\section{Introduction}

Let $(M,g)$ be a compact Riemannian manifold of dimension $n \geq 3$.
The Yamabe problem is concerned with finding metrics of constant scalar
curvature in the conformal class of $g$. This problem leads to a semi-linear elliptic PDE for the conformal factor. More precisely, a conformal metric of the form $u^{\frac{4}{n-2}} \, g$ has constant scalar curvature $c$ if and only if
\begin{equation}
\label{yamabe.pde}
\frac{4(n-1)}{n-2} \, \Delta_g u - R_g \, u + c \, u^{\frac{n+2}{n-2}} = 0,
\end{equation}
where $\Delta_g$ is the Laplace operator with respect to $g$ and $R_g$ denotes the scalar curvature of $g$. Every solution of (\ref{yamabe.pde}) is a critical point of the functional
\begin{equation}
\label{yamabe.functional}
E_g(u) = \frac{\int_M \big ( \frac{4(n-1)}{n-2} \, |du|_g^2 + R_g \, u^2
\big ) \, dvol_g}{\big ( \int_M
u^{\frac{2n}{n-2}} \, dvol_g \big )^{\frac{n-2}{n}}}.
\end{equation}
In this paper, we address the question whether the set of all solutions to the Yamabe PDE is compact in the $C^2$-topology. It has been conjectured that this should be true unless $(M,g)$ is conformally equivalent to the round sphere (see \cite{Schoen1},\cite{Schoen2},\cite{Schoen3}). The case of the round sphere $S^n$ is special in that (\ref{yamabe.pde}) is invariant under the action of the conformal group on $S^n$, which is non-compact. It follows from a theorem of Obata \cite{Obata} that every solution of the Yamabe PDE on $S^n$ is minimizing, and the space of all solutions to the Yamabe PDE on $S^n$
can be identified with the unit ball $B^{n+1}$. Note that the round sphere is the only compact manifold for which the set of minimizing solutions is non-compact.

The Compactness Conjecture has been verified in low dimensions and in the locally conformally flat case. If $(M,g)$ is locally conformally flat, compactness follows from work of R.~Schoen \cite{Schoen1},\cite{Schoen2}. Moreover, Schoen proposed a strategy, based on the Pohozaev identity, for
proving the conjecture in the non-locally conformally flat case. In \cite{Li-Zhu}, Y.Y.~Li and M.~Zhu followed this strategy to prove compactness in dimension
$3$. O.~Druet \cite{Druet} proved the conjecture in dimensions $4$ and $5$.

The case $n \geq 6$ is more subtle, and requires a careful analysis of
the local properties of the background metric $g$ near a blow-up point.
The Compactness Conjecture is closely related to the Weyl Vanishing
Conjecture, which  asserts that the Weyl tensor should vanish to an order
greater than $\frac{n-6}{2}$ at a blow-up point (see \cite{Schoen3}). The
Weyl Vanishing Conjecture has been verified in dimensions $6$ and $7$ by
F.~Marques \cite{Marques} and, independently, by Y.Y.~Li and L.~Zhang
\cite{Li-Zhang1}. Using these results and the positive mass theorem, these
authors were able to prove compactness for $n \leq 7$. Moreover, Li and Zhang
showed that compactness holds in all dimensions provided that $|W_g(p)|
+ |\nabla W_g(p)| > 0$ for all $p \in M$. In dimensions $10$ and $11$, it
is sufficient to assume that $|W_g(p)| + |\nabla W_g(p)| + |\nabla^2 W_g(p)| > 0$ for all $p \in M$
(see \cite{Li-Zhang2}).

Very recently, M.~Khuri, F.~Marques and R.~Schoen \cite{Khuri-Marques-Schoen}
proved the Weyl Vanishing Conjecture up to dimension $24$. This result,
combined with the positive mass theorem, implies the Compactness Conjecture for
those dimensions. After proving sharp pointwise estimates, they reduce
these questions to showing a certain quadratic
form is positive definite. It turns out the quadratic form has negative
eigenvalues if $n \geq 25$.

In a recent paper \cite{Brendle1}, it was shown that the Compactness Conjecture
fails for $n \geq 52$. More precisely, given any integer $n \geq 52$, there
exists a smooth Riemannian metric $g$ on $S^n$ such that set of constant scalar
curvature metrics in the conformal class of $g$ is non-compact. Moreover, the
blowing-up sequences obtained in \cite{Brendle1} form exactly one bubble. The
construction relies on a gluing procedure based on some local model metric.
These local models are directions in which the quadratic form of
\cite{Khuri-Marques-Schoen} is negative definite. We refer to \cite{Brendle2} for a
survey of this and related results.

In the present paper, we extend these counterexamples to the dimensions
$25 \leq n \leq 51$. Our main theorem is:

\begin{theorem}
\label{main.theorem}
Assume that $25 \leq n \leq 51$. Then there exists a Riemannian metric
$g$ on $S^n$ (of class $C^\infty$) and a sequence of positive functions
$v_\nu \in C^\infty(S^n)$ ($\nu \in \mathbb{N}$) with the following
properties:
\begin{itemize}
\item[(i)] $g$ is not conformally flat
\item[(ii)] $v_\nu$ is a solution of the Yamabe PDE (\ref{yamabe.pde})
for all $\nu \in \mathbb{N}$
\item[(iii)] $E_g(v_\nu) < Y(S^n)$ for all $\nu \in \mathbb{N}$, and
$E_g(v_\nu) \to Y(S^n)$ as $\nu \to \infty$
\item[(iv)] $\sup_{S^n} v_\nu \to \infty$ as $\nu \to \infty$
\end{itemize}
(Here, $Y(S^n)$ denotes the Yamabe energy of the round metric on $S^n$.)
\end{theorem}

We note that O.~Druet and E.~Hebey \cite{Druet-Hebey1} have constructed
blow-up examples for perturbations of (\ref{yamabe.pde}) (see also \cite{Druet-Hebey2}).

In Section 2, we describe how the problem can be reduced to finding critical points of a certain function $\mathcal{F}_g(\xi,\varepsilon)$, where $\xi$ is a vector in $\mathbb{R}^n$ and $\varepsilon$ is a positive real number. This idea has been used by many authors (see, e.g., \cite{Ambrosetti}, \cite{Ambrosetti-Malchiodi}, \cite{Berti-Malchiodi}, \cite{Brendle1}). In Section 3, we show that the function
$\mathcal{F}_g(\xi,\varepsilon)$ can be approximated by an auxiliary
function $F(\xi,\varepsilon)$. In Section 4, we prove that the function
$F(\xi,\varepsilon)$ has a critical point, which is a strict local
minimum. Finally, in Section 5, we use a perturbation argument to
construct critical points of the function
$\mathcal{F}_g(\xi,\varepsilon)$. From this the non-compactness result
follows.

The authors would like to thank Professor Richard Schoen for constant support and encouragement. The first author was supported by the Alfred P. Sloan foundation and by the National Science Foundation under grant
DMS-0605223. The second author was supported by CNPq-Brazil, FAPERJ and the Stanford Mathematics Department. \\

\section{Lyapunov-Schmidt reduction}

In this section, we collect some basic results established in
\cite{Brendle1}. Let
\[\mathcal{E} = \bigg \{ w \in L^{\frac{2n}{n-2}}(\mathbb{R}^n) \cap
W_{loc}^{1,2}(\mathbb{R}^n): \int_{\mathbb{R}^n} |dw|^2 < \infty \bigg
\}.\] By Sobolev's inequality, there exists a constant $K$, depending
only on $n$, such that
\[\bigg ( \int_{\mathbb{R}^n} |w|^{\frac{2n}{n-2}} \bigg
)^{\frac{n-2}{n}} \leq K \, \int_{\mathbb{R}^n} |dw|^2\]
for all $w \in \mathcal{E}$. We define a norm on $\mathcal{E}$ by
$\|w\|_{\mathcal{E}}^2 = \int_{\mathbb{R}^n} |dw|^2$. It is easy to see
that $\mathcal{E}$, equipped with this norm, is complete. \\

Given any pair $(\xi,\varepsilon) \in \mathbb{R}^n \times (0,\infty)$,
we define a function $u_{(\xi,\varepsilon)}:
\mathbb{R}^n \to \mathbb{R}$ by
\[u_{(\xi,\varepsilon)}(x) = \Big ( \frac{\varepsilon}{\varepsilon^2 +
|x - \xi|^2} \Big )^{\frac{n-2}{2}}.\]
The function $u_{(\xi,\varepsilon)}$ satisfies the elliptic PDE
\[\Delta u_{(\xi,\varepsilon)} + n(n-2) \,
u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} = 0.\]
It is well known that
\[\int_{\mathbb{R}^n} u_{(\xi,\varepsilon)}^{\frac{2n}{n-2}} = \Big (
\frac{Y(S^n)}{4n(n-1)} \Big )^{\frac{n}{2}}\]
for all $(\xi,\varepsilon) \in \mathbb{R}^n \times (0,\infty)$. We next
define
\[\varphi_{(\xi,\varepsilon,0)}(x) =
\Big ( \frac{\varepsilon}{\varepsilon^2 + |x - \xi|^2} \Big
)^{\frac{n+2}{2}} \, \frac{\varepsilon^2 - |x - \xi|^2}{\varepsilon^2 +
|x - \xi|^2}\]
and
\[\varphi_{(\xi,\varepsilon,k)}(x) =
\Big ( \frac{\varepsilon}{\varepsilon^2 + |x - \xi|^2} \Big
)^{\frac{n+2}{2}} \, \frac{2\varepsilon \, (x_k - \xi_k)}{\varepsilon^2
+ |x - \xi|^2}\]
for $k = 1,\hdots,n$. Finally, we define a closed subspace
$\mathcal{E}_{(\xi,\varepsilon)} \subset \mathcal{E}$ by
\[\mathcal{E}_{(\xi,\varepsilon)} = \bigg \{ w \in \mathcal{E}:
\int_{\mathbb{R}^n} \varphi_{(\xi,\varepsilon,k)} \, w = 0 \quad
\text{for $k = 0,1,\hdots,n$} \bigg \}.\]
Clearly, $u_{(\xi,\varepsilon)} \in \mathcal{E}_{(\xi,\varepsilon)}$.

\begin{proposition}
\label{linearized.operator}
Consider a Riemannian metric on $\mathbb{R}^n$ of the form $g(x) =
\exp(h(x))$, where $h(x)$ is a trace-free symmetric two-tensor on
$\mathbb{R}^n$ satisfying  $h(x) = 0$ for $|x| \geq 1$. There exists a
positive constant $\alpha_0 \leq 1$, depending only on $n$, with the
following significance: if $|h(x)| + |\partial h(x)| + |\partial^2 h(x)|
\leq \alpha_0$ for all $x \in \mathbb{R}^n$, then, given any pair
$(\xi,\varepsilon) \in \mathbb{R}^n \times (0,\infty)$ and any function
$f \in L^{\frac{2n}{n+2}}(\mathbb{R}^n)$, there exists a unique function
$w=G_{(\xi,\varepsilon)}(f) \in \mathcal{E}_{(\xi,\varepsilon)}$ such that
\[\int_{\mathbb{R}^n} \Big ( \langle dw,d\psi \rangle_g
+ \frac{n-2}{4(n-1)} \, R_g \, w \, \psi - n(n+2) \,
u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, w \, \psi \Big ) =
\int_{\mathbb{R}^n} f \, \psi\]
for all test functions $\psi \in \mathcal{E}_{(\xi,\varepsilon)}$.
Moreover, we have $\|w\|_{\mathcal{E}} \leq C \,
\|f\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)}$, where $C$ is a constant that
depends only on $n$.
\end{proposition}

\begin{proposition}
\label{fixed.point.argument}
Consider a Riemannian metric on $\mathbb{R}^n$ of the form $g(x) =
\exp(h(x))$, where $h(x)$ is a trace-free symmetric two-tensor on
$\mathbb{R}^n$ satisfying $h(x) = 0$ for $|x| \geq 1$. Moreover, let
$(\xi,\varepsilon) \in \mathbb{R}^n \times (0,\infty)$. There exists a
positive constant $\alpha_1 \leq \alpha_0$, depending only on $n$, with
the following significance: if $|h(x)| + |\partial h(x)| + |\partial^2
h(x)| \leq \alpha_1$ for all $x \in \mathbb{R}^n$, then there exists a
function $v_{(\xi,\varepsilon)} \in \mathcal{E}$ such that
$v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} \in
\mathcal{E}_{(\xi,\varepsilon)}$
and
\[\int_{\mathbb{R}^n} \Big ( \langle dv_{(\xi,\varepsilon)},d\psi
\rangle_g + \frac{n-2}{4(n-1)} \, R_g \, v_{(\xi,\varepsilon)} \, \psi -
n(n-2) \, |v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} \,
v_{(\xi,\varepsilon)} \, \psi \Big ) = 0\] for all test functions $\psi
\in \mathcal{E}_{(\xi,\varepsilon)}$. Moreover, we have the estimate
\begin{align*}
&\|v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}\|_{\mathcal{E}} \\
&\leq C \, \Big \| \Delta_g u_{(\xi,\varepsilon)} - \frac{n-2}{4(n-1)}
\, R_g \, u_{(\xi,\varepsilon)} + n(n-2) \,
u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \Big
\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)},
\end{align*}
where $C$ is a constant that depends only on $n$.
\end{proposition}

We next define a function $\mathcal{F}_g: \mathbb{R}^n \times (0,\infty)
\to \mathbb{R}$ by
\begin{align*}
\mathcal{F}_g(\xi,\varepsilon) &= \int_{\mathbb{R}^n}
\Big ( |dv_{(\xi,\varepsilon)}|_g^2 + \frac{n-2}{4(n-1)} \, R_g \,
v_{(\xi,\varepsilon)}^2 - (n-2)^2 \,
|v_{(\xi,\varepsilon)}|^{\frac{2n}{n-2}} \Big ) \\
&- 2(n-2) \, \Big ( \frac{Y(S^n)}{4n(n-1)} \Big )^{\frac{n}{2}}.
\end{align*}
If we choose $\alpha_1$ small enough, then we obtain the following
result: \\

\begin{proposition}
\label{reduction.to.a.finite.dimensional.problem}
The function $\mathcal{F}_g$ is continuously differentiable. Moreover,
if $(\bar{\xi},\bar{\varepsilon})$ is a critical point of the function
$\mathcal{F}_g$, then
the function $v_{(\bar{\xi},\bar{\varepsilon})}$ is a non-negative weak
solution of the equation
\[\Delta_g v_{(\bar{\xi},\bar{\varepsilon})} - \frac{n-2}{4(n-1)} \, R_g
\, v_{(\bar{\xi},\bar{\varepsilon})} + n(n-2) \,
v_{(\bar{\xi},\bar{\varepsilon})}^{\frac{n+2}{n-2}} = 0.\]
\end{proposition}

\section{An estimate for the energy of a ``bubble"}

Throughout this paper, we fix a real number $\tau$ and a multi-linear
form $W: \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}^n \times
\mathbb{R}^n \to \mathbb{R}$. The number $\tau$ depends only on the
dimension $n$. The exact choice of $\tau$ will be postponed until
Section 4. We assume that $W_{ijkl}$ satisfies all the algebraic
properties of the Weyl tensor. Moreover, we assume that some components
of $W$ are non-zero, so that
\[\sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 > 0.\]
For abbreviation, we put
\[H_{ik}(x) = \sum_{p,q=1}^n W_{ipkq} \, x_p \, x_q\]
and
\[\overline{H}_{ik}(x) = f(|x|^2) \, H_{ik}(x),\]
where $f(s) = \tau + 5s - s^2 + \frac{1}{20} \, s^3$. It is easy to see
that $H_{ik}(x)$ is trace-free, $\sum_{i=1}^n x_i \, H_{ik}(x) = 0$, and
$\sum_{i=1}^n \partial_i H_{ik}(x) = 0$ for all $x \in \mathbb{R}^n$. \\

We consider a Riemannian metric of the form $g(x) = \exp(h(x))$, where
$h(x)$ is a trace-free symmetric two-tensor on $\mathbb{R}^n$ satisfying
$h(x) = 0$ for $|x| \geq 1$,
\[|h(x)| + |\partial h(x)| + |\partial^2 h(x)| \leq \alpha_1\]
for all $x \in \mathbb{R}^n$, and
\[h_{ik}(x) = \mu \, \lambda^6 \, f(\lambda^{-2} \, |x|^2) \, H_{ik}(x)\]
for $|x| \leq \rho$. We assume that the parameters $\lambda$, $\mu$, and
$\rho$ are chosen such that
$\mu \leq 1$ and $\lambda \leq \rho \leq 1$. Note that
$\sum_{i=1}^n x_i \, h_{ik}(x) = 0$ and $\sum_{i=1}^n \partial_i
h_{ik}(x) = 0$ for $|x| \leq \rho$.

