\label{main th}

There exists a minimizing couple $(u,\ell)$ of the following problem in both variables
\begin{equation}
    \label{pb min}
    \min\left \{E_\varepsilon(u,\ell), \, u \in  H^{1}({\mathcal C}) \text{ such that } 0\leqslant u \leqslant 1 , \, u = 1 \text{ on } \partial{{\mathcal C}} \text{ and } \ell \in \mathrm{Hom}^\Lambda(\gamma_0,\gamma_1) \right\}.
\end{equation}

Moreover, the minimizer $u$ is $\alpha$-Hölder continuous for all $0<\alpha<1$ with the estimate:
\[\|u\|_{C^{0,\alpha}(\mathcal{C})} \leqslant C_\alpha \frac{1+\Lambda c_\varepsilon ^{-1}}{\varepsilon^{\alpha}}.\]

\begin{proposition}
\label{prop 1.2}
For all $u \in H^1(\C)\cap C(\overline{\Ccal})$ there exists a minimizer for the problem \eqref{pb min u fixe}.
\end{proposition}

\begin{theorem}
\label{main th}

There exists a minimizing couple $(u,\ell)$ of the following problem in both variables
\begin{equation}
    \label{pb min}
    \min\left \{E_\eps(u,\ell), \, u \in  H^{1}(\C) \text{ such that } 0\leqslant u \leqslant 1 , \, u = 1 \text{ on } \partial{\C} \text{ and } \ell \in \curve^\Lambda(\gamma_0,\gamma_1) \right\}.
\end{equation}

Moreover, the minimizer $u$ is $\alpha$-Hölder continuous for all $0<\alpha<1$ with the estimate:
\[\|u\|_{C^{0,\alpha}(\mathcal{C})} \leqslant C_\alpha \frac{1+\Lambda c_\varepsilon ^{-1}}{\varepsilon^{\alpha}}.\]

\end{theorem}

This paper is devoted to the study of the functional introduced in \cite{bonnivard2025phasefieldapproximationplateaus} to approximate a Plateau problem, which consists in finding a surface of minimal area spanning a collection of closed curves $\gamma_0,..., \gamma_n$ contained in the boundary of an open, bounded, convex set $\Ccal_0 \subset \mathbb{R}^3$.

More precisely, the competitor surfaces are defined as the images of homotopies connecting the given curves. We define below the set of admissible homotopies connecting a curve $\gamma_i$ to $\gamma_j$:
\[
\mathrm{Hom}(\gamma_i,\gamma_j) : = \{ \ell \in \mathrm{Lip}([0,1] \times \mathbb{S}^{1}, \overline{\Ccal_0}) \text{ such that } \ell(0) = \gamma_i \text{ and } \ell(1) = \gamma_{j} \}.
\]
Note that we require more than mere homotopies: we specifically consider Lipschitz homotopies. For any homotopy $\ell \in \mathrm{Hom}(\gamma_i,\gamma_j)$, we define the associated surface $\surf_\ell$ as its image:
\begin{equation}\label{Def:surface_image}
	\surf_{\ell} := \ell([0,1] \times \mathbb{S}^{1}) \subset \overline{\Ccal_0}.
\end{equation}
We emphasize that the surface $\surf_{\ell}$ is $\mathcal{H}^{2}$-rectifiable and has finite $\mathcal{H}^{2}$-measure. Moreover, for any $u\in H^1(\mathbb{R}^3)$, the trace of $u$ on $\surf_{\ell}$ is well defined, since the set of points that are not Lebesgue points of $u$ is $\mathcal{H}^2$-negligible.
However, to obtain the existence result, we need to impose additional properties on the competitors.
For $\Lambda>0$, we consider
\[
\mathrm{Hom}^\Lambda(\gamma_i,\gamma_j) : = \{ \ell \in \mathrm{Hom}(\gamma_i,\gamma_j) \text{ such that }  \;\mathrm{Lip}(\ell)\leq \Lambda \text{ and } \surf_\ell \text{ is } \Lambda-\text{upper Ahlfors regular} \ \}.\]

Let us recall the definition of upper Ahlfors regularity.
\begin{definition}
Let $E\subset \mathbb{R}^3$ and $\Lambda>0$. The set $E$ is said to be $\Lambda$-upper Ahlfors regular if for all $x\in E$ and all $r>0$ we have
\[\mathcal{H}^{2}(E\cap B(x,r)) \leqslant \Lambda \pi r^2.\]
\end{definition}

The definition of our functional is based on the notion of a generalized geodesic distance between curves, associated with a given weight function $u$, which is defined as follows.
\begin{definition}\label{Def:geoDistance}
	Let $(\delta_\varepsilon)_{\varepsilon>0}$ be a sequence of positive numbers and $u \in C(\overline{\Ccal_0})$. We define the geodesic distance between $\gamma_i$ and $\gamma_j$:
	\begin{align*}
			&d_{u}(\gamma_i,\gamma_j) := \inf\left \{  \int_{\surf_{\ell}}( u^2 + \delta_\varepsilon) d\mathcal{H}^{2} \ |\ \ell \in \mathrm{Hom}^\Lambda(\gamma_i,\gamma_j)\right \}.
	\end{align*}
\end{definition}
Note that, in \cite{bonnivard2025phasefieldapproximationplateaus}, the geodesic distance is defined for Lipschitz homotopies $\ell$. The assumptions of uniform Lipschitz regularity and uniform upper Ahlfors regularity of the images are not required there. However, in the present work, we need to impose these assumptions in order to obtain the compactness result necessary for the existence theorem.

Let $\mathcal{C}$ an smooth open bounded convex set containing the closure $\overline{\Ccal_0}$. This set $\mathcal{C}$ will serve as our domain of study. Indeed, since we rely on PDE techniques, it is convenient to work within a smooth bounded domain. We now recall the definition of the functional introduced in \cite{bonnivard2025phasefieldapproximationplateaus}:
	\begin{equation}
		\label{functionalGeneral}
		F_{\varepsilon}(u) := \varepsilon \int_{\mathcal{C}}|\nabla u|^{2} dx + \frac{1}{4\varepsilon}\int_{\mathcal{C}}(1-u)^{2}dx + \frac{1}{c_{\varepsilon}}\sum_{i=1}^{n}d_{u}(\gamma_0, \gamma_{i}).
	\end{equation}
When, the sequences of positive numbers $(\delta_\varepsilon)$ and $(c_\varepsilon)$ are assumed to converge to zero, and to satisfy that $\delta_\varepsilon/c_\varepsilon$ converges to zero as $\varepsilon \to 0$, \cite{bonnivard2025phasefieldapproximationplateaus} establishes that this functional approximate some Plateau's problem through a $\Gamma$-convergence result.

For simplicity, we will assume in the following that the prescribed boundary consists of only two curves, $\gamma_0$ and $\gamma_1$. However, the results established in the present article remain valid when the boundary contains more than two curves.

