon}(k_{1}W^{\prime}_{1})\,W_{2}^{\prime}\,dt&=&\displaystyle
3\,k_{1}\,\int_{(-c,c)}\left(\left(W_{1}-(1+W_{2})-V\right)^{2}-W_{1}^{2}\right)\,W^{\prime}_{1}\,W_{2}^{\prime}\,dt\\\\[11.38109pt]
&+&\varepsilon^{5}\,(\ln\varepsilon)^{4}\,L(k_{1},\partial_{s}k_{1},\partial_{s}^{2}k_{1}),\\\\[11.38109pt]
&=&\displaystyle
3\,k_{1}\,\int_{(-c,c)}(1-W_{2}^{2})\,W^{\prime}_{1}\,W_{2}^{\prime}\,dt\\\\[11.38109pt]
&+&\varepsilon^{5}\,(\ln\varepsilon)^{4}\,L(k_{1},\partial_{s}k_{1},\partial_{s}^{2}k_{1}),\end{array}$

where $L$ changes from one line to the next but still has bounded
coefficients, and we conclude that

$\begin{array}[]{rllll}\displaystyle\int_{(-c,c)}\mathfrak{L}_{\varepsilon}(k_{1}W^{\prime}_{1})\,W_{2}^{\prime}\,dt&=&\displaystyle-\varepsilon\,k_{1}\,e^{-\sqrt{2}\,(f_{2}-f_{1})}\,6\,\int_{\mathbf{R}}e^{-\sqrt{2}\,x}(1-W^{2})\,W\,W^{\prime}\,dx\\\\[11.38109pt]
&+&\varepsilon^{5}\,(\ln\varepsilon)^{4}\,L(k_{1},\partial_{s}k_{1},\partial_{s}^{2}k_{1}).\end{array}$

A similar analysis can be done for the projection of
$\mathfrak{L}_{\varepsilon}(k_{2}\,W^{\prime}_{2})$ over $W^{\prime}_{1}$. The
computation is exactly the same but one need to be careful that, this time the
signs in front of

$e^{-\sqrt{2}\,(f_{2}-f_{1})}\,6\,\int_{\mathbf{R}}e^{-\sqrt{2}\,x}(1-W^{2})\,W\,W^{\prime}\,dx,$

are changed. ∎

The next result shows that, in some sense, eigenfunctions of
$\mathfrak{L}_{\varepsilon}$ associated to non-positive eigenvalues are
essentially of the form $k_{1}\,W^{\prime}_{1}+k_{2}\,W_{2}^{\prime}$.

######  Lemma 7.6.

There exists $\alpha>0$ independend of $\varepsilon$ such that

$\|\eta\|_{\mathcal{C}^{2,\alpha}_{\varepsilon}}\leq
C\,\varepsilon^{4}\,(\ln\varepsilon)^{4}\,\left(\left\|k_{1}\right\|_{\mathcal{C}^{2,\alpha}}+\left\|k_{2}\right\|_{\mathcal{C}^{2,\alpha}}\right).$
(7.2)
$\begin{array}[]{rllll}\displaystyle\int_{(-c,c)}\mathfrak{L}_{\varepsilon}\eta\,W_{2}^{\prime}\,dt&=&C\,\varepsilon^{4}\,(\ln\varepsilon)^{4}\,L(k_{1},k_{2}),\\\\[8.53581pt]