Given any pair $(\xi,\varepsilon) \in \mathbb{R}^n \times (0,\infty)$,
there exists a unique function $v_{(\xi,\varepsilon)}$ such that $v_{(\xi,\varepsilon)}
- u_{(\xi,\varepsilon)} \in \mathcal{E}_{(\xi,\varepsilon)}$
and
\[\int_{\mathbb{R}^n} \Big ( \langle dv_{(\xi,\varepsilon)},d\psi
\rangle_g + \frac{n-2}{4(n-1)} \, R_g \, v_{(\xi,\varepsilon)} \, \psi -
n(n-2) \, |v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} \,
v_{(\xi,\varepsilon)} \, \psi \Big ) = 0\] for all test functions $\psi
\in \mathcal{E}_{(\xi,\varepsilon)}$ (see Proposition \ref{fixed.point.argument}).

For abbreviation, let
\[\Omega = \bigg \{ (\xi,\varepsilon) \in \mathbb{R}^n \times
\mathbb{R}: |\xi| < 1, \, \frac{1}{2} < \varepsilon < 2 \bigg \}.\]

The following result is proved in the Appendix A of \cite{Brendle1}. A similar formula is derived in \cite{Ambrosetti-Malchiodi}.

\begin{proposition}
\label{Taylor.expansion.of.scalar.curvature}
Consider a Riemannian metric on $\mathbb{R}^n$ of the form $g(x) =
\exp(h(x))$, where $h(x)$ is a trace-free symmetric two-tensor on
$\mathbb{R}^n$ satisfying $|h(x)| \leq 1$ for all $x \in \mathbb{R}^n$.
Let $R_g$ be the scalar curvature of $g$. There exists a constant $C$,
depending only on $n$, such that
\begin{align*}
&\Big | R_g - \partial_i \partial_k h_{ik}
+ \partial_i(h_{il} \, \partial_k h_{kl}) - \frac{1}{2} \, \partial_i
h_{il} \, \partial_k h_{kl} + \frac{1}{4} \, \partial_l h_{ik} \,
\partial_l h_{ik} \Big | \\
&\leq C \, |h|^2 \, |\partial^2 h| + C \, |h| \, |\partial h|^2.
\end{align*}
\end{proposition}

\vspace{2mm}

\begin{proposition}
\label{estimate.for.error.term}
Assume that $(\xi,\varepsilon) \in \lambda \, \Omega$. Then we have
\begin{align*}
&\Big \| \Delta_g u_{(\xi,\varepsilon)}
- \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)}
+ n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \Big
\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} \\
&\leq C \, \lambda^8 \, \mu + C \, \Big ( \frac{\lambda}{\rho} \Big
)^{\frac{n-2}{2}}
\end{align*}
and
\begin{align*}
&\Big \| \Delta_g u_{(\xi,\varepsilon)}
- \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)}
+ n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \\
&\hspace{10mm} + \sum_{i,k=1}^n \mu \, \lambda^6 \, f(\lambda^{-2} \,
|x|^2) \, H_{ik}(x) \, \partial_i \partial_k u_{(\xi,\varepsilon)} \Big
\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} \\
&\leq C \, \lambda^{\frac{8(n+2)}{n-2}} \, \mu^2 + C \, \Big (
\frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}}.
\end{align*}
\end{proposition}

\textbf{Proof.}
Note that $\sum_{i=1}^n \partial_i h_{ik}(x) = 0$ for $|x| \leq \rho$.
Hence, it follows from Proposition
\ref{Taylor.expansion.of.scalar.curvature} that
\[|R_g(x)| \leq C \, |h(x)|^2 \, |\partial^2 h(x)| + C \, |\partial
h(x)|^2 \leq C \, \mu^2 \, (\lambda + |x|)^{14}\]
for $|x| \leq \rho$. This implies
\begin{align*}
&\Big | \Delta_g u_{(\xi,\varepsilon)} - \frac{n-2}{4(n-1)} \, R_g \,
u_{(\xi,\varepsilon)}
+ n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \Big | \\
&= \Big | \sum_{i,k=1}^n \partial_i \big [ (g^{ik} - \delta_{ik}) \,
\partial_k u_{(\xi,\varepsilon)} \big ]
- \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)} \Big | \\
&\leq C \, \lambda^{\frac{n-2}{2}} \, \mu \, (\lambda + |x|)^{8-n}
\end{align*}
and
\begin{align*}
&\Big | \Delta_g u_{(\xi,\varepsilon)}
- \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)}
+ n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} + \sum_{i,k=1}^n
h_{ik} \, \partial_i \partial_k u_{(\xi,\varepsilon)} \Big | \\
&= \Big | \sum_{i,k=1}^n \partial_i \big [ (g^{ik} - \delta_{ik} +
h_{ik}) \, \partial_k u_{(\xi,\varepsilon)} \big ] - \frac{n-2}{4(n-1)}
\, R_g \, u_{(\xi,\varepsilon)} \Big | \\
&\leq C \, \lambda^{\frac{n-2}{2}} \, \mu^2 \, (\lambda + |x|)^{16-n} \\
&\leq C \, \lambda^{\frac{n-2}{2}} \, \mu^2 \, (\lambda +
|x|)^{\frac{8(n+2)}{n-2}-n}
\end{align*}
for $|x| \leq \rho$. From this the assertion follows. \\

\begin{corollary}
\label{estimate.for.v.1}
The function $v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}$ satisfies
the estimate
\[\|v_{(\xi,\varepsilon)} -
u_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)}
\leq C \, \lambda^8 \, \mu + C \, \Big ( \frac{\lambda}{\rho} \Big
)^{\frac{n-2}{2}}\]
whenever $(\xi,\varepsilon) \in \lambda \, \Omega$.
\end{corollary}

\textbf{Proof.}
It follows from Proposition \ref{fixed.point.argument} that
\begin{align*}
&\|v_{(\xi,\varepsilon)} -
u_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \\
&\leq C \, \Big \| \Delta_g u_{(\xi,\varepsilon)}
- \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)}
+ n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \Big
\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)},
\end{align*}
where $C$ is a constant that depends only on $n$. Hence, the assertion
follows from Proposition \ref{estimate.for.error.term}. \\

We next establish a more precise estimate for the function $v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}$.
Applying Proposition \ref{linearized.operator} with $h = 0$, we conclude that there exists a unique function $w_{(\xi,\varepsilon)} \in
\mathcal{E}_{(\xi,\varepsilon)}$ such that
\begin{align*}
&\int_{\mathbb{R}^n} \Big ( \langle dw_{(\xi,\varepsilon)},d\psi \rangle
- n(n+2) \, u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \,
w_{(\xi,\varepsilon)} \, \psi \Big ) \\
&= -\int_{\mathbb{R}^n} \sum_{i,k=1}^n \mu \, \lambda^6 \,
f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k
u_{(\xi,\varepsilon)} \, \psi
\end{align*}
for all test functions $\psi \in \mathcal{E}_{(\xi,\varepsilon)}$.

\begin{proposition}
\label{properties.of.w}
The function $w_{(\xi,\varepsilon)}$ is smooth. Moreover, if $(\xi,\varepsilon) \in \lambda \, \Omega$, then the function
$w_{(\xi,\varepsilon)}$ satisfies the estimates
\begin{align*}
&|w_{(\xi,\varepsilon)}(x)| \leq C \, \lambda^{\frac{n-2}{2}} \, \mu \,
(\lambda + |x|)^{10-n} \\
&|\partial w_{(\xi,\varepsilon)}(x)| \leq C \, \lambda^{\frac{n-2}{2}}
\, \mu \, (\lambda + |x|)^{9-n} \\
&|\partial^2 w_{(\xi,\varepsilon)}(x)| \leq C \, \lambda^{\frac{n-2}{2}}
\, \mu \, (\lambda + |x|)^{8-n}
\end{align*}
for all $x \in \mathbb{R}^n$.
\end{proposition}

\textbf{Proof.}
There exist real numbers $b_k(\xi,\varepsilon)$ such that
\begin{align*}
&\int_{\mathbb{R}^n} \Big ( \langle dw_{(\xi,\varepsilon)},d\psi \rangle
- n(n+2) \, u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \,
w_{(\xi,\varepsilon)} \, \psi \Big ) \\
&= -\int_{\mathbb{R}^n} \sum_{i,k=1}^n \mu \, \lambda^6 \,
f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k
u_{(\xi,\varepsilon)} \, \psi \\
&+ \sum_{k=0}^n b_k(\xi,\varepsilon) \, \int_{\mathbb{R}^n} \varphi_{(\xi,\varepsilon,k)} \, \psi
\end{align*}
for all test functions $\psi \in \mathcal{E}$. Hence, standard elliptic regularity theory implies that $w_{(\xi,\varepsilon)}$ is smooth.

It remains to prove quantitative estimates for $w_{(\xi,\varepsilon)}$. To that end, we consider a pair $(\xi,\varepsilon) \in \lambda \, \Omega$. One readily verifies that
\[\Big \| \sum_{i,k=1}^n \mu \, \lambda^6 \,
f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k
u_{(\xi,\varepsilon)} \Big \|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} \leq C \, \lambda^8 \, \mu.\] As a consequence, the function $w_{(\xi,\varepsilon)}$ satisfies $\|w_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \leq C \, \lambda^8 \, \mu$. Moreover, we have $\sum_{k=0}^n |b_k(\xi,\varepsilon)| \leq C \, \lambda^8 \, \mu$. This implies
\begin{align*}
&\big | \Delta w_{(\xi,\varepsilon)} + n(n+2) \, u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, w_{(\xi,\varepsilon)} \big | \\
&= \bigg | \sum_{i,k=1}^n \mu \, \lambda^6 \,
f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k
u_{(\xi,\varepsilon)} - \sum_{k=0}^n b_k(\xi,\varepsilon) \, \int_{\mathbb{R}^n} \varphi_{(\xi,\varepsilon,k)} \bigg | \\
&\leq C \, \lambda^{\frac{n-2}{2}} \, \mu \, (\lambda+|x|)^{8-n}
\end{align*}
for all $x \in \mathbb{R}^n$. We claim that
\[\sup_{x \in \mathbb{R}^n} (\lambda + |x|)^{\frac{n-2}{2}} \, |w_{(\xi,\varepsilon)}(x)| \leq C \, \lambda^8 \, \mu.\]
To show this, we fix a point $x_0 \in \mathbb{R}^n$. Let $r = \frac{1}{2} \, (\lambda + |x_0|)$. Then
\[u_{(\xi,\varepsilon)}(x)^{\frac{4}{n-2}} \leq C \, r^{-2}\] and
\[\big | \Delta w_{(\xi,\varepsilon)} + n(n+2) \, u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, w_{(\xi,\varepsilon)} \big | \leq C \, \lambda^{\frac{n-2}{2}} \, \mu \, r^{8-n}\]
for all $x \in B_r(x_0)$. Hence, it follows from standard interior estimates that
\begin{align*}
r^{\frac{n-2}{2}} \, |w_{(\xi,\varepsilon)}(x_0)| &\leq C \, \|w_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(B_r(x_0))} \\
&+ C \, r^{\frac{n+2}{2}} \, \big \| \Delta w_{(\xi,\varepsilon)} + n(n+2) \, u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, w_{(\xi,\varepsilon)} \big \|_{L^\infty(B_r(x_0))} \\
&\leq C \, \lambda^8 \, \mu + C \, \lambda^{\frac{n-2}{2}} \, \mu \, r^{-\frac{n-18}{2}} \\
&\leq C \, \lambda^8 \, \mu.
\end{align*}
Therefore, we have
\[\sup_{x \in \mathbb{R}^n} (\lambda + |x|)^{\frac{n-2}{2}} \, |w_{(\xi,\varepsilon)}(x)| \leq C \, \lambda^8 \, \mu,\] as
claimed. Since $\sup_{x \in \mathbb{R}^n} |x|^{\frac{n-2}{2}} \, |w_{(\xi,\varepsilon)}(x)| < \infty$, we can express the function $w_{(\xi,\varepsilon)}$ in the form
\begin{equation}
\label{convolution.formula}
w_{(\xi,\varepsilon)}(x) = -\frac{1}{(n-2) \, |S^{n-1}|} \int_{\mathbb{R}^n} |x - y|^{2-n} \, \Delta w_{(\xi,\varepsilon)}(y) \, dy
\end{equation}
for all $x \in \mathbb{R}^n$.