In this paper, we are specifically interested in the decoupled functional defined below.
\begin{definition}
	For $u \in H^{1}({\mathcal C})$ such that $0\leqslant u \leqslant 1$, with $u = 1$ on $\partial{{\mathcal C}}$, and $\ell \in \mathrm{Hom}^\Lambda(\gamma_0,\gamma_1)$,
	\begin{align*}
		&E_{\varepsilon}(u,\ell) := \varepsilon\int_{\mathcal{C}} |\nabla u|^{2} dx + \frac{1}{4\varepsilon}\int_{\mathcal{C}}(1-u)^{2} dx + \frac{1}{c_{\varepsilon}}\int_{\surf_{\ell}} (u^2 + \delta_\varepsilon) d\mathcal{H}^{2}.
	\end{align*}
\end{definition}

\begin{remark}
	From Definition~\ref{Def:geoDistance} of the geodesic distance between closed curves, we know that for all $u\in H^{1}(\mathcal{C})$ such that $0\leqslant u \leqslant 1$ and $u = 1$ on $\partial{\mathcal{C}}$,
		\[F_{\varepsilon}(u) = \inf\left \{ E_{\varepsilon}(u,\ell)\ ,\ell \in \mathrm{Hom}^\Lambda(\gamma_0,\gamma_1)  \right \}.\]
\end{remark}

The main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\varepsilon$
\begin{equation*}
    \inf_{(u,\ell)} E_\varepsilon(u,\ell).
\end{equation*}

More precisely, the competitor surfaces are defined as the images of homotopies connecting the given curves. We define below the set of admissible homotopies connecting a curve $\gamma_i$ to $\gamma_j$:
\[
\mathrm{Hom}(\gamma_i,\gamma_j) : = \{ \ell \in \mathrm{Lip}([0,1] \times \mathbb{S}^{1}, \overline{\Ccal_0}) \text{ such that } \ell(0) = \gamma_i \text{ and } \ell(1) = \gamma_{j} \}.
\]
Note that we require more than mere homotopies: we specifically consider Lipschitz homotopies. For any homotopy $\ell \in \mathrm{Hom}(\gamma_i,\gamma_j)$, we define the associated surface $\surf_\ell$ as its image:
\begin{equation}\label{Def:surface_image}
 \surf_{\ell} := \ell([0,1] \times \mathbb{S}^{1}) \subset \overline{\Ccal_0}.
\end{equation}
We emphasize that the surface $\surf_{\ell}$ is $\mathcal{H}^{2}$-rectifiable and has finite $\mathcal{H}^{2}$-measure. Moreover, for any $u\in H^1(\mathbb{R}^3)$, the trace of $u$ on $\surf_{\ell}$ is well defined, since the set of points that are not Lebesgue points of $u$ is $\mathcal{H}^2$-negligible.
However, to obtain the existence result, we need to impose additional properties on the competitors.
For $\Lambda>0$, we consider
\[
\mathrm{Hom}^\Lambda(\gamma_i,\gamma_j) : = \{ \ell \in \mathrm{Hom}(\gamma_i,\gamma_j) \text{ such that } \;\mathrm{Lip}(\ell)\leq \Lambda \text{ and } \surf_\ell \text{ is } \Lambda-\text{upper Ahlfors regular} \ \}.\]

The definition of our functional is based on the notion of a generalized geodesic distance between curves, associated with a given weight function $u$, which is defined as follows.
\begin{definition}\label{Def:geoDistance}
 Let $(\delta_\varepsilon)_{\varepsilon>0}$ be a sequence of positive numbers and $u \in C(\overline{\Ccal_0})$. We define the geodesic distance between $\gamma_i$ and $\gamma_j$:
 \begin{align*}
 &d_{u}(\gamma_i,\gamma_j) := \inf\left \{ \int_{\surf_{\ell}}( u^2 + \delta_\varepsilon) d\mathcal{H}^{2} \ |\ \ell \in \mathrm{Hom}^\Lambda(\gamma_i,\gamma_j)\right \}.
 \end{align*}
\end{definition}
Note that, in \cite{bonnivard2025phasefieldapproximationplateaus}, the geodesic distance is defined for Lipschitz homotopies $\ell$. The assumptions of uniform Lipschitz regularity and uniform upper Ahlfors regularity of the images are not required there. However, in the present work, we need to impose these assumptions in order to obtain the compactness result necessary for the existence theorem.

Let $\mathcal{C}$ an smooth open bounded convex set containing the closure $\overline{\Ccal_0}$. This set $\mathcal{C}$ will serve as our domain of study. Indeed, since we rely on PDE techniques, it is convenient to work within a smooth bounded domain. We now recall the definition of the functional introduced in \cite{bonnivard2025phasefieldapproximationplateaus}:
 \begin{equation}
 \label{functionalGeneral}
 F_{\varepsilon}(u) := \varepsilon \int_{\mathcal{C}}|\nabla u|^{2} dx + \frac{1}{4\varepsilon}\int_{\mathcal{C}}(1-u)^{2}dx + \frac{1}{c_{\varepsilon}}\sum_{i=1}^{n}d_{u}(\gamma_0, \gamma_{i}).
 \end{equation}
When, the sequences of positive numbers $(\delta_\varepsilon)$ and $(c_\varepsilon)$ are assumed to converge to zero, and to satisfy that $\delta_\varepsilon/c_\varepsilon$ converges to zero as $\varepsilon \to 0$, \cite{bonnivard2025phasefieldapproximationplateaus} establishes that this functional approximate some Plateau's problem through a $\Gamma$-convergence result.

In this paper, we are specifically interested in the decoupled functional defined below.
\begin{definition}
 For $u \in H^{1}({\mathcal C})$ such that $0\leqslant u \leqslant 1$, with $u = 1$ on $\partial{{\mathcal C}}$, and $\ell \in \mathrm{Hom}^\Lambda(\gamma_0,\gamma_1)$,
 \begin{align*}
 &E_{\varepsilon}(u,\ell) := \varepsilon\int_{\mathcal{C}} |\nabla u|^{2} dx + \frac{1}{4\varepsilon}\int_{\mathcal{C}}(1-u)^{2} dx + \frac{1}{c_{\varepsilon}}\int_{\surf_{\ell}} (u^2 + \delta_\varepsilon) d\mathcal{H}^{2}.
 \end{align*}
\end{definition}

\begin{remark}
 From Definition~\ref{Def:geoDistance} of the geodesic distance between closed curves, we know that for all $u\in H^{1}(\mathcal{C})$ such that $0\leqslant u \leqslant 1$ and $u = 1$ on $\partial{\mathcal{C}}$,
 \[F_{\varepsilon}(u) = \inf\left \{ E_{\varepsilon}(u,\ell)\ ,\ell \in \mathrm{Hom}^\Lambda(\gamma_0,\gamma_1) \right \}.\]
\end{remark}

The main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\varepsilon$
\begin{equation*}
 \inf_{(u,\ell)} E_\varepsilon(u,\ell).
\end{equation*}

\begin{definition}\label{Def:geoDistance}
	Let $(\delta_\eps)_{\eps>0}$ be a sequence of positive numbers and $u \in C(\overline{\Ccal_0})$. We define the geodesic distance between $\gamma_i$ and $\gamma_j$:
	\begin{align*}
			&d_{u}(\gamma_i,\gamma_j) := \inf\left \{  \int_{\surf_{\ell}}( u^2 + \delta_\eps) d\Haus \ |\ \ell \in \curve^\Lambda(\gamma_i,\gamma_j)\right \}.
	\end{align*}
\end{definition}

The main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\varepsilon$
\begin{equation*}
 \inf_{(u,\ell)} E_\varepsilon(u,\ell).
\end{equation*}

There exists a minimizing couple $(u,\ell)$ of the following problem in both variables
\begin{equation}
 \label{pb min}
 \min\left \{E_\eps(u,\ell), \, u \in H^{1}(\C) \text{ such that } 0\leqslant u \leqslant 1 , \, u = 1 \text{ on } \partial{\C} \text{ and } \ell \in \curve^\Lambda(\gamma_0,\gamma_1) \right\}.
\end{equation}

The study of this energy is motivated by the $\Gamma$-convergence result established in \cite{bonnivard2025phasefieldapproximationplateaus}, where this decoupled functional was introduced both in the proof of the $\Gamma$-convergence and for numerical applications. Recall that the phase-field approach they proposed is a generalization to Plateau’s problem of the method introduced in \cite{bonnivard2015approximation} for Steiner’s problem. A similar study of the existence of solutions for the decoupled functional in the Steiner case was carried out in \cite{bonnivard2018phase}. The present work is inspired by this paper, but there are some differences that we explain below.