We are now able to use a bootstrap argument to prove the desired estimate for $w_{(\xi,\varepsilon)}$. It follows from (\ref{convolution.formula}) that
\[\sup_{x \in \mathbb{R}^n} (\lambda + |x|)^\beta \, |w_{(\xi,\varepsilon)}(x)| \leq C \, \sup_{x \in \mathbb{R}^n} (\lambda + |x|)^{\beta+2} \, |\Delta w_{(\xi,\varepsilon)}(x)|\]
for all $0 < \beta < n-2$. Since
\begin{align*}
|\Delta w_{(\xi,\varepsilon)}(x)|
&\leq n(n+2) \, u_{(\xi,\varepsilon)}(x)^{\frac{4}{n-2}} \, |w_{(\xi,\varepsilon)}(x)| \\
&+ C \, \lambda^{\frac{n-2}{2}} \, \mu \, (\lambda+|x|)^{8-n}
\end{align*}
for all $x \in \mathbb{R}^n$, we conclude that
\begin{align*}
\sup_{x \in \mathbb{R}^n} (\lambda + |x|)^\beta \, |w_{(\xi,\varepsilon)}(x)| &\leq C \, \lambda^2 \, \sup_{x \in \mathbb{R}^n} (\lambda + |x|)^{\beta-2} \, |w_{(\xi,\varepsilon)}(x)| \\ &+ C \, \lambda^{\beta-\frac{n-18}{2}} \, \mu
\end{align*}
for all $0 < \beta \leq n-10$. Iterating this inequality, we obtain
\[\sup_{x \in \mathbb{R}^n} (\lambda + |x|)^{n-10} \, |w_{(\xi,\varepsilon)}(x)| \leq C \, \lambda^{\frac{n-2}{2}} \, \mu.\]
The estimates for the first and second derivatives of $w_{(\xi,\varepsilon)}$ follow now from standard interior estimates. \\

\begin{corollary}
\label{estimate.for.v.2}
The function $v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} -
w_{(\xi,\varepsilon)}$ satisfies the estimate
\[\|v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} -
w_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)}
\leq C \, \lambda^{\frac{8(n+2)}{n-2}} \, \mu^{\frac{n+2}{n-2}} + C \,
\Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}}\]
whenever $(\xi,\varepsilon) \in \lambda \, \Omega$.
\end{corollary}

\textbf{Proof.}
Consider the functions
\[B_1 = \sum_{i,k=1}^n \partial_i \big [ (g^{ik} - \delta_{ik}) \,
\partial_k w_{(\xi,\varepsilon)} \big ] - \frac{n-2}{4(n-1)} \, R_g \,
w_{(\xi,\varepsilon)}\]
and
\[B_2 = \sum_{i,k=1}^n \mu \, \lambda^6 \, f(\lambda^{-2} \, |x|^2) \,
H_{ik}(x) \, \partial_i \partial_k u_{(\xi,\varepsilon)}.\]
By definition of $w_{(\xi,\varepsilon)}$, we have
\begin{align*}
&\int_{\mathbb{R}^n} \Big ( \langle dw_{(\xi,\varepsilon)},d\psi \rangle_g
+ \frac{n-2}{4(n-1)} \, R_g \, w_{(\xi,\varepsilon)} \, \psi - n(n+2) \,
u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, w_{(\xi,\varepsilon)} \, \psi
\Big ) \\
&= -\int_{\mathbb{R}^n} (B_1 + B_2) \, \psi
\end{align*}
for all functions $\psi \in \mathcal{E}_{(\xi,\varepsilon)}$. Since
$w_{(\xi,\varepsilon)} \in \mathcal{E}_{(\xi,\varepsilon)}$, it follows
that
\[w_{(\xi,\varepsilon)} = -G_{(\xi,\varepsilon)}(B_1 + B_2).\]
Moreover, we have
\[v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} = G_{(\xi,\varepsilon)}
\big ( B_3 + n(n-2) \, B_4 \big ),\]
where
\[B_3 = \Delta_g u_{(\xi,\varepsilon)} - \frac{n-2}{4(n-1)} \, R_g \,
u_{(\xi,\varepsilon)} + n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}}\]
and
\[B_4 = |v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} \, v_{(\xi,\varepsilon)}
- u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}}  - \frac{n+2}{n-2} \,
u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, (v_{(\xi,\varepsilon)} -
u_{(\xi,\varepsilon)}).\]
Thus, we conclude that
\[v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} - w_{(\xi,\varepsilon)}
= G_{(\xi,\varepsilon)} \big ( B_1 + B_2 + B_3 + n(n-2) \, B_4 \big ),\]
where $G_{(\xi,\varepsilon)}$ denotes the solution operator constructed in Proposition \ref{linearized.operator}. As a consequence, we obtain
\[\|v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} - w_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \leq C \,
\big \| B_1 + B_2 + B_3 + n(n-2) \, B_4 \big \|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)}.\]
It follows from Proposition \ref{properties.of.w} that
\[|B_1(x)| \leq C \, \lambda^{\frac{n-2}{2}} \, \mu^2 \, (\lambda + |x|)^{16-n} \leq C \, \lambda^{\frac{n-2}{2}} \, \mu^2 \, (\lambda+|x|)^{\frac{8(n+2)}{n-2}-n}\]
for $|x| \leq \rho$ and
\[|B_1(x)| \leq C \, \lambda^{\frac{n-2}{2}} \, \mu \, |x|^{8-n}\]
for $|x| \geq \rho$. This implies
\[\|B_1\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} \leq C \,
\lambda^{\frac{8(n+2)}{n-2}} \, \mu^2 + C \, \rho^8 \, \mu \, \Big (
\frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}}.\]
Moreover, observe that
\[\|B_2 + B_3\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} \leq C \,
\lambda^{\frac{8(n+2)}{n-2}} \, \mu^2 + C \, \Big ( \frac{\lambda}{\rho}
\Big )^{\frac{n-2}{2}}\]
by Proposition \ref{estimate.for.error.term}. Finally, Corollary \ref{estimate.for.v.1} implies that
\begin{align*}
\|B_4\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)}
&\leq C \, \|v_{(\xi,\varepsilon)} -
u_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)}^{\frac{n+2}{n-2}}
\\
&\leq C \, \lambda^{\frac{8(n+2)}{n-2}} \, \mu^{\frac{n+2}{n-2}} + C \,
\Big ( \frac{\lambda}{\rho} \Big )^{\frac{n+2}{2}}.
\end{align*}
Putting these facts together, we obtain
\[\|v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} -
w_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)}
\leq C \, \lambda^{\frac{8(n+2)}{n-2}} \, \mu^{\frac{n+2}{n-2}} + C \,
\Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}}.
\]
This completes the proof. \\

\begin{proposition}
\label{term.1}
We have
\begin{align*}
&\bigg | \int_{\mathbb{R}^n} \Big ( |dv_{(\xi,\varepsilon)}|_g^2
- |du_{(\xi,\varepsilon)}|_g^2 + \frac{n-2}{4(n-1)} \, R_g \,
(v_{(\xi,\varepsilon)}^2 - u_{(\xi,\varepsilon)}^2) \Big ) \\
&\hspace{10mm} + \int_{\mathbb{R}^n} n(n-2) \,
(|v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} -
u_{(\xi,\varepsilon)}^{\frac{4}{n-2}}) \, u_{(\xi,\varepsilon)} \,
v_{(\xi,\varepsilon)} \\
&\hspace{10mm} - \int_{\mathbb{R}^n} n(n-2) \,
(|v_{(\xi,\varepsilon)}|^{\frac{2n}{n-2}} -
u_{(\xi,\varepsilon)}^{\frac{2n}{n-2}}) \\
&\hspace{10mm} - \int_{\mathbb{R}^n} \sum_{i,k=1}^n \mu \, \lambda^6 \,
f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k
u_{(\xi,\varepsilon)} \, w_{(\xi,\varepsilon)} \bigg | \\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^{\frac{2n}{n-2}} + C \,
\lambda^8 \, \mu \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}} +
C \, \Big ( \frac{\lambda}{\rho} \Big )^{n-2}
\end{align*}
whenever $(\xi,\varepsilon) \in \lambda \, \Omega$.
\end{proposition}

\textbf{Proof.}
By definition of $v_{(\xi,\varepsilon)}$, we have
\begin{align*}
&\int_{\mathbb{R}^n} \Big ( |dv_{(\xi,\varepsilon)}|_g^2 - \langle
du_{(\xi,\varepsilon)},dv_{(\xi,\varepsilon)} \rangle_g
+ \frac{n-2}{4(n-1)} \, R_g \, v_{(\xi,\varepsilon)} \,
(v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}) \Big ) \\
&\hspace{10mm} - \int_{\mathbb{R}^n} n(n-2) \,
|v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} \, v_{(\xi,\varepsilon)} \,
(v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}) = 0.
\end{align*}
Using Proposition \ref{estimate.for.error.term} and Corollary
\ref{estimate.for.v.1}, we obtain
\begin{align*}
&\bigg | \int_{\mathbb{R}^n} \Big ( \langle
du_{(\xi,\varepsilon)},dv_{(\xi,\varepsilon)} \rangle_g
- |du_{(\xi,\varepsilon)}|_g^2 + \frac{n-2}{4(n-1)} \, R_g \,
u_{(\xi,\varepsilon)} \, (v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)})
\Big ) \\
&\hspace{10mm} - \int_{\mathbb{R}^n} n(n-2) \,
u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \, (v_{(\xi,\varepsilon)} -
u_{(\xi,\varepsilon)}) \\
&\hspace{10mm} - \int_{\mathbb{R}^n} \sum_{i,k=1}^n \mu \, \lambda^6 \,
f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k
u_{(\xi,\varepsilon)} \, (v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)})
\bigg | \\
&\leq \Big \| \Delta_g u_{(\xi,\varepsilon)}
- \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)}
+ n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \\
&\hspace{10mm} + \sum_{i,k=1}^n \mu \, \lambda^6 \, f(\lambda^{-2} \,
|x|^2) \, H_{ik}(x) \, \partial_i \partial_k u_{(\xi,\varepsilon)} \Big
\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} \\
&\hspace{5mm} \cdot \|v_{(\xi,\varepsilon)} -
u_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^3 + C \, \lambda^8 \, \mu \,
\Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}} + C \, \Big (
\frac{\lambda}{\rho} \Big )^{n-2}.
\end{align*}
Moreover, we have
\begin{align*}
&\bigg | \int_{\mathbb{R}^n} \sum_{i,k=1}^n \mu \, \lambda^6 \,
f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k
u_{(\xi,\varepsilon)} \, (v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}
- w_{(\xi,\varepsilon)}) \bigg | \\
&\leq C \, \lambda^8 \, \mu \, \|v_{(\xi,\varepsilon)} -
u_{(\xi,\varepsilon)} -
w_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^{\frac{2n}{n-2}} + C \,
\lambda^8 \, \mu \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}}
\end{align*}
by Corollary \ref{estimate.for.v.2}. Putting these facts together, the
assertion follows. \\

\begin{proposition}
\label{term.2}
We have
\begin{align*}
&\bigg | \int_{\mathbb{R}^n} (|v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} -
u_{(\xi,\varepsilon)}^{\frac{4}{n-2}}) \, u_{(\xi,\varepsilon)} \,
v_{(\xi,\varepsilon)} - \frac{2}{n} \int_{\mathbb{R}^n}
(|v_{(\xi,\varepsilon)}|^{\frac{2n}{n-2}} -
u_{(\xi,\varepsilon)}^{\frac{2n}{n-2}}) \bigg | \\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^{\frac{2n}{n-2}} + C \, \Big
( \frac{\lambda}{\rho} \Big )^n
\end{align*}
whenever $(\xi,\varepsilon) \in \lambda \, \Omega$.
\end{proposition}

\textbf{Proof.}
We have the pointwise estimate
\begin{align*}
&\Big | (|v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} -
u_{(\xi,\varepsilon)}^{\frac{4}{n-2}}) \, u_{(\xi,\varepsilon)} \,
v_{(\xi,\varepsilon)} - \frac{2}{n} \,
(|v_{(\xi,\varepsilon)}|^{\frac{2n}{n-2}} -
u_{(\xi,\varepsilon)}^{\frac{2n}{n-2}}) \Big | \\
&\leq C \, |v_{(\xi,\varepsilon)} -
u_{(\xi,\varepsilon)}|^{\frac{2n}{n-2}},
\end{align*}
where $C$ is a constant that depends only on $n$. This implies
\begin{align*}
&\bigg | \int_{\mathbb{R}^n} (|v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} -
u_{(\xi,\varepsilon)}^{\frac{4}{n-2}}) \, u_{(\xi,\varepsilon)} \,
v_{(\xi,\varepsilon)} - \frac{2}{n} \int_{\mathbb{R}^n}
(|v_{(\xi,\varepsilon)}|^{\frac{2n}{n-2}} -
u_{(\xi,\varepsilon)}^{\frac{2n}{n-2}}) \bigg | \\
&\leq C \, \|v_{(\xi,\varepsilon)} -
u_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)}^{\frac{2n}{n-2}}
\\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^{\frac{2n}{n-2}} + C \, \Big
( \frac{\lambda}{\rho} \Big )^n
\end{align*}
by Corollary \ref{estimate.for.v.1}. \\

\begin{proposition}
\label{term.3}
We have
\begin{align*}
&\bigg | \int_{\mathbb{R}^n} \Big ( |du_{(\xi,\varepsilon)}|_g^2 +
\frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)}^2 - n(n-2) \,
u_{(\xi,\varepsilon)}^{\frac{2n}{n-2}} \Big ) \\
&\hspace{10mm} - \int_{B_\rho(0)} \frac{1}{2} \, \sum_{i,k,l=1}^n
h_{il} \, h_{kl} \, \partial_i u_{(\xi,\varepsilon)} \, \partial_k
u_{(\xi,\varepsilon)} \\
&\hspace{10mm} + \int_{B_\rho(0)} \frac{n-2}{16(n-1)} \,
\sum_{i,k,l=1}^n (\partial_l h_{ik})^2 \,
u_{(\xi,\varepsilon)}^2 \bigg | \\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^3 + C \, \Big (
\frac{\lambda}{\rho} \Big )^{n-2}
\end{align*}
whenever $(\xi,\varepsilon) \in \lambda \, \Omega$.
\end{proposition}

\textbf{Proof.}
Note that
\begin{align*}
&\Big | g^{ik}(x) - \delta_{ik} + h_{ik}(x) - \frac{1}{2} \,
\sum_{l=1}^n h_{il}(x) \, h_{kl}(x) \Big | \\
&\leq C \, |h(x)|^3 \leq C \, \mu^3 \, (\lambda + |x|)^{24} \leq C \,
\mu^3 \, (\lambda+|x|)^{\frac{16n}{n-2}}
\end{align*}
for $|x| \leq \rho$. This implies
\begin{align*}
&\bigg | \int_{\mathbb{R}^n} \big ( |du_{(\xi,\varepsilon)}|_g^2 -
|du_{(\xi,\varepsilon)}|^2 \big ) + \int_{\mathbb{R}^n} \sum_{i,k=1}^n
h_{ik} \, \partial_i u_{(\xi,\varepsilon)} \, \partial_k u_{(\xi,\varepsilon)} \\
&\hspace{10mm} - \int_{B_\rho(0)} \frac{1}{2} \sum_{i,k,l=1}^n h_{il}
\, h_{kl} \, \partial_i u_{(\xi,\varepsilon)} \, \partial_k
u_{(\xi,\varepsilon)} \bigg | \\
&\leq C \, \lambda^{n-2} \, \mu^3 \, \int_{B_\rho(0)} (\lambda+|x|)^{\frac{16n}{n-2}+2-2n}
+ C \, \lambda^{n-2} \, \int_{\mathbb{R}^n \setminus B_\rho(0)} (\lambda + |x|)^{2-2n} \\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^3 + C \, \Big ( \frac{\lambda}{\rho} \Big )^{n-2}.
\end{align*}
By Proposition \ref{Taylor.expansion.of.scalar.curvature}, the scalar
curvature of $g$ satisfies the estimate
\begin{align*}
&\Big | R_g(x) + \frac{1}{4} \sum_{i,k,l=1}^n (\partial_l h_{ik}(x))^2
\Big | \\
&\leq C \, |h(x)|^2 \, |\partial^2 h(x)| + C \, |h(x)| \, |\partial
h(x)|^2 \\
&\leq C \, \mu^3 \, (\lambda + |x|)^{22} \leq C \, \mu^3 \, (\lambda +
|x|)^{\frac{16n}{n-2}-2}
\end{align*}
for $|x| \leq \rho$. This implies
\begin{align*}
&\bigg | \int_{\mathbb{R}^n} R_g \, u_{(\xi,\varepsilon)}^2 +
\int_{B_\rho(0)} \frac{1}{4} \sum_{i,k,l=1}^n (\partial_l h_{ik})^2
\, u_{(\xi,\varepsilon)}^2 \bigg | \\
&\leq C \, \lambda^{n-2} \, \mu^3 \, \int_{B_\rho(0)} (\lambda+|x|)^{\frac{16n}{n-2}+2-2n}
+ C \, \lambda^{n-2} \, \int_{\mathbb{R}^n \setminus B_\rho(0)} (\lambda + |x|)^{4-2n} \\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^3 + C \, \rho^2 \, \Big (
\frac{\lambda}{\rho} \Big )^{n-2}.
\end{align*}
At this point, we use the formula
\begin{align*}
&\partial_i u_{(\xi,\varepsilon)} \, \partial_k u_{(\xi,\varepsilon)} -
\frac{n-2}{4(n-1)} \, \partial_i \partial_k (u_{(\xi,\varepsilon)}^2) \\
&= \frac{1}{n} \, \Big ( |du_{(\xi,\varepsilon)}|^2 - \frac{n-2}{4(n-1)}
\, \Delta (u_{(\xi,\varepsilon)}^2) \Big ) \, \delta_{ik}.
\end{align*}
Since $h_{ik}$ is trace-free, we obtain
\[\sum_{i,k=1}^n h_{ik} \, \partial_i u_{(\xi,\varepsilon)} \,
\partial_k u_{(\xi,\varepsilon)} = \frac{n-2}{4(n-1)} \sum_{i,k=1}^n
h_{ik} \, \partial_i \partial_k (u_{(\xi,\varepsilon)}^2),\]
hence
\[\int_{\mathbb{R}^n} \sum_{i,k=1}^n h_{ik} \, \partial_i
u_{(\xi,\varepsilon)} \, \partial_k u_{(\xi,\varepsilon)} =
\int_{\mathbb{R}^n} \frac{n-2}{4(n-1)} \sum_{i,k=1}^n \partial_i
\partial_k h_{ik} \, u_{(\xi,\varepsilon)}^2.\]
Since $\sum_{i=1}^n \partial_i h_{ik}(x) = 0$ for $|x| \leq \rho$, it
follows that
\[\bigg | \int_{\mathbb{R}^n} \sum_{i,k=1}^n h_{ik} \, \partial_i
u_{(\xi,\varepsilon)} \, \partial_k u_{(\xi,\varepsilon)} \bigg |
\leq C \int_{\mathbb{R}^n \setminus B_\rho(0)} u_{(\xi,\varepsilon)}^2 \leq C \, \rho^2 \, \Big ( \frac{\lambda}{\rho} \Big )^{n-2}.\]
Putting these facts together, the assertion follows. \\

\begin{corollary}
\label{key.estimate}
The function $\mathcal{F}_g(\xi,\varepsilon)$ satisfies the estimate
\begin{align*}
&\bigg | \mathcal{F}_g(\xi,\varepsilon) - \int_{B_\rho(0)}
\frac{1}{2} \, \sum_{i,k,l=1}^n h_{il} \, h_{kl} \, \partial_i
u_{(\xi,\varepsilon)} \, \partial_k u_{(\xi,\varepsilon)} \\
&\hspace{10mm} + \int_{B_\rho(0)} \frac{n-2}{16(n-1)} \,
\sum_{i,k,l=1}^n (\partial_l h_{ik})^2 \,
u_{(\xi,\varepsilon)}^2 \\
&\hspace{10mm} - \int_{\mathbb{R}^n} \sum_{i,k=1}^n \mu \, \lambda^6 \,
f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i\partial_k
u_{(\xi,\varepsilon)} \, w_{(\xi,\varepsilon)} \bigg | \\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^{\frac{2n}{n-2}}
+ C \, \lambda^8 \, \mu \, \Big ( \frac{\lambda}{\rho} \Big
)^{\frac{n-2}{2}} + C \, \Big ( \frac{\lambda}{\rho} \Big )^{n-2}
\end{align*}
whenever $(\xi,\varepsilon) \in \lambda \, \Omega$.
\end{corollary}

\section{Finding a critical point of an auxiliary function}

We define a function $F: \mathbb{R}^n \times (0,\infty) \to \mathbb{R}$
as follows: given any pair $(\xi,\varepsilon) \in \mathbb{R}^n \times
(0,\infty)$, we define
\begin{align*}
F(\xi,\varepsilon)
&= \int_{\mathbb{R}^n} \frac{1}{2} \sum_{i,k,l=1}^n \overline{H}_{il}(x)
\, \overline{H}_{kl}(x) \, \partial_i u_{(\xi,\varepsilon)}(x) \,
\partial_k u_{(\xi,\varepsilon)}(x) \\
&- \int_{\mathbb{R}^n} \frac{n-2}{16(n-1)} \, \sum_{i,k,l=1}^n
(\partial_l \overline{H}_{ik}(x))^2 \, u_{(\xi,\varepsilon)}(x)^2 \\
&+ \int_{\mathbb{R}^n} \sum_{i,k=1}^n \overline{H}_{ik}(x) \, \partial_i
\partial_k u_{(\xi,\varepsilon)}(x) \, z_{(\xi,\varepsilon)}(x),
\end{align*}
where $z_{(\xi,\varepsilon)} \in \mathcal{E}_{(\xi,\varepsilon)}$
satisfies the relation
\begin{align*}
&\int_{\mathbb{R}^n} \Big ( \langle dz_{(\xi,\varepsilon)},d\psi \rangle
- n(n+2) \, u_{(\xi,\varepsilon)}(x)^{\frac{4}{n-2}} \,
z_{(\xi,\varepsilon)} \, \psi \Big ) \\
&= -\int_{\mathbb{R}^n} \sum_{i,k=1}^n \overline{H}_{ik} \, \partial_i
\partial_k u_{(\xi,\varepsilon)} \, \psi
\end{align*}
for all test functions $\psi \in \mathcal{E}_{(\xi,\varepsilon)}$. Our
goal in this section is to show that the function $F(\xi,\varepsilon)$
has a critical point.

\begin{proposition}
\label{symmetry}
The function $F(\xi,\varepsilon)$ satisfies $F(\xi,\varepsilon) =
F(-\xi,\varepsilon)$ for all $(\xi,\varepsilon) \in \mathbb{R}^n \times
(0,\infty)$. Consequently, we have $\frac{\partial}{\partial \xi_p}
F(0,\varepsilon) = 0$ and $\frac{\partial^2}{\partial \varepsilon \,
\partial \xi_p} F(0,\varepsilon) = 0$ for all $\varepsilon > 0$ and $p =
1, \hdots, n$.
\end{proposition}

\textbf{Proof.}
This follows immediately from the relation $\overline{H}_{ik}(-x) =
\overline{H}_{ik}(x)$. \\

\begin{proposition}
\label{integral.identity.1}
We have
\begin{align*}
&\int_{\partial B_r(0)} \sum_{i,k,l=1}^n (\partial_l H_{ik}(x))^2 \, x_p
\, x_q \\
&= \frac{2}{n(n+2)} \, |S^{n-1}| \, \sum_{i,k,l=1}^n (W_{ipkl} +
W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \, r^{n+3} \\
&+ \frac{1}{n(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} +
W_{ilkj})^2 \, \delta_{pq} \, r^{n+3}
\end{align*}
and
\begin{align*}
&\int_{\partial B_r(0)} \sum_{i,k=1}^n H_{ik}(x)^2 \, x_p \, x_q \\
&= \frac{2}{n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,k,l=1}^n (W_{ipkl} +
W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \, r^{n+5} \\
&+ \frac{1}{2n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} +
W_{ilkj})^2 \, \delta_{pq} \, r^{n+5}.
\end{align*}
\end{proposition}

\textbf{Proof.}
See \cite{Brendle1}, Proposition 16. \\

\begin{proposition}
\label{integral.identity.2}
We have
\begin{align*}
&\int_{\partial B_r(0)} \sum_{i,k,l=1}^n (\partial_l
\overline{H}_{ik}(x))^2 \, x_p \, x_q \\
&= \frac{2}{n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,k,l=1}^n (W_{ipkl} +
W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \\
&\hspace{10mm} \cdot r^{n+3} \, \Big [ (n+4) \, f(r^2)^2 + 8r^2 \,
f(r^2) \, f'(r^2) + 4r^4 \, f'(r^2)^2 \Big ] \\
&+ \frac{1}{n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} +
W_{ilkj})^2 \, \delta_{pq} \\
&\hspace{10mm} \cdot r^{n+3} \, \Big [ (n+4) \, f(r^2)^2 + 4r^2 \,
f(r^2) \, f'(r^2) + 2r^4 \, f'(r^2)^2 \Big ].
\end{align*}
\end{proposition}

\textbf{Proof.}
Using the identity
\[\partial_l \overline{H}_{ik}(x) = f(|x|^2) \, \partial_l H_{ik}(x) + 2
\, f'(|x|^2) \, H_{ik}(x) \, x_l\]
and Euler's theorem, we obtain
\begin{align*}
&\sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \\
&= f(|x|^2)^2 \, \sum_{i,k,l=1}^n (\partial_l H_{ik}(x))^2 \\
&+ 4 \, f(|x|^2) \, f'(|x|^2) \, \sum_{i,k,l=1}^n H_{ik}(x) \, x_l \, \partial_l H_{ik}(x) \\
&+ 4 \, |x|^2 \, f'(|x|^2)^2 \, \sum_{i,k=1}^n H_{ik}(x)^2 \\
&= f(|x|^2)^2 \, \sum_{i,k,l=1}^n (\partial_l H_{ik}(x))^2 \\
&+ \big [ 8 \, f(|x|^2) \, f'(|x|^2) + 4 \, |x|^2 \, f'(|x|^2)^2 \big ]
\, \sum_{i,k=1}^n H_{ik}(x)^2.
\end{align*}
Hence, the assertion follows from the previous proposition. \\

\begin{corollary}
\label{integral.identity.3}
We have
\begin{align*}
&\int_{\partial B_r(0)} \sum_{i,k,l=1}^n (\partial_l
\overline{H}_{ik}(x))^2 \\ &= \frac{1}{n(n+2)} \, |S^{n-1}| \,
\sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 \\
&\hspace{10mm} \cdot r^{n+1} \, \Big [ (n+2) \, f(r^2)^2 + 4 \, r^2 \,
f(r^2) \, f'(r^2)
+ 2 \, r^4 \, f'(r^2)^2 \Big ].
\end{align*}
\end{corollary}

\vspace{2mm}

\begin{proposition}
\label{formula.for.F}
We have
\begin{align*}
F(0,\varepsilon) &= -\frac{n-2}{16n(n-1)(n+2)} \, |S^{n-1}| \,
\sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 \\
&\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 +
r^2)^{2-n} \, r^{n+1} \\
&\hspace{15mm} \cdot \Big [ (n+2) \, f(r^2)^2 + 4 \, r^2 \, f(r^2) \,
f'(r^2)
+ 2 \, r^4 \, f'(r^2)^2 \Big ] \, dr.
\end{align*}
\end{proposition}

\textbf{Proof.}
Note that $z_{(0,\varepsilon)}(x) = 0$ for all $x \in \mathbb{R}^n$.
This implies
\[F(0,\varepsilon) = -\int_{\mathbb{R}^n} \frac{n-2}{16(n-1)} \,
\varepsilon^{n-2} \, (\varepsilon^2 + |x|^2)^{2-n} \, \sum_{i,k,l=1}^n
(\partial_l \overline{H}_{ik}(x))^2.\]
Using Corollary \ref{integral.identity.3}, we obtain
\begin{align*}
&\int_{\mathbb{R}^n} \varepsilon^{n-2} \, (\varepsilon^2 + |x|^2)^{2-n}
\, \sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \\
&= \frac{1}{n(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} +
W_{ilkj})^2 \\
&\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 +
r^2)^{2-n} \, r^{n+1} \\
&\hspace{15mm} \cdot \Big [ (n+2) \, f(r^2)^2 + 4 \, r^2 \, f(r^2) \,
f'(r^2)
+ 2 \, r^4 \, f'(r^2)^2 \Big ].
\end{align*}
This proves the assertion. \\

\begin{proposition}
\label{i}
The function $F(0,\varepsilon)$ can be written in the form
\begin{align*}
F(0,\varepsilon) &= -\frac{n-2}{16n(n-1)(n+2)} \, |S^{n-1}| \,
\sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 \\
&\hspace{10mm} \cdot I(\varepsilon^2) \, \int_0^\infty (1 + r^2)^{2-n}
\, r^{n+7} \, dr,
\end{align*}
where
\begin{align*}
I(s) &= \frac{n-12}{n+6} \, \frac{n-10}{n+4} \, (n-8) \, \tau^2 \, s^2 +
10 \, \frac{n-12}{n+6} \, (n-10) \, \tau \, s^3 \\
&+ \Big ( 25 \, \frac{n-12}{n+6} \, (n+8) - 2(n-12) \, \tau \Big ) \,
s^4 + \Big ( \frac{n+8}{10} \, \tau - 10(n+12) \Big ) \, s^5 \\
&+ \frac{n+8}{n-14} \, \frac{3n+52}{2} \, s^6 - \frac{n+8}{n-14} \,
\frac{n+10}{n-16} \, \frac{n+24}{10} \, s^7 \\
&+ \frac{n+8}{n-14} \, \frac{n+10}{n-16} \, \frac{n+12}{n-18} \,
\frac{n+32}{400} \, s^8.
\end{align*}
\end{proposition}

\textbf{Proof.} It is straightforward to check that
\begin{align*}
&(n+2) \, f(s)^2 + 4s \, f(s) \, f'(s) + 2s^2 \, f'(s)^2 \\
&= (n+2)\tau^2 + 10(n+4)\tau \, s + \Big ( 25(n+8) - 2(n+6)\tau \Big )
\, s^2 \\
&+ \Big ( \frac{n+8}{10} \, \tau - 10(n+12) \Big ) \, s^3 +
\frac{3n+52}{2} \, s^4 - \frac{n+24}{10} \, s^5
+ \frac{n+32}{400} \, s^6.
\end{align*}
This implies
\begin{align*}
&\int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{2-n} \,
r^{n+1} \\
&\hspace{10mm} \cdot \Big [ (n+2) \, f(r^2)^2 + 4 \, r^2 \, f(r^2) \,
f'(r^2)
+ 2r^4 \, f'(r^2)^2 \Big ] \, dr \\
&= (n+2)\tau^2 \, \varepsilon^4 \int_0^\infty (1+r^2)^{2-n} \, r^{n+1}
\, dr \\
&+ 10(n+4)\tau \, \varepsilon^6 \int_0^\infty (1+r^2)^{2-n} \, r^{n+3}
\, dr \\
&+ \Big ( 25(n+8) - 2(n+6)\tau \Big ) \, \varepsilon^8 \int_0^\infty
(1+r^2)^{2-n} \, r^{n+5} \, dr \\
&+ \Big ( \frac{n+8}{10} \, \tau - 10(n+12) \Big ) \, \varepsilon^{10}
\int_0^\infty (1+r^2)^{2-n} \, r^{n+7} \, dr \\
&+ \frac{3n+52}{2} \, \varepsilon^{12} \int_0^\infty (1+r^2)^{2-n} \,
r^{n+9} \, dr \\
&- \frac{n+24}{10} \, \varepsilon^{14} \int_0^\infty (1+r^2)^{2-n} \,
r^{n+11} \, dr \\
&+ \frac{n+32}{400} \, \varepsilon^{16} \int_0^\infty (1+r^2)^{2-n} \,
r^{n+13} \, dr.
\end{align*}
Using the identity
\[\int_0^\infty (1+r^2)^{2-n} \, r^{\beta+2} \, dr
= \frac{\beta+1}{2n-\beta-7} \int_0^\infty (1+r^2)^{2-n}
\, r^\beta \, dr,\]
we obtain
\begin{align*}
&\int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{2-n} \,
r^{n+1} \\
&\hspace{10mm} \cdot \Big [ (n+2) \, f(r^2)^2 + 4 \, r^2 \, f(r^2) \,
f'(r^2)
+ 2r^4 \, f'(r^2)^2 \Big ] \, dr \\
&= I(\varepsilon^2) \, \int_0^\infty (1+r^2)^{2-n} \, r^{n+7} \, dr.
\end{align*}
This completes the proof. \\