We begin by studying the following minimization problem
\begin{equation}
\label{pb min l fixe}
 \inf_{u} E_\varepsilon(u,\ell), \text{ where } \ell \in \curve^\Lambda(\gamma_0,\gamma_1).
\end{equation}
The following Proposition will be proven in Section~\ref{section l fixe}.
\begin{proposition}
For any given $\ell \in \curve^\Lambda(\gamma_0, \gamma_1)$, there exists a unique minimizer $u \in H^1(\Ccal)$ of $ E_\varepsilon(\cdot,l)$ which is globally $C^{0,\alpha}$, for all $0< \alpha <1$: there exists a constant $C_\alpha>0$ such that
\[\|u \|_{C^{0,\alpha}(\mathcal{C})} \leqslant C_\alpha \frac{1+\Lambda c_\varepsilon ^{-1}}{\varepsilon^{\alpha}}.\]
\end{proposition}
The assumption that the surface $S_\ell$ is $\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity, and hence the existence of minimizers in this setting. We then apply classical results from elliptic regularity to derive Hölder estimates for the minimizer. Note that these estimates depend explicitly on $\Lambda$ and $\varepsilon$, but are independent of the specific surface $S_\ell$. This part is strongly inspired by \cite{bonnivard2018phase}.

\begin{proposition}
\label{prop holder}
Let $\ell \in~\curve^\Lambda(\gamma_0,\gamma_1)$ and let $u$ be the minimizer of $E_\varepsilon^\ell$. Then, $u$ is $\alpha$-Hölder continuous for all $0<\alpha<1$ with the estimate:
\[\|u\|_{C^{0,\alpha}(\mathcal{C})} \leqslant C_\alpha \frac{1+\Lambda c_\varepsilon ^{-1}}{\varepsilon^{\alpha}}.\]
\end{proposition}

\begin{lemma}
\label{lem 2.10}
Recall the definition of the parameter $\eta_0 = \dist(\partial \Ccal_0,\partial \Ccal)$ introduced in the previous Lemma.
Let $\ell \in \curve^\Lambda(\gamma_0, \gamma_1)$, $0< \rho < \min(1,\frac{\eta_0}{4})$ and $x_0 \in \mathcal{C}$ such that $\dist(x_0,\partial \mathcal{C}) \geqslant \frac{\eta_0}{2} $. Then, $u \in W^{1,p}(B(x_0,\rho))$ for all $3\leqslant p < \infty$ and we have the following estimate on the rescaled function $u_\rho (x) := u(x_0+\rho x)$
\begin{equation}
 \|\nabla u_\rho\|_{L^p(B_1)} \leqslant C_p\left( \frac{\rho^2}{\varepsilon^2}+1+\frac{\Lambda \rho}{c_\varepsilon \varepsilon}\right).
\end{equation}
\end{lemma}

\medskip
\emph{Step 3.} We define the rescaled function $u_\rho$ on $B_2$ by $u_\rho(x) := u(x_0+\rho x)$. Then, for all $\varphi \in C^\infty_c(B_2)$
\begin{align*}
 \int_{B_2}\nabla u_\rho \cdot \nabla\varphi \dm x &= \frac{1}{\rho}\int_{B(x_0,2\rho)}\nabla u \cdot \nabla\varphi_\rho \dm x,\\
 &= \frac{1}{\rho4\varepsilon^2}\int_{B(x_0,2\rho)}(1- u)\varphi_\rho \dm x - \frac{1}{\rho c_\varepsilon \varepsilon}B[\ell](u,\varphi_\rho),\\
 &= \frac{\rho^2}{4\varepsilon^2}\int_{B_2}(1- u_\rho)\varphi \dm x - \frac{1}{\rho c_\varepsilon \varepsilon}\langle T_\rho,\varphi \rangle.
\end{align*}
Hence, $u_\rho$ satisfies the following PDE in $\mathcal{D}'(B_2)$
\[-\Delta u_\rho = \frac{\rho^2}{4\varepsilon^2}(1- u_\rho) - \frac{1}{\rho c_\varepsilon \varepsilon} T_\rho.\]
We denote $w_\rho := u_\rho + \frac{1}{\rho c_\varepsilon \varepsilon}v_\rho \in H^1(B_2)\cap L^\infty(B_2)$. And we deduce that
\[-\Delta w_\rho = \frac{\rho^2}{4\varepsilon^2}(1- u_\rho) \text{ in } \mathcal{D}'(B_2). \]
Thus, \cite[Corollary 8.36]{gilbargelliptic} yields that $w_\rho \in C_{\mathrm{loc}}^{1,\alpha}(B_2)$ for all $\alpha >0$ and
\begin{align*}
 \|\nabla w_\rho\|_{L^\infty(B_1)} \leqslant \|w_\rho\|_{C^{1,\alpha}(B_1)} &\leqslant C\left(\| w_\rho\|_{L^\infty(B_1)} +\frac{\rho^2}{4\varepsilon^2}\|1- u_\rho\|_{L^\infty(B_1)}\right),\\
 &= C\left(\| u_\rho\|_{L^\infty(B_1)}+ \frac{1}{\rho c_\varepsilon \varepsilon}\| v_\rho\|_{L^\infty(B_1)} +\frac{\rho^2}{4\varepsilon^2}\right),\\
 &\leqslant C\left(1 + \frac{\Lambda \rho}{c_\varepsilon \varepsilon} + \frac{\rho^2}{4\varepsilon^2}\right).
\end{align*}
Finally, we can conclude that $u_\rho = w_\rho - \frac{1}{\rho c_\varepsilon \varepsilon}v_\rho \in W^{1,p}(B_1)$ with the desired estimate
\[ \|\nabla u_\rho\|_{L^p(B_1)} \leqslant C_p\left( \frac{\rho^2}{\varepsilon^2}+1+\frac{\Lambda \rho}{c_\varepsilon \varepsilon}\right).\]
This concludes the proof of this Lemma.
\end{proof}

Now let's consider the case where $\dist(x_0, \partial \mathcal{C}) > \frac{\eta_0}{2}> \varepsilon$, in particular this implies that $B(x_0,\varepsilon) \subset \C$. In this case, Lemma~\ref{lem 2.10}, applied with $\rho = \varepsilon$ and $p = \frac{3}{1-\alpha}$ yields the estimate on the rescaled function $u_\varepsilon$
 \[\|\nabla u_\varepsilon\|_{L^p(B_1)} \leqslant C_p\left(1+ \frac{\Lambda}{c_\varepsilon}\right).\]
 Since we have shown that the $L^\infty$ norm of $u$ is smaller than $1$, we deduce that the $L^p$ norm of the scaled function $u_\varepsilon$ is bounded by the volume of the domain. And we conclude by applying the Sobolev embedding Theorem (see for instance \cite[Theorem 4.12]{adams2003sobolev}),
 \begin{equation}
 \|u_\varepsilon\|_{C^{0,\alpha}(B_1)} \leqslant C_\alpha\left(1+ \frac{\Lambda}{c_\varepsilon}\right).
 \end{equation}
 Hence, scaling back leads to
 \begin{equation}
 \frac{|u(x)-u(y)|}{|x-y|^\alpha} \leqslant \frac{1}{\varepsilon^\alpha} \|u_\varepsilon\|_{C^{0,\alpha}(B_1)} \leqslant \frac{C_\alpha}{\varepsilon^\alpha}\left(1+ \frac{\Lambda}{c_\varepsilon}\right).
 \end{equation}
\end{itemize}
Finally, since $\|u\|_{L^\infty(\Ccal)} \leqslant 1$, we get the desired estimate on the Hölder norm of $u$
\[ \|u\|_{C^{0,\alpha}(\mathcal{C})} \leqslant C_\alpha \frac{1+\Lambda c_\varepsilon ^{-1}}{\epsilon^{\alpha}}.\]
This achieves the proof of the Hölder regularity of the solution $u$ of $\min E_\eps(u,\ell)$, when $\ell$ is fixed in $\curve^\Lambda$.
\end{proof}

The study of this energy is motivated by the $\Gamma$-convergence result established in \cite{bonnivard2025phasefieldapproximationplateaus}, where this decoupled functional was introduced both in the proof of the $\Gamma$-convergence and for numerical applications. Recall that the phase-field approach they proposed is a generalization to Plateau’s problem of the method introduced in \cite{bonnivard2015approximation} for Steiner’s problem. A similar study of the existence of solutions for the decoupled functional in the Steiner case was carried out in \cite{bonnivard2018phase}. The present work is inspired by this paper, but there are some differences that we explain below.