In the next step, we compute the Hessian of $F$ at $(0,\varepsilon)$. \\

\begin{proposition}
\label{Hessian.of.F.1}
The second order partial derivatives of the function
$F(\xi,\varepsilon)$ are given by
\begin{align*}
\frac{\partial^2}{\partial \xi_p \, \partial \xi_q} F(0,\varepsilon)
&= \int_{\mathbb{R}^n} (n-2)^2 \, \varepsilon^{n-2} \, (\varepsilon^2 +
|x|^2)^{-n} \, \sum_{l=1}^n \overline{H}_{pl}(x) \, \overline{H}_{ql}(x) \\
&- \int_{\mathbb{R}^n} \frac{(n-2)^2}{4} \, \varepsilon^{n-2} \,
(\varepsilon^2 + |x|^2)^{-n} \, \sum_{i,k,l=1}^n (\partial_l
\overline{H}_{ik}(x))^2 \, x_p \, x_q \\
&+ \int_{\mathbb{R}^n} \frac{(n-2)^2}{8(n-1)} \, \varepsilon^{n-2} \,
(\varepsilon^2 + |x|^2)^{1-n} \, \sum_{i,k,l=1}^n (\partial_l
\overline{H}_{ik}(x))^2 \, \delta_{pq}.
\end{align*}
\end{proposition}

\textbf{Proof.}
See \cite{Brendle1}, Proposition 21. \\

\begin{proposition}
\label{Hessian.of.F.2}
The second order partial derivatives of the function
$F(\xi,\varepsilon)$ are given by
\begin{align*}
&\frac{\partial^2}{\partial \xi_p \, \partial \xi_q} F(0,\varepsilon) \\
&= -\frac{2(n-2)^2}{n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,k,l=1}^n
(W_{ipkl} + W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \\
&\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 +
r^2)^{-n} \, r^{n+5} \, \Big [ 2 \, f(r^2) \, f'(r^2) + r^2 \, f'(r^2)^2
\Big ] \, dr \\
&- \frac{(n-2)^2}{2n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n
(W_{ijkl} + W_{ilkj})^2 \, \delta_{pq} \\
&\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 +
r^2)^{-n} \, r^{n+5} \, \Big [
2 \, f(r^2) \, f'(r^2) + r^2 \, f'(r^2)^2 \Big ] \, dr \\
&+ \frac{(n-2)^2}{4n(n-1)(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n
(W_{ijkl} + W_{ilkj})^2 \, \delta_{pq} \\
&\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 +
r^2)^{1-n} \, r^{n+5} \, f'(r^2)^2 \, dr.
\end{align*}
\end{proposition}

\textbf{Proof.}
Using the identity
\begin{align*}
&\int_{\partial B_r(0)} \sum_{l=1}^n \overline{H}_{pl}(x) \,
\overline{H}_{ql}(x) \\
&= \int_{\partial B_r(0)} \sum_{i,j,k,l,m=1}^n W_{ipkl} \, W_{jqml} \,
x_i \, x_j \, x_k \, x_m \, f(|x|^2)^2 \\
&= \frac{1}{n(n+2)} \, |S^{n-1}| \\
&\hspace{10mm} \cdot \sum_{i,j,k,l,m=1}^n W_{ipkl} \, W_{jqml} \,
(\delta_{ij} \, \delta_{km} + \delta_{ik} \, \delta_{jm} + \delta_{im}
\, \delta_{jk}) \, r^{n+3} \, f(r^2)^2 \\
&= \frac{1}{2n(n+2)} \, |S^{n-1}| \, \sum_{i,k,l=1}^n (W_{ipkl} +
W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \, r^{n+3} \, f(r^2)^2,
\end{align*}
we obtain
\begin{align*}
&\int_{\mathbb{R}^n} \varepsilon^{n-2}
\, (\varepsilon^2 + |x|^2)^{-n} \, \sum_{i,k,l=1}^n \overline{H}_{pl}(x)
\, \overline{H}_{ql}(x) \\
&= \frac{1}{2n(n+2)} \, |S^{n-1}| \,
\sum_{i,k,l=1}^n (W_{ipkl} + W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \\
&\hspace{5mm} \cdot \int_0^\infty
\varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{-n} \, r^{n+3} \, f(r^2)^2
\, dr.
\end{align*}
Similarly, it follows from Proposition \ref{integral.identity.2} that
\begin{align*}
&\int_{\mathbb{R}^n} \varepsilon^{n-2} \, (\varepsilon^2 + |x|^2)^{-n}
\, \sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \, x_p \, x_q \\
&= \frac{2}{n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,k,l=1}^n (W_{ipkl} +
W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \\
&\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 +
r^2)^{-n} \, r^{n+3} \\
&\hspace{15mm} \cdot \Big [ (n+4) \, f(r^2)^2 + 8r^2 \, f(r^2) \,
f'(r^2) + 4r^4 \, f'(r^2)^2 \Big ] \, dr \\
&+ \frac{1}{n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} +
W_{ilkj})^2 \, \delta_{pq} \\
&\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 +
r^2)^{-n} \, r^{n+3} \\
&\hspace{15mm} \cdot \Big [ (n+4) \, f(r^2)^2 + 4r^2 \, f(r^2) \,
f'(r^2) + 2r^4 \, f'(r^2)^2 \Big ] \, dr.
\end{align*}
Moreover, we have
\begin{align*}
&\int_{\mathbb{R}^n} \varepsilon^{n-2} \, (\varepsilon^2 + |x|^2)^{1-n}
\, \sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \, \delta_{pq} \\
&= \frac{1}{n(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} +
W_{ilkj})^2 \, \delta_{pq} \\
&\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 +
r^2)^{1-n} \, r^{n+1} \\
&\hspace{15mm} \cdot \Big [ (n+2) \, f(r^2)^2 + 4 \, r^2 \, f(r^2)
\, f'(r^2) + 2 \, r^4 \, f'(r^2)^2 \Big ] \, dr
\end{align*}
by Corollary \ref{integral.identity.3}. A straightforward calculation yields
\begin{align*}
&(\varepsilon^2+r^2)^{1-n} \, r^{n+1} \, \big [ (n+2) \, f(r^2)^2 + 4r^2
\, f(r^2) \, f'(r^2) \big ] \\
&= 2(n-1) \, (\varepsilon^2+r^2)^{-n} \, r^{n+3} \, f(r^2)^2 +
\frac{d}{dr} \big [ (\varepsilon^2+r^2)^{1-n} \, r^{n+2} \, f(r^2)^2 \big ].
\end{align*}
This implies
\begin{align*}
&\int_{\mathbb{R}^n} \varepsilon^{n-2} \, (\varepsilon^2 + |x|^2)^{1-n}
\, \sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \, \delta_{pq} \\
&= \frac{2(n-1)}{n(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} +
W_{ilkj})^2 \, \delta_{pq} \\
&\hspace{5mm} \cdot
\int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{-n} \, r^{n+3}
\, f(r^2)^2 \, dr \\
&+ \frac{2}{n(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} +
W_{ilkj})^2 \, \delta_{pq} \\
&\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 +
r^2)^{1-n} \, r^{n+5} \, f'(r^2)^2 \, dr.
\end{align*}
Putting these facts together, the assertion follows. \\

\begin{proposition}
\label{j}
We have
\begin{align*}
&\int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{-n} \,
r^{n+5} \, \Big [ 2 \, f(r^2) \, f'(r^2) + r^2 \, f'(r^2)^2 \Big ] \, dr \\
&= J(\varepsilon^2) \, \int_0^\infty (1+r^2)^{-n} \, r^{n+9} \, dr,
\end{align*}
where
\begin{align*}
J(s) &= 10 \, \frac{n-10}{n+8} \, \frac{n-8}{n+6} \, \tau \, s^2 +
\frac{n-10}{n+8} \, (75-4\tau) \, s^3 \\
&+ \Big ( \frac{3}{10} \, \tau - 50 \Big ) \, s^4 + \frac{23}{2} \,
\frac{n+10}{n-12} \, s^5 - \frac{11}{10} \, \frac{n+10}{n-12} \,
\frac{n+12}{n-14} \, s^6 \\
&+ \frac{3}{80} \, \frac{n+10}{n-12} \, \frac{n+12}{n-14} \,
\frac{n+14}{n-16} \, s^7.
\end{align*}
\end{proposition}

\textbf{Proof.}
Note that
\begin{align*}
&2 \, f(s) \, f'(s) + s \, f'(s)^2 \\
&= 10\tau + (75-4\tau) \, s + \Big ( \frac{3}{10} \, \tau - 50 \Big ) \,
s^2 + \frac{23}{2} \, s^3
- \frac{11}{10} \, s^4 + \frac{3}{80} \, s^5.
\end{align*}
This implies
\begin{align*}
&\int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{-n} \,
r^{n+5} \, \Big [ 2 \, f(r^2) \, f'(r^2)
+ r^2 \, f'(r^2)^2 \Big ] \, dr \\
&= 10\tau \, \varepsilon^4 \, \int_0^\infty (1 + r^2)^{-n} \, r^{n+5} \,
dr \\
&+ (75-4\tau) \, \varepsilon^6 \, \int_0^\infty (1 + r^2)^{-n} \,
r^{n+7} \, dr \\
&+ \Big ( \frac{3}{10} \, \tau - 50 \Big ) \, \varepsilon^8 \,
\int_0^\infty (1 + r^2)^{-n} \, r^{n+9} \, dr \\
&+ \frac{23}{2} \, \varepsilon^{10} \, \int_0^\infty (1 + r^2)^{-n} \,
r^{n+11} \, dr \\
&- \frac{11}{10} \, \varepsilon^{12} \, \int_0^\infty (1 + r^2)^{-n} \,
r^{n+13} \, dr \\
&+ \frac{3}{80} \, \varepsilon^{14} \, \int_0^\infty (1 + r^2)^{-n} \,
r^{n+15} \, dr.
\end{align*}
Hence, the assertion follows from the identity
\[\int_0^\infty (1+r^2)^{-n} \, r^{\beta+2} \, dr
= \frac{\beta+1}{2n-\beta-3} \int_0^\infty (1+r^2)^{-n}
\, r^\beta \, dr.\]

\begin{proposition}
Assume that $25 \leq n \leq 51$. Then we can choose $\tau \in
\mathbb{R}$ such that $I'(1) = 0$, $I''(1) < 0$, and $J(1) < 0$.
\end{proposition}

\textbf{Proof.}
The condition $I'(1) = 0$ is equivalent to
\[a_n \, \tau^2 + b_n \, \tau + c_n = 0,\]
where
\begin{align*}
a_n &= 2 \, \frac{n-12}{n+6} \, \frac{n-10}{n+4} \, (n-8) \\
b_n &= 30 \, \frac{n-12}{n+6} \, (n-10) - 8(n-12) + \frac{n+8}{2} \\
c_n &= 100 \, \frac{n-12}{n+6} \, (n+8) - 50(n+12) + 3 \,
\frac{n+8}{n-14} \, (3n+52) \\ &- 7 \, \frac{n+8}{n-14} \,
\frac{n+10}{n-16} \, \frac{n+24}{10} + \frac{n+8}{n-14} \,
\frac{n+10}{n-16} \, \frac{n+12}{n-18} \, \frac{n+32}{50}.
\end{align*}
By inspection, one verifies that $49 \, a_n - 7 \, b_n + c_n < 0$ for
$25 \leq n \leq 51$. Since $a_n$ is positive, there exists a unique real
number $\tau < -7$ such that $a_n \, \tau^2 + b_n \, \tau + c_n = 0$.
Moreover, we have
\[I''(1) - I'(1) = \alpha_n \, \tau + \beta_n\]
and
\[J(1) = \gamma_n \, \tau + \delta_n,\]
where
\begin{align*}
\alpha_n &= 30 \, \frac{n-12}{n+6} \, (n-10) - 16(n-12) +
\frac{3(n+8)}{2} \\
\beta_n &= 200 \, \frac{n-12}{n+6} \, (n+8) - 150(n+12) + 12 \,
\frac{n+8}{n-14} \, (3n+52) \\ &- 35 \, \frac{n+8}{n-14} \,
\frac{n+10}{n-16} \, \frac{n+24}{10} + 3 \, \frac{n+8}{n-14} \,
\frac{n+10}{n-16} \, \frac{n+12}{n-18} \, \frac{n+32}{25} \\[2mm]
\gamma_n &= 10 \, \frac{n-10}{n+8} \, \frac{n-8}{n+6} -
\frac{4(n-10)}{n+8} + \frac{3}{10} \\
\delta_n &= 75 \, \frac{n-10}{n+8} - 50 + \frac{23}{2} \,
\frac{n+10}{n-12} - \frac{11}{10} \, \frac{n+10}{n-12} \,
\frac{n+12}{n-14} \\ &+ \frac{3}{80} \, \frac{n+10}{n-12} \,
\frac{n+12}{n-14} \, \frac{n+14}{n-16}.
\end{align*}
By inspection, one verifies that $7\alpha_n > \beta_n > 0$ and
$7\gamma_n > \delta_n > 0$ for $25 \leq n \leq 51$. This implies
$I''(1) = \alpha_n \, \tau + \beta_n < -7\alpha_n + \beta_n < 0$ and
$J(1) = \gamma_n \, \tau + \delta_n < -7\gamma_n + \delta_n < 0$. This
completes the proof. \\

\begin{corollary}
\label{strict.local.minimum}
Assume that $\tau$ is chosen such that $I'(1) = 0$, $I''(1) < 0$, and
$J(1) < 0$. Then the function $F(\xi,\varepsilon)$ has a strict local
minimum at $(0,1)$.
\end{corollary}

\textbf{Proof.}
Since $I'(1) = 0$, we have $\frac{\partial}{\partial \varepsilon} F(0,1)
= 0$. Therefore, $(0,1)$ is a critical point of the function
$F(\xi,\varepsilon)$. Since $J(1) < 0$, we have
\[\int_0^\infty (1 + r^2)^{-n} \, r^{n+5} \, \Big [ 2 \, f(r^2) \,
f'(r^2) + r^2 \, f'(r^2)^2 \Big ] \, dr < 0\]
by Proposition \ref{j}. Hence, it follows from Proposition
\ref{Hessian.of.F.2} that the matrix $\frac{\partial^2}{\partial \xi_p
\, \partial \xi_q} F(0,1)$ is positive definite. Using
Proposition \ref{i} and the inequality $I''(0) < 0$, we obtain
$\frac{\partial^2}{\partial \varepsilon^2} F(0,1) > 0$. Consequently,
the function $F(\xi,\varepsilon)$ has a strict local minimum at $(0,1)$. \\

\section{Proof of the main theorem}

\begin{proposition}
\label{perturbation.argument}
Assume that $25 \leq n \leq 51$. Moreover, let $g$ be a smooth metric on
$\mathbb{R}^n$ of the form
$g(x) = \exp(h(x))$, where $h(x)$ is a trace-free symmetric two-tensor
on $\mathbb{R}^n$ such
that $|h(x)| + |\partial h(x)| + |\partial^2 h(x)| \leq \alpha \leq
\alpha_1$ for all $x \in \mathbb{R}^n$,
$h(x) = 0$ for $|x| \geq 1$, and
\[h_{ik}(x) = \mu \, \lambda^6 \, f(\lambda^{-2} \, |x|^2) \,
H_{ik}(x)\] for $|x| \leq \rho$. If $\alpha$ and $\rho^{2-n} \, \mu^{-2}
\, \lambda^{n-18}$ are
sufficiently small, then there exists a positive function $v$ such that
\[\Delta_g v - \frac{n-2}{4(n-1)} \, R_g \, v + n(n-2) \,
v^{\frac{n+2}{n-2}} = 0,\]
\[\int_{\mathbb{R}^n} v^{\frac{2n}{n-2}} < \Big ( \frac{Y(S^n)}{4n(n-1)}
\Big )^{\frac{n}{2}},\] and
$\sup_{|x| \leq \lambda} v(x) \geq c \, \lambda^{\frac{2-n}{2}}$. Here,
$c$ is a positive constant that depends only on $n$.
\end{proposition}

\textbf{Proof.}
By Corollary \ref{strict.local.minimum}, the function
$F(\xi,\varepsilon)$ has a strict local minimum at $(0,1)$. It follows
from Proposition
\ref{formula.for.F} that $F(0,1) < 0$. Hence, we can find an open set
$\Omega' \subset \Omega$ such that $(0,1) \in \Omega'$ and
\[F(0,1) < \inf_{(\xi,\varepsilon) \in \partial \Omega'}
F(\xi,\varepsilon) < 0.\]
Using Corollary \ref{key.estimate}, we obtain
\begin{align*}
&|\mathcal{F}_g(\lambda\xi,\lambda\varepsilon) - \lambda^{16} \, \mu^2
\, F(\xi,\varepsilon)| \\
&\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^{\frac{2n}{n-2}} + C \,
\lambda^8 \, \mu \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}} +
C \, \Big ( \frac{\lambda}{\rho} \Big )^{n-2}
\end{align*}
for all $(\xi,\varepsilon) \in \Omega$. This implies
\begin{align*}
&|\lambda^{-16} \, \mu^{-2} \,
\mathcal{F}_g(\lambda\xi,\lambda\varepsilon) - F(\xi,\varepsilon)| \\
&\leq C \, \lambda^{\frac{32}{n-2}} \, \mu^{\frac{4}{n-2}} + C \,
\rho^{\frac{2-n}{2}} \, \mu^{-1} \, \lambda^{\frac{n-18}{2}} + C \,
\rho^{2-n} \, \mu^{-2} \, \lambda^{n-18}
\end{align*}
for all $(\xi,\varepsilon) \in \Omega$.
Hence, if $\rho^{2-n} \, \mu^{-2} \, \lambda^{n-18}$ is sufficiently
small, then we have
\[\mathcal{F}_g(0,\lambda) < \inf_{(\xi,\varepsilon) \in \partial
\Omega'} \mathcal{F}_g(\lambda\xi,\lambda\varepsilon) < 0.\]
Consequently, there exists a point $(\bar{\xi},\bar{\varepsilon}) \in
\Omega'$ such that
\[\mathcal{F}_g(\lambda\bar{\xi},\lambda\bar{\varepsilon}) =
\inf_{(\xi,\varepsilon) \in \Omega'}
\mathcal{F}_g(\lambda\xi,\lambda\varepsilon) < 0.\]
By Proposition \ref{reduction.to.a.finite.dimensional.problem}, the
function $v = v_{(\lambda\bar{\xi},\lambda\bar{\varepsilon})}$ is a
non-negative weak solution of the partial differential equation
\[\Delta_g v - \frac{n-2}{4(n-1)} \, R_g \, v + n(n-2) \,
v^{\frac{n+2}{n-2}} = 0.\]
Using a result of N.~Trudinger, we conclude that $v$ is smooth (see
\cite{Trudinger}, Theorem 3 on p. 271). Moreover, we have
\begin{align*}
2(n-2) \int_{\mathbb{R}^n} v^{\frac{2n}{n-2}}
&= 2(n-2) \, \Big ( \frac{Y(S^n)}{4n(n-1)} \Big )^{\frac{n}{2}} +
\mathcal{F}_g(\lambda\bar{\xi},\lambda\bar{\varepsilon}) \\
&< 2(n-2) \, \Big ( \frac{Y(S^n)}{4n(n-1)} \Big )^{\frac{n}{2}}.
\end{align*}
Finally, it follows from Proposition \ref{fixed.point.argument} that
$\|v -
u_{(\lambda\bar{\xi},\lambda\bar{\varepsilon})}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)}
\leq C \, \alpha$.
This implies
\[|B_\lambda(0)|^{\frac{n-2}{2n}} \, \sup_{|x| \leq \lambda} v(x) \geq
\|v\|_{L^{\frac{2n}{n-2}}(B_\lambda(0))} \geq
\|u_{(\lambda\bar{\xi},\lambda\bar{\varepsilon})}\|_{L^{\frac{2n}{n-2}}(B_\lambda(0))}
- C \, \alpha.\]
Hence, if $\alpha$ is sufficiently small, then we obtain
$\lambda^{\frac{n-2}{2}} \, \sup_{|x| \leq \lambda} v(x) \geq c$. \\

\begin{proposition}
Let $25 \leq n \leq 51$. Then there exists a smooth metric $g$ on
$\mathbb{R}^n$ with the following properties:
\begin{itemize}
\item[(i)] $g_{ik}(x) = \delta_{ik}$ for $|x| \geq \frac{1}{2}$
\item[(ii)] $g$ is not conformally flat
\item[(iii)] There exists a sequence of non-negative smooth functions
$v_\nu$ ($\nu \in \mathbb{N}$) such that
\[\Delta_g v_\nu - \frac{n-2}{4(n-1)} \, R_g \, v_\nu + n(n-2) \,
v_\nu^{\frac{n+2}{n-2}} = 0\]
for all $\nu \in \mathbb{N}$,
\[\int_{\mathbb{R}^n} v_\nu^{\frac{2n}{n-2}} < \Big (
\frac{Y(S^n)}{4n(n-1)} \Big )^{\frac{n}{2}}\]
for all $\nu \in \mathbb{N}$, and $\sup_{|x| \leq 1} v_\nu(x) \to
\infty$ as $\nu \to \infty$.
\end{itemize}
\end{proposition}

\textbf{Proof.}
Choose a smooth cutoff function $\eta: \mathbb{R} \to \mathbb{R}$ such
that $\eta(t) = 1$ for $t \leq 1$ and $\eta(t) = 0$ for $t \geq 2$. We
 define a trace-free symmetric two-tensor on $\mathbb{R}^n$ by
\[h_{ik}(x) = \sum_{N=N_0}^\infty \eta(4N^2 \, |x - y_N|) \, 2^{-4N} \,
f(2^{N} \, |x - y_N|^2) \, H_{ik}(x - y_N),\]
where $y_N = (\frac{1}{N},0,\hdots,0) \in \mathbb{R}^n$. It is
straightforward to verify that $h(x)$ is $C^\infty$ smooth.
Moreover, if $N_0$ is sufficiently large, then we have $h(x) = 0$ for
$|x| \geq \frac{1}{2}$ and $|h(x)| + |\partial h(x)| + |\partial^2 h(x)|
\leq \alpha$ for all $x \in \mathbb{R}^n$.
(Here, $\alpha$ is the constant that appears in Proposition
\ref{perturbation.argument}.) We now define a Riemannian metric $g$ by
$g(x) = \exp(h(x))$. The assertion is then a consequence of Proposition
\ref{perturbation.argument}. \\


\section{Science objective of the Cherenkov telescope}
\label{sec:CTScience}
The CT is a novel instrument and no previous version was flown or even built before. Therefore, the main objective for the telescope is to raise the technical readiness level and to understand the occurring backgrounds in such an instrument before utilizing it in future space missions such as POEMMA. At any given time, the science goals of the CT depend on its pointing direction.
If the instrument is pointed below the limb, there are two objectives that EUSO-SPB2 aims to achieve: i) in principle, detect upwards going EAS sourced from neutrinos from transient astrophysical events (see \ref{subsec:ToO}) and ii) measure optical background signals for the observation of Earth-skimming neutrino events.\\
The background observations are particularly crucial for future missions as currently no measurements in this wavelength range or with this time resolution have been conducted from suborbital space. The night sky background has a significant impact on the detection threshold and the subsequent event reconstruction, which indicates that a detailed measurement of its variation over time and telescope pointing direction over the spectral response of the SiPMs is crucial for the event rate estimation of future experiments. In addition, EUSO-SPB2 will be able to identify known and unknown background sources that could influence the detection of the signal of upwards going EASs for example atmospheric events. Finally, the impact of direct hits on the focal surface cased by charged particles will be studied as they produce the same signature as Cherenkov light from upwards going EASs. EUSO-SPB2 will use the bi-focal technique to identify such events correctly.

\subsection{Target of Opportunity}
\label{subsec:ToO}
The sensitivity of EUSO-SPB2 to the diffuse cosmogenic neutrino flux was investigated but due to the limited field of view and relatively short mission time, EUSO-SPB2 is not sensitive to the diffuse neutrino flux even assuming the most optimistic UHECR source evolution (corresponding to the cosmological evolution of Active Galactic Nuclei), which is already excluded by the IceCube and Auger experiments. For more details how the sensitivity was estimated, see \cite{NeutrinoSensitivity}. One possibility for EUSO-SPB2 to still observe neutrinos is to point the instrument in the direction of a transient astrophysical event (e.g. binary neutron star mergers after receiving an alert). The EUSO-SPB2 sensitivity to such events, called "target of opportunity" (ToO) events, was investigated in detail in \cite{ToO_poemma}.\\
The ToO events can be divided in two classes based on their characteristic time duration. Short bursts are defined as events which last around 1000~s while long bursts can last from days to even weeks. For the latter, the neutrino flux is averaged over the duration of the event while for the former, it is assumed that the instrument is positioned at the ideal location for the source at the beginning of the event and the flux is integrated over the duration of the event.\\
The left panel of fig. \ref{fig:ToOSensitivity} shows EUSO-SPB2's sensitivity for Binary Neutron star mergers \cite{Fang_BNNMerger} scaled for a distance of 0.8~Mpc and 3~Mpc. Possible sources that may correspond to these distances could be located in M31, and NGC 253 and M82 (star burst) respectively. The red line shows a flight duration of 30 days while the black line is for 100 days. The right panel shows the sensitivity for emissions from short gamma-ray burst assuming a moderate emission as discussed in \cite{KMMK_sGRB}. The distances chosen are 3~Mpc and 40~Mpc which correspond to the distances of GW170817 and GRB170817A respectively.
\begin{figure}
    \centering
    \includegraphics[width=.9\textwidth]{fig/SPB2_ToO_Sen.pdf}
    \caption{The EUSO-SPB2 all-flavor 90\% unified confidence level sensitivity to ToO neutrino flux per decade in energy. In blue the all-flavor fluency for different models, \cite{Fang_BNNMerger} for the long and \cite{KMMK_sGRB} for the short scenario. For the long duration events a 30 and 100 days flight is considered (red and black line respectively).}
    \label{fig:ToOSensitivity}
\end{figure}
The sensitivities shown in fig. \ref{fig:ToOSensitivity} are not taking the balloon trajectory  into account. For the long scenario the effect of the sun and moon is considered including a longer twilight period due to the balloon altitude. The short scenario does not contain these restrains. Further, although EUSO-SPB2 is sensitive to neutrinos produced in these events, the short mission duration of up to 100 days makes the probability of such an event occurring in close enough range during operations unlikely.

\subsection{Above-the-limb Extensive Air Showers}
\label{subsec:AboveLimb}
When the instrument is pointed above the limb, EUSO-SPB2 will be able to, for the first time, measure Cherenkov emission from EAS from suborbital space. The trajectory of the above-the-limb cosmic ray events is shown in panel A of fig. \ref{fig:AboveLimb}. The trajectories of these events provide enough atmosphere for a primary cosmic ray to produce an EAS which is detectable via its Cherenkov emission as long as the primary energy is larger than a few PeV. This is due to the highly forward beam nature of Cherenkov radiation. As the shower mostly develops in rarefied atmosphere, the atmospheric attenuation is minimized but the Cherenkov emission is also limited due to increased thresholds. Details of how to accurately simulate the Cherenkov signal of such showers is discussed in \cite{aboveLimb}.
\begin{figure}[ht]
    \centering
    \includegraphics[width=.9\textwidth]{fig/aboveLimb.png}
    \caption{A) Schematic trajectory of upwards going EAS initiated by cosmic rays, observable at the detector via Cherenkov emission of the shower B) The cumulative, energy dependent, expected event rate of above-the-limb cosmic rays, assuming a detection threshold of 10~photons~m$^{-2}$ns$^{-1}$ (See \cite{aboveLimb} for details). The cosmic ray fluxes are taken from \cite{Ralf_book}.}
    \label{fig:AboveLimb}
\end{figure}

The resulting event rate of the above-the-limb cosmic rays for EUSO-SPB2 is displayed in panel B of fig. \ref{fig:AboveLimb}. The cumulative event rate is more than 100 events per hour for primary energies above 1~PeV. Such a high rate makes it possible to collect a statistically relevant dataset in a relatively short observation time, thereby providing a guaranteed signal which has similar properties to those signals from air showers sourced from Earth-skimming neutrinos. Frequent observation of such events will help evaluate and refine the detection and reconstruction techniques of the experiment.
\section{Conclusion}
EUSO-SPB2 is the follow up project of EUSO-SPB1 and represents a pathfinder for the POEMMA mission and, as such, is equipped with two telescopes. One telescope is optimized for the fluorescence detection of UHECRs while the other is optimized to measure the Cherenkov emission from EAS sourced from either Earth skimming neutrinos or above-the-limb cosmic rays. EUSO-SPB2 is scheduled to fly as a NASA SPB payload from Wanaka, NZ in early 2023.

EUSO-SPB2 will measure, for the first time, EAS from above using the fluorescence technique. Extensive simulation studies have shown that the expected event rate is 0.12 UHECR events per hour with the FT and the possibility to search for ANITA event-like candidates are still under study.
For the first time, a Cherenkov Telescope based on SiPMs will be flown on a stratospheric balloon raising the technological readiness level. EUSO-SPB2 will study the background signals encountered while looking for signals from upward going EASs sourced from below the limb. Although studies have revealed that EUSO-SPB2 has very limited sensitivity towards the diffuse cosmogenic neutrino flux, being significantly less competitive than existing ground based experiments, the instrument still has the capability to detect neutrinos emitted by astrophysical events if they happen within our cosmic neighbourhood. Pointing the CT above the limb allows to record more than 100 events per hour of above-the-limb direct cosmic rays. As this signal is similar to that from potential neutrino induced showers, these events allow to evaluate the detection technique.