To prove Theorem~\ref{main th}, we first consider the optimization problems with either $u$ or $\ell$ fixed. As usual, establishing the existence of minimizers requires both a compactness result and the lower semi-continuity of the functional. The compactness follows from the choice of the class of competitors, so the main issue is to establish lower semi-continuity.

We begin by studying the following minimization problem
\begin{equation}
\label{pb min l fixe}
    \inf_{u} E_\varepsilon(u,\ell), \text{ where } \ell \in \mathrm{Hom}^\Lambda(\gamma_0,\gamma_1).
\end{equation}
The following Proposition will be proven in Section~\ref{section l fixe}.
\begin{proposition}
For any given $\ell \in \mathrm{Hom}^\Lambda(\gamma_0, \gamma_1)$, there exists a unique minimizer $u \in H^1(\mathcal{C})$ of $ E_\varepsilon(\cdot,l)$ which is globally $C^{0,\alpha}$, for all $0< \alpha <1$: there exists a constant $C_\alpha>0$ such that
\[\|u \|_{C^{0,\alpha}(\mathcal{C})} \leqslant C_\alpha \frac{1+\Lambda c_\varepsilon ^{-1}}{\varepsilon^{\alpha}}.\]
\end{proposition}
The assumption that the surface $S_\ell$ is $\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity, and hence the existence of minimizers in this setting. We then apply classical results from elliptic regularity to derive Hölder estimates for the minimizer. Note that these estimates depend explicitly on $\Lambda$ and $\varepsilon$, but are independent of the specific surface $S_\ell$. This part is strongly inspired by \cite{bonnivard2018phase}.

Then, in Section~\ref{section u fixe} we consider the minimization problem
\begin{equation}
\label{pb min u fixe}
    \inf_{\ell \in \mathrm{Hom}^\Lambda} E_\varepsilon(u,\ell), \text{ where } u \in H^1(\mathcal{C})\cap C(\overline{\mathcal{C}}),
\end{equation}
and prove the following Proposition.
\begin{proposition}
\label{prop 1.2}
For all $u \in H^1({\mathcal C})\cap C(\overline{\mathcal{C}})$ there exists a minimizer for the problem \eqref{pb min u fixe}.
\end{proposition}

As previously mentionned, the main difficulty in the proof of Proposition~\ref{prop 1.2} lies in establishing the lower semicontinuity of the functional, which relies on the following Lemma.

 \begin{lemma}
 \label{golab}
Let $\ell_n : [0,1] \times  \mathbb{S}^1 \to \mathbb{R}^3$ be a sequence of uniformly Lipschitz functions, which converges uniformly to a Lipschitz function $\ell  : [0,1] \times  \mathbb{S}^1 \to \mathbb{R}^3$.
Then, for all open set $A$

$$\liminf \mathcal{H}^{2} \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} S_{\ell_n}(A) \geqslant \mathcal{H}^{2} \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} S_\ell(A),$$
where $S_\ell$ (respectively $S_{\ell_n}$) denotes the image of the Lipschitz function $\ell$ (respectively $\ell_n)$.
\end{lemma}

This Lemma, proven in Section~\ref{section u fixe}, constitutes a central and original contribution of the present article and relies crucially on the specific definition of the class of surfaces under consideration. It can be viewed as a generalization of Go\l{}\k{a}b’s theorem to the two-dimensional Hausdorff measure, and its proof follows a similar strategy. In Go\l{}\k{a}b’s theorem, the first step consists in establishing the rectifiability of the limit set; in our setting, this property is immediate, since the surfaces are assumed to be images of Lipschitz functions. We then study the weak limit of the measures $(\mathcal{H}^{2} \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} S_{\ell_n})$ and, using the tangent plane of the limit surface, derive a lower bound on the upper density of the limit measure to conclude. The key point of our proof relies on a topological argument, stating that a Lipschitz image, which is uniformly close to a disk, must have at least almost the density of a disk (see the proof of Lemma~\ref{golab}).

We emphasize that the proof of this Lemma does not rely on the specific definition set $\mathbb{S}^1 \times [0,1]$; it only assumes it to be a two-dimensional subset of $\mathbb{R}^3$. Moreover, the result is also expected to remain valid in dimensions higher than two, although the proof would then require more delicate topological arguments.

Finally, in Section~\ref{section rien fixe}, Theorem~\ref{main th} is proved by combining these results, in particular the Hölder regularity and the preceding Lemma.

\textbf{Acknowledgments.}
 I would like to thank Antoine Lemenant for suggesting this problem to me as well as Matthieu Bonnivard for their help in this project.
This work was partially supported by the IUF grant of Antoine Lemenant and by the ANR project STOIQUES.

To prove Theorem~\ref{main th}, the authors “first consider the optimization problems with either $u$ or $\ell$ fixed.” Existence requires “both a compactness result and the lower semi-continuity of the functional”; “the compactness follows from the choice of the class of competitors,” so “the main issue is to establish lower semi-continuity.”

1) **Fix $\ell$**: study \eqref{pb min l fixe}. For any $\ell\in\mathrm{Hom}^\Lambda(\gamma_0,\gamma_1)$ there is a “unique minimizer $u\in H^1(\mathcal C)$” which is “globally $C^{0,\alpha}$,” with the stated estimate. Here, “the assumption that the surface $S_\ell$ is $\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity,” and then “classical results from elliptic regularity” give Hölder estimates (depending on $\Lambda$ and $\varepsilon$ but “independent of the specific surface $S_\ell$”).

2) **Fix $u$**: study \eqref{pb min u fixe}. For $u\in H^1(\mathcal C)\cap C(\overline{\mathcal C})$ there exists a minimizer in $\ell$. The “main difficulty” is again lower semicontinuity, which “relies on” Lemma~\ref{golab}, a Go\l{}\k{a}b-type lower semicontinuity statement for $\mathcal H^2\llcorner S_{\ell_n}$ under uniform convergence of uniformly Lipschitz parametrizations. Its proof “follows a similar strategy” to Go\l{}\k{a}b: rectifiability is “immediate,” then one studies “the weak limit of the measures,” and “using the tangent plane of the limit surface, derive[s] a lower bound on the upper density of the limit measure to conclude”; the “key point” is “a topological argument” that a Lipschitz image uniformly close to a disk has “at least almost the density of a disk.”

3) **Conclusion**: “Finally, in Section~\ref{section rien fixe}, Theorem~\ref{main th} is proved by combining these results, in particular the Hölder regularity and the preceding Lemma.”

To prove the main theorem, the authors “first consider the optimization problems with either $u$ or $\ell$ fixed.” Existence requires “both a compactness result and the lower semi-continuity of the functional”; “the compactness follows from the choice of the class of competitors,” so “the main issue is to establish lower semi-continuity.”

1) **Fix $\ell$**: study \eqref{pb min l fixe}. For any $\ell\in\mathrm{Hom}^\Lambda(\gamma_0,\gamma_1)$ there is a “unique minimizer $u\in H^1(\mathcal C)$” which is “globally $C^{0,\alpha}$,” with the stated estimate. Here, “the assumption that the surface $S_\ell$ is $\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity,” and then “classical results from elliptic regularity” give Hölder estimates (depending on $\Lambda$ and $\varepsilon$ but “independent of the specific surface $S_\ell$”).