\footnotesize{\textit{Acknowledgment:}
This work was partially supported by Basic Science Interdisciplinary Research Projects of
RIKEN and JSPS KAKENHI Grant (22340063, 23340081, and 24244042), by
the Italian Ministry of Foreign Affairs	and International Cooperation,
by the Italian Space Agency through the ASI INFN agreements n. 2017-8-H.0 and n. 2021-8-HH.0,
by NASA award 11-APRA-0058, 16-APROBES16-0023, 17-APRA17-0066, NNX17AJ82G, NNX13AH54G, 80NSSC18K0246, 80NSSC18K0473, 80NSSC19K0626, and 80NSSC18K0464 in the USA,
by the French space agency CNES,
by the Deutsches Zentrum f\"ur Luft- und Raumfahrt,
the Helmholtz Alliance for Astroparticle Physics funded by the Initiative and Networking Fund
of the Helmholtz Association (Germany),
by Slovak Academy of Sciences MVTS JEM-EUSO, by National Science Centre in Poland grants 2017/27/B/ST9/02162 and
2020/37/B/ST9/01821,
by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC-2094-390783311,
by Mexican funding agencies PAPIIT-UNAM, CONACyT and the Mexican Space Agency (AEM),
as well as VEGA grant agency project 2/0132/17, by the MEYS project CZ.02.1.01/0.0/0.0/17\_049/0008422, and by State Space Corporation ROSCOSMOS and the Interdisciplinary Scientific and Educational School of Moscow University "Fundamental and Applied Space Research".
}

\section{Science Objective of the Fluorescence Telescope}
\label{sec:FTscience}
The primary scientific goal of the FT is to achieve the first measurement of an EAS induced by a primary cosmic ray via the fluorescence technique from suborbital space as well as the accurate reconstruction of the properties of such an event (direction and energy). Such a measurement would prove the feasibility of measuring UHECR from space using fluorescence emission--a technique proposed also for the POEMMA mission. To evaluate the capability of the FT to obtain this goal, extensive simulation work was performed. A detailed discussion of the FT performance can be found in \cite{FT-performance}.

For these simulations, CONEX \cite{CONEX} is used to simulate the shower profiles using EPOS-LHC \cite{EPOSLHC} assuming proton primaries. A total of 1.6 million showers were isotropically simulated landing on a disk with \unit[100]{km} radius. The center of the disk is the detector location projected on ground. The 100~km disk is needed to ensure that all shower geometries that deposit light inside the field of view are taken into account. The shower azimuth angle is uniformly distributed between $0^{\circ}$ and $360^{\circ}$ while the zenith angle is distributed flat in $\mbox{sin}\theta\mbox{cos}\theta$ ($0^{\circ}$-80$^\circ$). The energy of the simulated showers is sampled between $10^{17.8}$~eV and $10^{19.7}$~eV in 20 bins evenly spaced in $\mathrm{log}_{10}(E/\mathrm{eV})$. The background used in the simulation is estimated using the measurements from previous experiments appropriately scaled to the new detector characteristics. The Pierre Auger Observatory \cite{Auger_energy} energy spectrum is used to convert a trigger rate to a hourly event rate as shown in fig. \ref{fig:FTEventRate}.

\begin{figure}[h]
    \centering
    \includegraphics[width=.6\textwidth]{fig/EventRate_jeser.pdf}
    \caption{Expected event rate for the FT (See \cite{FT-performance} for more details).}
    \label{fig:FTEventRate}
\end{figure}
Integrating over the entire energy span, we expect to record $0.12\pm0.01$ events per observation hour which equates to roughly 0.6 events per night. Machine learning techniques will be used on board to identify these events and download them with the highest priority. This is necessary as a recovery of the whole data set is not guaranteed for a stratospheric balloon flight. More information is provided in \cite{FT-performance}.

Another interesting science objective for the FT concerns the search for the candidate upwards going events as reported by the Antarctic Impulse Transient Antenna (ANITA) \cite{ANITA_upwards}. The feasibility of such a search is still under investigation but preliminary simulations look promising, mostly due to the apparent strong intensity of those events.
\section{The EUSO-SPB2 Instrument and Mission}
\label{sec:MissionInstrument}
To achieve the science goals outlined earlier, EUSO-SPB2 is equipped with two telescopes: one optimized for the fluorescence detection technique (nadir pointing), and a second one optimized for the Cherenkov detection technique (pointing near the Earth limb).

The \textit{Fluorescence Telescope (FT)} camera has a modular design following the layout of previous instruments, particularly EUSO-SPB1, which flew in 2017. While EUSO-SPB1 was equipped with only one Photo Detection Modules (PDM), each consisting of 2304 individual pixels capable of single photo electron counting with a double pulse resolution of 6~ns, EUSO-SPB2 will fly 3 PDMs. Also the integration time was shortened from \unit[2.5]{$\mu$s} to \unit[1]{$\mu$s} in EUSO-SPB2 to increase the signal to noise ratio. The optical system used in EUSO-SPB1 was a 2 Fresnel lens system. To increase the light collection onto the focal surface EUSO-SPB2 design was switched to a Schmidt system consisting of 6 mirror segments and a 1~m diameter Schmidt corrector plate at the aperture providing a field of view of roughly 12$^\circ$ by 36$^\circ$. A detailed description of the FT is given in \cite{SPB2-FT}.

The \textit{Cherenkov Telescope (CT)} is a brand new instrument. To accommodate the very fast signals of the Cherenkov emission, the camera is composed of 512 Silicon Photomultipliers (SiPMs) pixels with a 10ns integration time and a spectral range between 400~nm and 800~nm.
The optical system of the CT is a Schmidt system similar to the FT but with only 4 mirror segments. The field of view of the CT is 6.4$^\circ$ in zenith and 12.8 $^\circ$ in azimuth. The mirror segments are aligned in such a way that a parallel light pulse from outside the telescope produces two spots in the camera. This bi-focal alignment allows to distinguish triggers of interest from single spot events caused by low energy cosmic rays striking the SiPM camera directly. A detailed description of the CT is given in \cite{SPB2-CT}.

EUSO-SPB2 is designed as a payload of a NASA Super Pressure Balloon, allowing a mission duration of up to 100 days at a nominal altitude of 33~km. EUSO-SPB2's planned launch is in March/April of 2023 from Wanaka, New Zealand. This launch window and location is chosen to allow the balloon to travel easterly around the globe following a stratospheric air circulation that develops twice a year in these latitudes ($\sim 45^{\circ}$S). The azimuth rotator will allow to point the solar panels during the day towards the sun for maximum battery charging time for the night-time data taking. The average observation time per night is around 5 hours over the entire flight. While the FT points nadir during the entire flight for the observation of EAS via the fluorescence technique, the CT has a pointing mechanism allowing to aim the telescope both above and below the Earth limb. Above the limb, the CT will record direct Cherenkov light from PeV cosmic rays early in the flight to verify the functionality of the instrument. Below the limb, the CT will search for signals corresponding to the upward $\tau$-induced EAS. In case that an astrophysical event alert is issued during the mission, the zenith and azimuth pointing capability of the CT will allow for a follow up search (see \ref{subsec:ToO}).
\section{Introduction}
With a record energy of more than 100~EeV, Ultra High Energy Cosmic Rays (UHECRs) are the most energetic particles known to exist. Although ground-based observatories have observed these energetic particles for decades, their source and acceleration mechanisms remain largely unknown due to their extremely low flux at Earth's surface (see \cite{UHECRoverview} and references therein).

Very High Energy (VHE) neutrinos (E>10~PeV) can also help to explain the most energetic processes in the universe as well as the evolution of astrophysical sources. Few of these VHE neutrinos have been detected so far due to their minuscule interaction cross sections, requiring very large amounts of target material to allow for observation.

A space-based experiment such as the proposed Probe for Multi Messenger Astrophysics (POEMMA) \cite{POEMMA-JCAP} could overcome the shortcomings of ground-based observation (by observing the Earth and its atmosphere from above, allowing for gargantuan increase in acceptance at the highest energies), thereby representing the next frontier in UHECR and VHE neutrino physics. Before such a detector can be built and launched, it is highly advantageous to develop pathfinder missions to raise the technological readiness and to verify the targeted detection techniques. A stratospheric balloon allows for such an investigation in a near space environment without the risk and cost of a fully realized space mission.

The Extreme Universe Space Observatory on a Super Pressure Balloon 2 (EUSO-SPB2) is the third (and most advanced) balloon mission undertaken by the EUSO collaboration and will build on the experiences of previous missions \cite{EUSOBalloon, EUSOSPB1}. The timeline of the evolution of the EUSO ballooning missions is shown in Fig. \ref{fig:EUSOTimeline}.
\begin{figure}[h!]
    \centering
    \includegraphics[width=0.75\textwidth]{fig/timeline.png}
    \caption{Evolution of the EUSO ballooning program towards a space based mission, starting from the first proof of concept of the technology with EUSO-Balloon in 2014 and updated payload in EUSO-SPB1 in 2017. EUSO-SPB2 will launch in 2023 and is the next step towards a space based mission like K-EUSO or POEMMA.}
    \label{fig:EUSOTimeline}
\end{figure}

EUSO-SPB2 has three main scientific objectives:
\begin{enumerate}
    \item Observing the first extensive air showers via the fluorescence technique from suborbital space.
    \item Observing Cherenkov light from upwards going extensive air showers initiated by cosmic rays.
    \item Measuring the background conditions for the detection of neutrino induced upwards going air showers.
    \item Searching for neutrinos from astrophysical transient events (e.g. binary neutron star mergers)
\end{enumerate}
The mission and the instrument will be detailed in section \ref{sec:MissionInstrument} before we discuss the different science objectives for the two telescopes in sections \ref{sec:FTscience} and \ref{sec:CTScience}, respectively.

\section*{Full Authors List: \Coll\ Collaboration}
\begin{sloppypar}
\scriptsize
\noindent
G.~Abdellaoui$^{ah}$,
S.~Abe$^{fq}$,
J.H.~Adams Jr.$^{pd}$,
D.~Allard$^{cb}$,
G.~Alonso$^{md}$,
L.~Anchordoqui$^{pe}$,
A.~Anzalone$^{eh,ed}$,
E.~Arnone$^{ek,el}$,
K.~Asano$^{fe}$,
R.~Attallah$^{ac}$,
H.~Attoui$^{aa}$,
M.~Ave~Pernas$^{mc}$,
M.~Bagheri$^{ph}$,
J.~Bal\'az{$^{la}$,
M.~Bakiri$^{aa}$,
D.~Barghini$^{el,ek}$,
S.~Bartocci$^{ei,ej}$,
M.~Battisti$^{ek,el}$,
J.~Bayer$^{dd}$,
B.~Beldjilali$^{ah}$,
T.~Belenguer$^{mb}$,
N.~Belkhalfa$^{aa}$,
R.~Bellotti$^{ea,eb}$,
A.A.~Belov$^{kb}$,
K.~Benmessai$^{aa}$,
M.~Bertaina$^{ek,el}$,
P.F.~Bertone$^{pf}$,
P.L.~Biermann$^{db}$,
F.~Bisconti$^{el,ek}$,
C.~Blaksley$^{ft}$,
N.~Blanc$^{oa}$,
S.~Blin-Bondil$^{ca,cb}$,
P.~Bobik$^{la}$,
M.~Bogomilov$^{ba}$,
E.~Bozzo$^{ob}$,
S.~Briz$^{pb}$,
A.~Bruno$^{eh,ed}$,
K.S.~Caballero$^{hd}$,
F.~Cafagna$^{ea}$,
G.~Cambi\'e$^{ei,ej}$,
D.~Campana$^{ef}$,
J-N.~Capdevielle$^{cb}$,
F.~Capel$^{de}$,
A.~Caramete$^{ja}$,
L.~Caramete$^{ja}$,
P.~Carlson$^{na}$,
R.~Caruso$^{ec,ed}$,
M.~Casolino$^{ft,ei}$,
C.~Cassardo$^{ek,el}$,
A.~Castellina$^{ek,em}$,
O.~Catalano$^{eh,ed}$,
A.~Cellino$^{ek,em}$,
K.~\v{C}ern\'{y}$^{bb}$,
M.~Chikawa$^{fc}$,
G.~Chiritoi$^{ja}$,
M.J.~Christl$^{pf}$,
R.~Colalillo$^{ef,eg}$,
L.~Conti$^{en,ei}$,
G.~Cotto$^{ek,el}$,
H.J.~Crawford$^{pa}$,
R.~Cremonini$^{el}$,
A.~Creusot$^{cb}$,
A.~de Castro G\'onzalez$^{pb}$,
C.~de la Taille$^{ca}$,
L.~del Peral$^{mc}$,
A.~Diaz Damian$^{cc}$,
R.~Diesing$^{pb}$,
P.~Dinaucourt$^{ca}$,
A.~Djakonow$^{ia}$,
T.~Djemil$^{ac}$,
A.~Ebersoldt$^{db}$,
T.~Ebisuzaki$^{ft}$,
L.~Eliasson$^{na}$,
J.~Eser$^{pb}$,
F.~Fenu$^{ek,el}$,
S.~Fern\'andez-Gonz\'alez$^{ma}$,
S.~Ferrarese$^{ek,el}$,
G.~Filippatos$^{pc}$,
 W.I.~Finch$^{pc}$
C.~Fornaro$^{en,ei}$,
M.~Fouka$^{ab}$,
A.~Franceschi$^{ee}$,
S.~Franchini$^{md}$,
C.~Fuglesang$^{na}$,
T.~Fujii$^{fg}$,
M.~Fukushima$^{fe}$,
P.~Galeotti$^{ek,el}$,
E.~Garc\'ia-Ortega$^{ma}$,
D.~Gardiol$^{ek,em}$,
G.K.~Garipov$^{kb}$,
E.~Gasc\'on$^{ma}$,
E.~Gazda$^{ph}$,
J.~Genci$^{lb}$,
A.~Golzio$^{ek,el}$,
C.~Gonz\'alez~Alvarado$^{mb}$,
P.~Gorodetzky$^{ft}$,
A.~Green$^{pc}$,
F.~Guarino$^{ef,eg}$,
C.~Gu\'epin$^{pl}$,
A.~Guzm\'an$^{dd}$,
Y.~Hachisu$^{ft}$,
A.~Haungs$^{db}$,
J.~Hern\'andez Carretero$^{mc}$,
L.~Hulett$^{pc}$,
D.~Ikeda$^{fe}$,
N.~Inoue$^{fn}$,
S.~Inoue$^{ft}$,
F.~Isgr\`o$^{ef,eg}$,
Y.~Itow$^{fk}$,
T.~Jammer$^{dc}$,
S.~Jeong$^{gb}$,
E.~Joven$^{me}$,
E.G.~Judd$^{pa}$,
J.~Jochum$^{dc}$,
F.~Kajino$^{ff}$,
T.~Kajino$^{fi}$,
S.~Kalli$^{af}$,
I.~Kaneko$^{ft}$,
Y.~Karadzhov$^{ba}$,
M.~Kasztelan$^{ia}$,
K.~Katahira$^{ft}$,
K.~Kawai$^{ft}$,
Y.~Kawasaki$^{ft}$,
A.~Kedadra$^{aa}$,
H.~Khales$^{aa}$,
B.A.~Khrenov$^{kb}$,
 Jeong-Sook~Kim$^{ga}$,
Soon-Wook~Kim$^{ga}$,
M.~Kleifges$^{db}$,
P.A.~Klimov$^{kb}$,
D.~Kolev$^{ba}$,
I.~Kreykenbohm$^{da}$,
J.F.~Krizmanic$^{pf,pk}$,
K.~Kr\'olik$^{ia}$,
V.~Kungel$^{pc}$,
Y.~Kurihara$^{fs}$,
A.~Kusenko$^{fr,pe}$,
E.~Kuznetsov$^{pd}$,
H.~Lahmar$^{aa}$,
F.~Lakhdari$^{ag}$,
J.~Licandro$^{me}$,
L.~L\'opez~Campano$^{ma}$,
F.~L\'opez~Mart\'inez$^{pb}$,
S.~Mackovjak$^{la}$,
M.~Mahdi$^{aa}$,
D.~Mand\'{a}t$^{bc}$,
M.~Manfrin$^{ek,el}$,
L.~Marcelli$^{ei}$,
J.L.~Marcos$^{ma}$,
W.~Marsza{\l}$^{ia}$,
Y.~Mart\'in$^{me}$,
O.~Martinez$^{hc}$,
K.~Mase$^{fa}$,
R.~Matev$^{ba}$,
J.N.~Matthews$^{pg}$,
N.~Mebarki$^{ad}$,
G.~Medina-Tanco$^{ha}$,
A.~Menshikov$^{db}$,
A.~Merino$^{ma}$,
M.~Mese$^{ef,eg}$,
J.~Meseguer$^{md}$,
S.S.~Meyer$^{pb}$,
J.~Mimouni$^{ad}$,
H.~Miyamoto$^{ek,el}$,
Y.~Mizumoto$^{fi}$,
A.~Monaco$^{ea,eb}$,
J.A.~Morales de los R\'ios$^{mc}$,
M.~Mastafa$^{pd}$,
S.~Nagataki$^{ft}$,
S.~Naitamor$^{ab}$,
T.~Napolitano$^{ee}$,
J.~M.~Nachtman$^{pi}$
A.~Neronov$^{ob,cb}$,
K.~Nomoto$^{fr}$,
T.~Nonaka$^{fe}$,
T.~Ogawa$^{ft}$,
S.~Ogio$^{fl}$,
H.~Ohmori$^{ft}$,
A.V.~Olinto$^{pb}$,
Y.~Onel$^{pi}$
G.~Osteria$^{ef}$,
A.N.~Otte$^{ph}$,
A.~Pagliaro$^{eh,ed}$,
W.~Painter$^{db}$,
M.I.~Panasyuk$^{kb}$,
B.~Panico$^{ef}$,
E.~Parizot$^{cb}$,
I.H.~Park$^{gb}$,
B.~Pastircak$^{la}$,
T.~Paul$^{pe}$,
M.~Pech$^{bb}$,
I.~P\'erez-Grande$^{md}$,
F.~Perfetto$^{ef}$,
T.~Peter$^{oc}$,
P.~Picozza$^{ei,ej,ft}$,
S.~Pindado$^{md}$,
L.W.~Piotrowski$^{ib}$,
S.~Piraino$^{dd}$,
Z.~Plebaniak$^{ek,el,ia}$,
A.~Pollini$^{oa}$,
E.M.~Popescu$^{ja}$,
R.~Prevete$^{ef,eg}$,
G.~Pr\'ev\^ot$^{cb}$,
H.~Prieto$^{mc}$,
M.~Przybylak$^{ia}$,
G.~Puehlhofer$^{dd}$,
M.~Putis$^{la}$,
P.~Reardon$^{pd}$,
M.H..~Reno$^{pi}$,
M.~Reyes$^{me}$,
M.~Ricci$^{ee}$,
M.D.~Rodr\'iguez~Fr\'ias$^{mc}$,
O.F.~Romero~Matamala$^{ph}$,
F.~Ronga$^{ee}$,
M.D.~Sabau$^{mb}$,
G.~Sacc\'a$^{ec,ed}$,
G.~S\'aez~Cano$^{mc}$,
H.~Sagawa$^{fe}$,
Z.~Sahnoune$^{ab}$,
A.~Saito$^{fg}$,
N.~Sakaki$^{ft}$,
H.~Salazar$^{hc}$,
J.C.~Sanchez~Balanzar$^{ha}$,
J.L.~S\'anchez$^{ma}$,
A.~Santangelo$^{dd}$,
A.~Sanz-Andr\'es$^{md}$,
M.~Sanz~Palomino$^{mb}$,
O.A.~Saprykin$^{kc}$,
F.~Sarazin$^{pc}$,
M.~Sato$^{fo}$,
A.~Scagliola$^{ea,eb}$,
T.~Schanz$^{dd}$,
H.~Schieler$^{db}$,
P.~Schov\'{a}nek$^{bc}$,
V.~Scotti$^{ef,eg}$,
M.~Serra$^{me}$,
S.A.~Sharakin$^{kb}$,
H.M.~Shimizu$^{fj}$,
K.~Shinozaki$^{ia}$,
T.~Shirahama$^{fn}$,
J.F.~Soriano$^{pe}$,
A.~Sotgiu$^{ei,ej}$,
I.~Stan$^{ja}$,
I.~Strharsk\'y$^{la}$,
N.~Sugiyama$^{fj}$,
D.~Supanitsky$^{ha}$,
M.~Suzuki$^{fm}$,
J.~Szabelski$^{ia}$,
N.~Tajima$^{ft}$,
T.~Tajima$^{ft}$,
Y.~Takahashi$^{fo}$,
M.~Takeda$^{fe}$,
Y.~Takizawa$^{ft}$,
M.C.~Talai$^{ac}$,
Y.~Tameda$^{fu}$,
C.~Tenzer$^{dd}$,
S.B.~Thomas$^{pg}$,
O.~Tibolla$^{he}$,
L.G.~Tkachev$^{ka}$,
T.~Tomida$^{fh}$,
N.~Tone$^{ft}$,
S.~Toscano$^{ob}$,
M.~Tra\"{i}che$^{aa}$,
Y.~Tsunesada$^{fl}$,
K.~Tsuno$^{ft}$,
S.~Turriziani$^{ft}$,
Y.~Uchihori$^{fb}$,
O.~Vaduvescu$^{me}$,
J.F.~Vald\'es-Galicia$^{ha}$,
P.~Vallania$^{ek,em}$,
L.~Valore$^{ef,eg}$,
G.~Vankova-Kirilova$^{ba}$,
T.~M.~Venters$^{pj}$,
C.~Vigorito$^{ek,el}$,
L.~Villase\~{n}or$^{hb}$,
B.~Vlcek$^{mc}$,
P.~von Ballmoos$^{cc}$,
M.~Vrabel$^{lb}$,
S.~Wada$^{ft}$,
J.~Watanabe$^{fi}$,
J.~Watts~Jr.$^{pd}$,
R.~Weigand Mu\~{n}oz$^{ma}$,
A.~Weindl$^{db}$,
L.~Wiencke$^{pc}$,
M.~Wille$^{da}$,
J.~Wilms$^{da}$,
D.~Winn$^{pm}$
T.~Yamamoto$^{ff}$,
J.~Yang$^{gb}$,
H.~Yano$^{fm}$,
I.V.~Yashin$^{kb}$,
D.~Yonetoku$^{fd}$,
S.~Yoshida$^{fa}$,
R.~Young$^{pf}$,
I.S~Zgura$^{ja}$,
M.Yu.~Zotov$^{kb}$,
A.~Zuccaro~Marchi$^{ft}$
}
\end{sloppypar}

{ \footnotesize
\noindent
$^{aa}$ Centre for Development of Advanced Technologies (CDTA), Algiers, Algeria \\
$^{ab}$ Dep. Astronomy, Centre Res. Astronomy, Astrophysics and Geophysics (CRAAG), Algiers, Algeria \\
$^{ac}$ LPR at Dept. of Physics, Faculty of Sciences, University Badji Mokhtar, Annaba, Algeria \\
$^{ad}$ Lab. of Math. and Sub-Atomic Phys. (LPMPS), Univ. Constantine I, Constantine, Algeria \\
$^{af}$ Department of Physics, Faculty of Sciences, University of M'sila, M'sila, Algeria \\
$^{ag}$ Research Unit on Optics and Photonics, UROP-CDTA, S\'etif, Algeria \\
$^{ah}$ Telecom Lab., Faculty of Technology, University Abou Bekr Belkaid, Tlemcen, Algeria \\
$^{ba}$ St. Kliment Ohridski University of Sofia, Bulgaria\\
$^{bb}$ Joint Laboratory of Optics, Faculty of Science, Palack\'{y} University, Olomouc, Czech Republic\\
$^{bc}$ Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic\\
$^{ca}$ Omega, Ecole Polytechnique, CNRS/IN2P3, Palaiseau, France\\
$^{cb}$ Universit de Paris, CNRS, AstroParticule et Cosmologie, F-75013 Paris, France\\
$^{cc}$ IRAP, Universit\'e de Toulouse, CNRS, Toulouse, France\\
$^{da}$ ECAP, University of Erlangen-Nuremberg, Germany\\
$^{db}$ Karlsruhe Institute of Technology (KIT), Germany\\
$^{dc}$ Experimental Physics Institute, Kepler Center, University of T\"ubingen, Germany\\
$^{dd}$ Institute for Astronomy and Astrophysics, Kepler Center, University of T\"ubingen, Germany\\
$^{de}$ Technical University of Munich, Munich, Germany\\
$^{ea}$ Istituto Nazionale di Fisica Nucleare - Sezione di Bari, Italy\\
$^{eb}$ Universita' degli Studi di Bari Aldo Moro and INFN - Sezione di Bari, Italy\\
$^{ec}$ Dipartimento di Fisica e Astronomia "Ettore Majorana", Universit di Catania, Italy\\
$^{ed}$ Istituto Nazionale di Fisica Nucleare - Sezione di Catania, Italy\\
$^{ee}$ Istituto Nazionale di Fisica Nucleare - Laboratori Nazionali di Frascati, Italy\\
$^{ef}$ Istituto Nazionale di Fisica Nucleare - Sezione di Napoli, Italy\\
$^{eg}$ UniversitaÕ di Napoli Federico II - Dipartimento di Fisica "Ettore Pancini", Italy\\
$^{eh}$ INAF - Istituto di Astrofisica Spaziale e Fisica Cosmica di Palermo, Italy\\
$^{ei}$ Istituto Nazionale di Fisica Nucleare - Sezione di Roma Tor Vergata, Italy\\
$^{ej}$ Universita' di Roma Tor Vergata - Dipartimento di Fisica, Roma, Italy\\
$^{ek}$ Istituto Nazionale di Fisica Nucleare - Sezione di Torino, Italy\\
$^{el}$ Dipartimento di Fisica, Universita' di Torino, Italy\\
$^{em}$ Osservatorio Astrofisico di Torino, Istituto Nazionale di Astrofisica, Italy\\
$^{en}$ Uninettuno University, Rome, Italy\\
$^{fa}$ Chiba University, Chiba, Japan\\
$^{fb}$ National Institutes for Quantum and Radiological Science and Technology (QST), Chiba, Japan\\
$^{fc}$ Kindai University, Higashi-Osaka, Japan\\
$^{fd}$ Kanazawa University, Kanazawa, Japan\\
$^{fe}$ Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Japan\\
$^{ff}$ Konan University, Kobe, Japan\\
$^{fg}$ Kyoto University, Kyoto, Japan\\
$^{fh}$ Shinshu University, Nagano, Japan \\
$^{fi}$ National Astronomical Observatory, Mitaka, Japan\\
$^{fj}$ Nagoya University, Nagoya, Japan\\
$^{fk}$ Institute for Space-Earth Environmental Research, Nagoya University, Nagoya, Japan\\
$^{fl}$ Graduate School of Science, Osaka City University, Japan\\
$^{fm}$ Institute of Space and Astronautical Science/JAXA, Sagamihara, Japan\\
$^{fn}$ Saitama University, Saitama, Japan\\
$^{fo}$ Hokkaido University, Sapporo, Japan \\
$^{fp}$ Osaka Electro-Communication University, Neyagawa, Japan\\
$^{fq}$ Nihon University Chiyoda, Tokyo, Japan\\
$^{fr}$ University of Tokyo, Tokyo, Japan\\
$^{fs}$ High Energy Accelerator Research Organization (KEK), Tsukuba, Japan\\
$^{ft}$ RIKEN, Wako, Japan\\
$^{ga}$ Korea Astronomy and Space Science Institute (KASI), Daejeon, Republic of Korea\\
$^{gb}$ Sungkyunkwan University, Seoul, Republic of Korea\\
$^{ha}$ Universidad Nacional Aut\'onoma de M\'exico (UNAM), Mexico\\
$^{hb}$ Universidad Michoacana de San Nicolas de Hidalgo (UMSNH), Morelia, Mexico\\
$^{hc}$ Benem\'{e}rita Universidad Aut\'{o}noma de Puebla (BUAP), Mexico\\
$^{hd}$ Universidad Aut\'{o}noma de Chiapas (UNACH), Chiapas, Mexico \\
$^{he}$ Centro Mesoamericano de F\'{i}sica Te\'{o}rica (MCTP), Mexico \\
$^{ia}$ National Centre for Nuclear Research, Lodz, Poland\\
$^{ib}$ Faculty of Physics, University of Warsaw, Poland\\
$^{ja}$ Institute of Space Science ISS, Magurele, Romania\\
$^{ka}$ Joint Institute for Nuclear Research, Dubna, Russia\\
$^{kb}$ Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Russia\\
$^{kc}$ Space Regatta Consortium, Korolev, Russia\\
$^{la}$ Institute of Experimental Physics, Kosice, Slovakia\\
$^{lb}$ Technical University Kosice (TUKE), Kosice, Slovakia\\
$^{ma}$ Universidad de Le\'on (ULE), Le\'on, Spain\\
$^{mb}$ Instituto Nacional de T\'ecnica Aeroespacial (INTA), Madrid, Spain\\
$^{mc}$ Universidad de Alcal\'a (UAH), Madrid, Spain\\
$^{md}$ Universidad Polit\'ecnia de madrid (UPM), Madrid, Spain\\
$^{me}$ Instituto de Astrof\'isica de Canarias (IAC), Tenerife, Spain\\
$^{na}$ KTH Royal Institute of Technology, Stockholm, Sweden\\
$^{oa}$ Swiss Center for Electronics and Microtechnology (CSEM), Neuch\^atel, Switzerland\\
$^{ob}$ ISDC Data Centre for Astrophysics, Versoix, Switzerland\\
$^{oc}$ Institute for Atmospheric and Climate Science, ETH Z\"urich, Switzerland\\
$^{pa}$ Space Science Laboratory, University of California, Berkeley, CA, USA\\
$^{pb}$ University of Chicago, IL, USA\\
$^{pc}$ Colorado School of Mines, Golden, CO, USA\\
$^{pd}$ University of Alabama in Huntsville, Huntsville, AL; USA\\
$^{pe}$ Lehman College, City University of New York (CUNY), NY, USA\\
$^{pf}$ NASA Marshall Space Flight Center, Huntsville, AL, USA\\
$^{pg}$ University of Utah, Salt Lake City, UT, USA\\
$^{ph}$ Georgia Institute of Technology, USA\\
$^{pi}$ University of Iowa, Iowa City, IA, USA\\
$^{pj}$ NASA Goddard Space Flight Center, Greenbelt, MD, USA\\
$^{pk}$ Center for Space Science \& Technology, University of Maryland, Baltimore County, Baltimore, MD, USA\\
$^{pl}$ Department of Astronomy, University of Maryland, College Park, MD, USA\\
$^{pm}$ Fairfield University, Fairfield, CT, USA
}
\end{document}