2) **Fix $u$**: study
\begin{equation}
\label{pb min u fixe}
    \inf_{\ell \in \curve^\Lambda} E_\varepsilon(u,\ell), \text{ where } u \in H^1(\Ccal)\cap C(\overline{\Ccal}),
\end{equation}
For $u\in H^1(\mathcal C)\cap C(\overline{\mathcal C})$ there exists a minimizer in $\ell$. The “main difficulty” is again lower semicontinuity, which “relies on” the following lemma.
\begin{lemma}
 \label{golab}
Let $\ell_n : [0,1] \times  \mathbb{S}^1 \to \R^3$ be a sequence of uniformly Lipschitz functions, which converges uniformly to a Lipschitz function $\ell  : [0,1] \times  \mathbb{S}^1 \to \R^3$.
Then, for all open set $A$

$$\liminf \Haus \mres S_{\ell_n}(A) \geqslant \Haus \mres S_\ell(A),$$
where $S_\ell$ (respectively $S_{\ell_n}$) denotes the image of the Lipschitz function $\ell$ (respectively $\ell_n)$.
\end{lemma}
Its proof “follows a similar strategy” to Go\l{}\k{a}b: rectifiability is “immediate,” then one studies “the weak limit of the measures,” and “using the tangent plane of the limit surface, derive[s] a lower bound on the upper density of the limit measure to conclude”; the “key point” is “a topological argument” that a Lipschitz image uniformly close to a disk has “at least almost the density of a disk.”

3) **Conclusion**: “Finally, next we conclude the proof of the main theorem by combining these results, in particular the Hölder regularity and the preceding lemma.”

Question:
Fix \(\varepsilon>0\), \(\Lambda>0\), positive numbers \(c_{\varepsilon}\) and \(\delta_{\varepsilon}\), a smooth bounded convex domain \(\mathcal C\), a set \(\mathcal C_0\) with \(\overline{\mathcal C_0}\subset \mathcal C\), and closed curves \(\gamma_0,\gamma_1\). For a Lipschitz homotopy \(\ell:[0,1]\times \mathbb S^1\to \overline{\mathcal C_0}\) with \(\ell(0)=\gamma_0\) and \(\ell(1)=\gamma_1\), let \(\Sigma_{\ell}=\ell([0,1]\times \mathbb S^1)\). Define \(\mathrm{Hom}^{\Lambda}(\gamma_0,\gamma_1)\) as the set of such \(\ell\) satisfying \(\mathrm{Lip}(\ell)\le \Lambda\) and such that \(\Sigma_{\ell}\) is \(\Lambda\)-upper Ahlfors regular (i.e. \(\mathcal H^2(\Sigma_{\ell}\cap B(x,r))\le \Lambda r^2\) for all balls). For admissible \(u\in H^1(\mathcal C)\) with \(0\le u\le 1\) and \(u=1\) on \(\partial\mathcal C\), define
\[
E_{\varepsilon}(u,\ell)=\varepsilon\int_{\mathcal C}|\nabla u|^2\,dx+\frac{1}{4\varepsilon}\int_{\mathcal C}(1-u)^2\,dx+\frac{1}{c_{\varepsilon}}\int_{\Sigma_{\ell}}(u^2+\delta_{\varepsilon})\,d\mathcal H^2.
\]
Which statement holds for the minimization problem over all admissible pairs \((u,\ell)\)?

There exists at least one admissible pair \((u,\ell)\) that minimizes \(E_{\varepsilon}(u,\ell)\) jointly among all pairs with \(u\in H^1(\mathcal C)\), \(0\le u\le 1\), \(u=1\) on \(\partial\mathcal C\), and \(\ell\in \mathrm{Hom}^{\Lambda}(\gamma_0,\gamma_1)\). Moreover, for such a minimizing pair, the function \(u\) belongs to \(C^{0,\alpha}(\mathcal C)\) for every \(0<\alpha<1\), and it satisfies
\[
\|u\|_{C^{0,\alpha}(\mathcal C)}\le C_{\alpha}\,\frac{1+\Lambda c_{\varepsilon}^{-1}}{\varepsilon^{\alpha}}.
\]

Choices:
B.
There exists at least one admissible pair \((u,\ell)\) that minimizes \(E_{\varepsilon}(u,\ell)\) jointly among all pairs with \(u\in H^1(\mathcal C)\), \(0\le u\le 1\), \(u=1\) on \(\partial\mathcal C\), and \(\ell\in \mathrm{Hom}(\gamma_0,\gamma_1)\) (that is, without imposing the uniform Lipschitz bound or the \(\Lambda\)-upper Ahlfors regularity on \(\Sigma_\ell\)). Moreover, for such a minimizing pair, the function \(u\) belongs to \(C^{0,\alpha}(\mathcal C)\) for every \(0<\alpha<1\), and it satisfies
\[
\|u\|_{C^{0,\alpha}(\mathcal C)}\le C_{\alpha}\,\frac{1+\Lambda c_{\varepsilon}^{-1}}{\varepsilon^{\alpha}}.
\]
C. There exists at least one admissible pair \((u,\ell)\) that minimizes \(E_{\varepsilon}(u,\ell)\) jointly among all pairs with \(u\in H^1(\mathcal C)\), \(0\le u\le 1\), \(u=1\) on \(\partial\mathcal C\), and \(\ell\in \mathrm{Hom}^{\Lambda}(\gamma_0,\gamma_1)\).
D.
For every admissible \(u\in H^1(\mathcal C)\) with \(0\le u\le 1\) and \(u=1\) on \(\partial\mathcal C\), there exists \(\ell\in \mathrm{Hom}^{\Lambda}(\gamma_0,\gamma_1)\) minimizing \(E_{\varepsilon}(u,\ell)\) over \(\ell\); and for every admissible \(\ell\in \mathrm{Hom}^{\Lambda}(\gamma_0,\gamma_1)\), there exists a minimizer \(u\) of \(E_{\varepsilon}(u,\ell)\) over \(u\). Consequently, there exists a pair \((u,\ell)\) such that \(u\) minimizes \(E_{\varepsilon}(\,b7\,,\ell)\) and \(\ell\) minimizes \(E_{\varepsilon}(u,\,\u00b7\,)\), and this pair minimizes \(E_{\varepsilon}\) jointly over all admissible pairs. Moreover, for such a pair,
\[
\|u\|_{C^{0,\alpha}(\mathcal C)}\le C_{\alpha}\,\frac{1+\Lambda c_{\varepsilon}^{-1}}{\varepsilon^{\alpha}}
\]
for every \(0<\alpha<1\).
E.
There exists at least one admissible pair \((u,\ell)\) that minimizes \(E_{\varepsilon}(u,\ell)\) jointly among all pairs with \(u\in H^1(\mathcal C)\), \(0\le u\le 1\), \(u=1\) on \(\partial\mathcal C\), and \(\ell\in \mathrm{Hom}^{\Lambda}(\gamma_0,\gamma_1)\). Moreover, for such a minimizing pair, the function \(u\) belongs to \(C^{0,\alpha}(\mathcal C)\) for every \(0<\alpha<1\), and it satisfies
\[
\|u\|_{C^{0,\alpha}(\mathcal C)}\le C_{\alpha}\,\frac{1+\Lambda c_{\varepsilon}^{-1}}{\varepsilon}.
\]


Question:
Let \(\mathcal C\) be a smooth open bounded convex set containing \(\overline{\mathcal C_0}\), let \(\gamma_0,\gamma_1\) be given closed curves, fix \(\Lambda>0\), and for a Lipschitz homotopy \(\ell:[0,1]\times \mathbb S^1\to \overline{\mathcal C_0}\) define its associated surface by \(\Sigma_\ell:=\ell([0,1]\times \mathbb S^1)\). Define
\[
\mathrm{Hom}^\Lambda(\gamma_0,\gamma_1):=\Big\{\ell\in \mathrm{Lip}([0,1]\times \mathbb S^1,\overline{\mathcal C_0}) : \ell(0)=\gamma_0,\ \ell(1)=\gamma_1,\ \mathrm{Lip}(\ell)\le \Lambda,\ \Sigma_\ell\text{ is }\Lambda\text{-upper Ahlfors regular}\Big\}.
\]
For \(\varepsilon>0\), positive numbers \(c_\varepsilon,\delta_\varepsilon\), and \(u\in H^1(\mathcal C)\) with \(0\le u\le 1\) and \(u=1\) on \(\partial\mathcal C\), define
\[
E_\varepsilon(u,\ell):=\varepsilon\int_{\mathcal C}|\nabla u|^2\,dx+\frac1{4\varepsilon}\int_{\mathcal C}(1-u)^2\,dx+\frac1{c_\varepsilon}\int_{\Sigma_\ell}(u^2+\delta_\varepsilon)\,d\mathcal H^2,
\]
for \(\ell\in \mathrm{Hom}^\Lambda(\gamma_0,\gamma_1)\). Which existence statement holds for this minimization problem?

There exists a minimizing pair \((u,\ell)\) for
\[
\min\Big\{E_\varepsilon(u,\ell):\ u\in H^1(\mathcal C),\ 0\le u\le 1,\ u=1\text{ on }\partial\mathcal C,\ \ell\in \mathrm{Hom}^\Lambda(\gamma_0,\gamma_1)\Big\}.
\]
Moreover, the minimizing function \(u\) is \(\alpha\)-H\"older continuous on \(\mathcal C\) for every \(0<\alpha<1\), and it satisfies
\[
\|u\|_{C^{0,\alpha}(\mathcal C)}\le C_\alpha\,\frac{1+\Lambda c_\varepsilon^{-1}}{\varepsilon^\alpha}.
\]

Choices:
B.
There exists a minimizing pair \((u,\ell)\) for
\[
\min\Big\{E_\varepsilon(u,\ell):\ u\in H^1(\mathcal C),\ 0\le u\le 1,\ u=1\text{ on }\partial\mathcal C,\ \ell\in \mathrm{Hom}(\gamma_0,\gamma_1)\Big\}.
\]
Moreover, the minimizing function \(u\) is \(\alpha\)-H\"older continuous on \(\mathcal C\) for every \(0<\alpha<1\), and it satisfies
\[
\|u\|_{C^{0,\alpha}(\mathcal C)}\le C_\alpha\,\frac{1+\Lambda c_\varepsilon^{-1}}{\varepsilon^\alpha}.
\]
C.
There exists a minimizing pair \((u,\ell)\) for
\[
\min\Big\{E_\varepsilon(u,\ell):\ u\in H^1(\mathcal C),\ 0\le u\le 1,\ u=1\text{ on }\partial\mathcal C,\ \ell\in \mathrm{Hom}^\Lambda(\gamma_0,\gamma_1)\Big\}.
\]
Moreover, the minimizing function \(u\) is \(\alpha\)-H\"older continuous on \(\mathcal C\) for every \(0<\alpha<1\).
D.
For every \(\ell\in \mathrm{Hom}^\Lambda(\gamma_0,\gamma_1)\) there exists a minimizer \(u\) of \(E_\varepsilon(u,\ell)\) under the constraints \(u\in H^1(\mathcal C)\), \(0\le u\le 1\), and \(u=1\) on \(\partial\mathcal C\); and for every such \(u\in H^1(\mathcal C)\) with \(0\le u\le 1\), \(u=1\) on \(\partial\mathcal C\), there exists \(\ell\in \mathrm{Hom}^\Lambda(\gamma_0,\gamma_1)\) minimizing \(E_\varepsilon(u,\ell)\). Hence there exists a unique minimizing pair \((u,\ell)\) for the full problem, and the minimizing function satisfies
\[
\|u\|_{C^{0,\alpha}(\mathcal C)}\le C_\alpha\,\frac{1+\Lambda c_\varepsilon^{-1}}{\varepsilon^\alpha}
\]
for every \(0<\alpha<1\).
E.
There exists a minimizing pair \((u,\ell)\) for
\[
\min\Big\{E_\varepsilon(u,\ell):\ u\in H^1(\mathcal C),\ 0\le u\le 1,\ u=1\text{ on }\partial\mathcal C,\ \ell\in \mathrm{Hom}^\Lambda(\gamma_0,\gamma_1)\Big\}.
\]
Moreover, the minimizing function \(u\) is Lipschitz continuous on \(\mathcal C\), with an estimate of the form
\[
\|u\|_{C^{0,1}(\mathcal C)}\le C\,\frac{1+\Lambda c_\varepsilon^{-1}}{\varepsilon}.
\]

restricted competitor class Hom^Lambda replaced by unrestricted Hom

dropped the quantitative Hölder estimate while keeping existence and Hölder continuity

separate existence for fixed variables upgraded to uniqueness of a joint minimizer

Hölder regularity for every alpha<1 strengthened to Lipschitz regularity


\label{thm:deux-bulles}
		There exists a solution $u: (-\infty, T_0] \to \mathcal{E}$ of \eqref{har} such that
		\begin{equation}
			\label{eq:mainthm}
			\lim_{t\to -\infty}\Big\|u(t) - \Big({-}iW + W_{\kappa|t|^{-\frac{2}{N-6}}}\Big)\Big\|_\mathcal{E} = 0,
		\end{equation}
		where $\kappa$ is an explicit constant, see \eqref{eq:kappa}.

\begin{mainthm}
		\label{thm:deux-bulles}
		There exists a solution $u: (-\infty, T_0] \to \cE$ of \eqref{har} such that
		\begin{equation}
			\label{eq:mainthm}
			\lim_{t\to -\infty}\Big\|u(t) - \Big({-}iW + W_{\kappa|t|^{-\frac{2}{N-6}}}\Big)\Big\|_\cE = 0,
		\end{equation}
		where $\kappa$ is an explicit constant, see \eqref{eq:kappa}.
	\end{mainthm}

\label{sec:intro}
	\subsection{Setting of the problem}
	\label{ssec:setting}
	We consider the focusing, energy-critical Hartree equation
	\begin{equation}
		\label{har}
		i \partial_t u + \Delta u  + \left( |x|^{-4}*|u|^2\right)u =0,\quad f(u) :=  \left( |x|^{-4} * |u|^2\right)u, \quad
		(t,x)\in \mathbb{R} \times \mathbb{R}^N.
	\end{equation}
	The Hartree equation arises in the study of Boson stars and other physical phenomena, please refer to
	\cite{pi}. In chemistry, it appears as a continuous-limit model for
	mesoscopic structures; see \cite{grc}.
	For reasons that will become clear later (see Remark~\ref{rem:mod}), in this paper we assume $ N\geq 7$.

	The \emph{energy functional} associated with this equation is defined for $u_0 \in \dot H^1(\mathbb{R}^N; \mathbb{C})$ by the formula
	\begin{equation}\label{energy}
		E(u_0) := \int_{\mathbb{R}^N} \frac 12|{\nabla} u_0(x)|^2 - F(u_0(x))\mathrm{\,d} x,
	\end{equation}
	where $F(u) := \frac{1}{4} \left( |x|^{-4}*|u|^2\right)|u|^2 $. A crucial property of the solutions of \eqref{har} is that the energy $E$ is a conservation law. The differential of $E$ is $\mathrm{D} E(u_0) = -\Delta u_0 - f(u_0)$, therefore, we have the following Hamiltonian form of equation \eqref{har}:
	\begin{equation}
		\label{eq:nlsH}
		\partial_t u(t) = -i \mathrm{D} E(u(t)).
	\end{equation}

	The Cauchy problem for \eqref{har} was developed in \cite{Caz:book,
		MiaoXZ:08:LWP for Har}. That is, if $u_0 \in \dot H^1(\mathbb{R}^N)$, there
	exists a unique solution defined in a maximal interval
	$I=\left(-T_-(u), T_+(u)\right)$. The name ``energy critical'' refers to the fact that
	the scaling
	\begin{equation}\label{scaling}
		u(t,x)\rightarrow
		u_{\lambda}(t,x)=\lambda^{-\frac{N-2}{2}}u\left(\lambda^{-2} t,
		\lambda^{-1} x\right), \; \lambda>0,
	\end{equation}
	makes the equation \eqref{har} and the energy
	\eqref{energy} invariant.

	There are many results for the energy-critical Hartree equation. For the defocusing case, Miao et al., using the induction on energy argument and the localized Morawetz identity, prove the global well-posedness and scattering of the radial solution in \cite{MiaoXZ:07:F e-critical radial Har}. Subsequently,
	Miao et al. use the induction on energy argument in both the frequency space and the spatial space simultaneously and the frequency-localized interaction Morawetz estimate to remove the radial assumption in \cite{MiaoXZ:09:DF e-critical nonradial Har}.

	While for the focusing case, the dynamics behavior becomes
	more complicated. It turns out that the explicit ground state
	\begin{equation}\label{w function}
	 W(x)= c_0\left(\frac{\lambda}{\lambda^2+|x|^2}\right)^{ -\frac{N-2}{2}}
		\; \text{with}\; c_0>0, \lambda>0,
	\end{equation}
	plays an important role in the dynamical behavior of solutions for
	\eqref{har}. The functions $\mathrm e^{i\theta}W_\lambda$ are called \emph{ground states} or \emph{bubbles} (of energy). They are the only radially symmetric solutions
	of the critical elliptic problem
	\begin{equation}
		\label{eq:elliptic}
		-\Delta u - f(u) = 0.
	\end{equation}
	According to \cite{MiaoWX:dynamic gHartree, MiaoXZ:09:e-critical radial Har}, we have the variational characterization of $W$, which can be proved by combining sharp Sobolev inequality \cite{Aubin, LiebL:book,	Talenti:best constant} with sharp Hardy-Littlewood-Sobolev inequality \cite{Lieb:sharp constant for HLS, LiebL:book}. Recently, the second and third authors et al.\ give an alternative proof of the existence of the extremizer of this sharp Hardy-Littlewood-Sobolev inequality in $\mathbb{R}^N$ by using the stereographic projection and sharp Hardy-Littlewood-Sobolev inequality on the sphere  in \cite{LLTX:Nondegeneracy}.

	Miao et al.\ make use of the concentration compactness principle and the rigidity argument, which are first introduced in NLS and NLW by C.~Kenig and F.~Merle in \cite{Kenigmerle:H1 critical NLS, Kenig-merle:wave},  to prove the so-called \emph{Threshold Conjecture} by completely classifying the dynamical behavior of solutions $u(t)$ of \eqref{har} \cite{	LiMZ:e-critical Har, MiaoXZ:09:e-critical radial Har} such that $E(u(t)) < E(W)$. Later, Miao et al.\ make use of the modulation analysis and the concentration compactness rigidity argument to prove the dynamical behavior of solutions $u(t)$ of \eqref{har} such that $E(u(t)) = E(W)$ in \cite{MiaoWX:dynamic gHartree}. Solutions slightly above the ground state energy threshold were studied in \cite{LLX:dynamics ecritical Hartree}. A much stronger statement about the dynamics of solutions is the \emph{Soliton Resolution Conjecture}, which predicts that a bounded (in an appropriate sense) solution decomposes asymptotically into a sum of energy bubbles at different scales and a radiation term (a solution of the linear Hartree equation). This was proved for the radial energy-critical wave equation in dimension $N \geq 3$ in \cite{DuyKenMerle:NLW, DuyKenMerle:NLWodd, DuyJiaKenMerle:NLW, JacekAndrew:NLW}. For \eqref{har} this problem is completely open. For other dynamics results of the Hartree equation, please refer to
	\cite{CaoG, GiV00, KriLR:m-critcal Har, KriMR:mass-subcritical har, LiZh:har:class, LLTX:dynamics ecritical Hartree, MiaoWX:dynamic gHartree, MXY:defocusing, MiaoXZ:07:F e-critical radial Har, MiaoXZ:08:LWP for Har, MiaoXZ:09:m-critical Har,
			MiaoXZ:09:p-subcritical Har, MiaoXZ:10:blowup, Na99d, ZhouTao: Threshold}.

	In this paper, we always assume that the initial data are radially symmetric, which is preserved by the flow. We denote $\mathcal{E}$ the space radially symmetric functions in $\dot H^1(\mathbb{R}^N; \mathbb{C})$.

	\subsection{Main results}
	In view of the Soliton Resolution Conjecture, solutions which exhibit no dispersion in one or both time directions play a distinguished role.
	One obvious example of such solutions are the ground states $\mathrm e^{i\theta}W_\lambda$, which is a trivial result.
	In this paper, we will consider the simplest non-trivial case, namely
	we will construct a global radial solution which approaches a sum of two bubbles in the energy space. One of the bubbles develops at scale $1$, whereas the length scale of the other converges to $0$ at rate $|t|^{-\frac{2}{N-6}}$. The phases of the two bubbles form the right angle. That is

\label{sec:intro}
 \subsection{Setting of the problem}
 \label{ssec:setting}
 We consider the focusing, energy-critical Hartree equation
 \begin{equation}
 \label{har}
 i \partial_t u + \Delta u + \left( |x|^{-4}*|u|^2\right)u =0,\quad f(u) := \left( |x|^{-4} * |u|^2\right)u, \quad
 (t,x)\in \mathbb{R} \times \mathbb{R}^N.
 \end{equation}
 The Hartree equation arises in the study of Boson stars and other physical phenomena, please refer to
 \cite{pi}. In chemistry, it appears as a continuous-limit model for
 mesoscopic structures; see \cite{grc}.
 For reasons that will become clear later (see Remark~\ref{rem:mod}), in this paper we assume $ N\geq 7$.

The \emph{energy functional} associated with this equation is defined for $u_0 \in \dot H^1(\mathbb{R}^N; \mathbb{C})$ by the formula
 \begin{equation}\label{energy}
 E(u_0) := \int_{\mathbb{R}^N} \frac 12|{\nabla} u_0(x)|^2 - F(u_0(x))\mathrm{\,d} x,
 \end{equation}
 where $F(u) := \frac{1}{4} \left( |x|^{-4}*|u|^2\right)|u|^2 $. A crucial property of the solutions of \eqref{har} is that the energy $E$ is a conservation law. The differential of $E$ is $\mathrm{D} E(u_0) = -\Delta u_0 - f(u_0)$, therefore, we have the following Hamiltonian form of equation \eqref{har}:
 \begin{equation}
 \label{eq:nlsH}
 \partial_t u(t) = -i \mathrm{D} E(u(t)).
 \end{equation}

The Cauchy problem for \eqref{har} was developed in \cite{Caz:book,
 MiaoXZ:08:LWP for Har}. That is, if $u_0 \in \dot H^1(\mathbb{R}^N)$, there
 exists a unique solution defined in a maximal interval
 $I=\left(-T_-(u), T_+(u)\right)$. The name ``energy critical'' refers to the fact that
 the scaling
 \begin{equation}\label{scaling}
 u(t,x)\rightarrow
 u_{\lambda}(t,x)=\lambda^{-\frac{N-2}{2}}u\left(\lambda^{-2} t,
 \lambda^{-1} x\right), \; \lambda>0,
 \end{equation}
 makes the equation \eqref{har} and the energy
 \eqref{energy} invariant.

While for the focusing case, the dynamics behavior becomes
 more complicated. It turns out that the explicit ground state
 \begin{equation}\label{w function}
 W(x)= c_0\left(\frac{\lambda}{\lambda^2+|x|^2}\right)^{ -\frac{N-2}{2}}
 \; \text{with}\; c_0>0, \lambda>0,
 \end{equation}
 plays an important role in the dynamical behavior of solutions for
 \eqref{har}. The functions $\mathrm e^{i\theta}W_\lambda$ are called \emph{ground states} or \emph{bubbles} (of energy). They are the only radially symmetric solutions
 of the critical elliptic problem
 \begin{equation}
 \label{eq:elliptic}
 -\Delta u - f(u) = 0.
 \end{equation}
 According to \cite{MiaoWX:dynamic gHartree, MiaoXZ:09:e-critical radial Har}, we have the variational characterization of $W$, which can be proved by combining sharp Sobolev inequality \cite{Aubin, LiebL:book, Talenti:best constant} with sharp Hardy-Littlewood-Sobolev inequality \cite{Lieb:sharp constant for HLS, LiebL:book}. Recently, the second and third authors et al.\ give an alternative proof of the existence of the extremizer of this sharp Hardy-Littlewood-Sobolev inequality in $\mathbb{R}^N$ by using the stereographic projection and sharp Hardy-Littlewood-Sobolev inequality on the sphere in \cite{LLTX:Nondegeneracy}.

In this paper, we always assume that the initial data are radially symmetric, which is preserved by the flow. We denote $\mathcal{E}$ the space radially symmetric functions in $\dot H^1(\mathbb{R}^N; \mathbb{C})$.

\subsection{Main results}
 In view of the Soliton Resolution Conjecture, solutions which exhibit no dispersion in one or both time directions play a distinguished role.
 One obvious example of such solutions are the ground states $\mathrm e^{i\theta}W_\lambda$, which is a trivial result.
 In this paper, we will consider the simplest non-trivial case, namely
 we will construct a global radial solution which approaches a sum of two bubbles in the energy space. One of the bubbles develops at scale $1$, whereas the length scale of the other converges to $0$ at rate $|t|^{-\frac{2}{N-6}}$. The phases of the two bubbles form the right angle. That is

\begin{equation}\label{energy}
		E(u_0) := \int_{\bR^N} \frac 12|\grad u_0(x)|^2 - F(u_0(x))\ud x,
	\end{equation}

\begin{equation}
			\label{eq:kappa}
			\kappa := \Big(\frac{2C_1}{3(N-6)C_2}\Big)^{\frac{2}{N-6}},
		\end{equation}

\begin{equation}
		\label{har}
		i \partial_t u + \Delta u  + \left( |x|^{-4}*|u|^2\right)u =0,\quad f(u) :=  \left( |x|^{-4} * |u|^2\right)u, \quad
		(t,x)\in \bR \times \bR^N.
	\end{equation}

\begin{remark}
        \label{rem:mod}
        The proof below crucially relies on our choice of the orthogonality conditions \eqref{eq:orth}, which requires $W, \Lambda W \in \dot H^{-1}(\bR^N)$, equivalently $N \geq 7$.
    \end{remark}

\subsection{Outline of the proof}
 The overall structure is similar to the earlier work of the first author on the energy-critical NLS equations \cite{Jacek:nls}. The difference between Hartree equation here with NLS in \cite{Jacek:nls} is the nonlocal interaction, which is more complex to analyze. The main tools are the modulation analysis, bootstrap argument and topological argument.

Resuming \eqref{eq:B1-estim}, \eqref{eq:B2-estim}, \eqref{eq:B3-estim-rough} and \eqref{eq:B4-estim-rough}, we have
 \begin{equation}
 \label{eq:B-estim}
 |B_1| + |B_2| + |B_3| + |B_4| \lesssim |t|^{-\frac{N-2}{N-6}}.
 \end{equation}
 This and the form of the matrix $(m_{jk})$ directly imply $|\zeta'| + |\mu'| \lesssim |t|^{-\frac{N-2}{N-6}}$, hence \eqref{eq:mod-zeta} and \eqref{eq:mod-mu}.
 Note that the coefficients in the third and the fourth row of the matrix $(m_{jk})$ let us gain an additional factor $|t|^{-\frac{2}{N-6}}$.
 We obtain $\big|\lambda\lambda' - \|W\|_{L^2}^{-2}B_4\big| \lesssim |t|^{-\frac{N-1}{N-6}}$, which implies \eqref{eq:mod-l} thanks to \eqref{eq:B4-estim}.
 Similarly, \eqref{eq:B3-estim} yields \eqref{eq:mod-th}, which finishes the proof.
 \end{proof}
 \begin{remark}
 A computation similar to the proof of \eqref{eq:B1-estim-1} shows that $|K| \lesssim \|g\|_\cE^2 \leq |t|^{-\frac{N-1}{N-6}}$,
 so we obtain the following simple consequence of Lemma~\ref{lem:mod}:
 \begin{equation}
 \label{eq:param-all}
 |\zeta'(t)| + \Big|\frac{\mu'(t)}{\mu(t)}\Big| + |\theta'(t)| + \Big|\frac{\lambda'(t)}{\lambda(t)}\Big| \lesssim |t|^{-1},
 \end{equation}
 (for the last term, this bound is sharp).
 \end{remark}
 \subsection{Control of the stable and unstable component}
 An important step is to control
 the stable and unstable components $a_1^\pm(t) = \la \alpha_{\zeta(t), \mu(t)}^\pm, g(t)\ra$ and $a_2^\pm(t)= \la \alpha_{\theta(t), \lambda(t)}^\pm, g(t)\ra$. Recall that $\nu > 0$ is the positive eigenvalue of the linearized flow, see \eqref{eq:Y1Y2}.
 \begin{lemma}
 \label{lem:proper}
 Under assumptions of Lemma~\ref{lem:mod}, for $t \in [T, T_1]$ there holds
 \begin{align}
 \big| \dd t a_1^+(t) - \frac{\nu}{\mu(t)^2}a_1^+(t)\big| &\leq \frac{c}{\mu(t)^2}|t|^{-\frac{N}{2(N-6)}}, \label{eq:proper-1p} \\
 \big| \dd t a_1^-(t) + \frac{\nu}{\mu(t)^2}a_1^-(t)\big| &\leq \frac{c}{\mu(t)^2}|t|^{-\frac{N}{2(N-6)}}, \label{eq:proper-1m} \\
 \big| \dd t a_2^+(t) - \frac{\nu}{\lambda(t)^2}a_2^+(t)\big| &\leq \frac{c}{\lambda(t)^2}|t|^{-\frac{N}{2(N-6)}}, \label{eq:proper-2p} \\
 \big| \dd t a_2^-(t) + \frac{\nu}{\lambda(t)^2}a_2^-(t)\big| &\leq \frac{c}{\lambda(t)^2}|t|^{-\frac{N}{2(N-6)}}, \label{eq:proper-2m}
 \end{align}
 with $c \to 0$ as $|T_0| \to +\infty$.
 \end{lemma}
 \begin{proof}
 The proof is analogous to that of Lemma $3.4$ in \cite{Jacek:nls}, so we omit the proof.
 \end{proof}

\section{Bootstrap argument}
 \label{sec:boot}
 We turn to the heart of the proof, which consists in establishing bootstrap estimates.
 We consider a solution $u(t)$, decomposed according to \eqref{eq:decompose}, \eqref{eq:param-rough} and \eqref{eq:orth}.
 The initial data at time $T \leq T_0$ is chosen as follows.
 \begin{lemma}
 \label{lem:initial}
 There exists $T_0 < 0$ such that for all $T \leq T_0$ and for all $\lambda^0$, $a_1^0$, $a_2^0$
 satisfying
 \begin{equation}
 \label{eq:initial-assum}
 \big|\lambda^0 -\kappa T^{-\frac{2}{N-6}}\big| \leq \frac 12 |T|^{-\frac{5}{2(N-6)}},\qquad |a_1^0| \leq \frac 12 |T|^{-\frac{N}{2(N-6)}},\qquad |a_2^0| \leq \frac 12 |T|^{-\frac{N}{2(N-6)}},
 \end{equation}
 there exists $g^0 \in X^1$ satisfying
 \begin{gather}
 \label{eq:initial-orth}
 \la \Lambda W, g^0\ra = \la iW, g^0\ra = \la i\Lambda W_{\lambda^0}, g^0\ra = \la {-}W_{\lambda^0}, g^0\ra = 0, \\
 \label{eq:initial-unstable}
 \la \alpha_{-\frac{\pi}{2},1}^-, g^0\ra = 0,\quad \la \alpha_{-\frac{\pi}{2},1}^+, g^0\ra = a_1^0,\quad
 \la \alpha_{0,\lambda^0}^-, g^0\ra = 0,\quad \la \alpha_{0, \lambda^0}^+, g^0\ra = a_2^0, \\
 \label{eq:initial-size}
 \|g^0\|_{\cE} \lesssim |T|^{-\frac{N}{2(N-6)}}.
 \end{gather}
 This $g^0$ is continuous for the $X^1$ topology with respect to $\lambda^0$, $a_1^0$ and $a_2^0$.
 \end{lemma}
 \begin{remark}
 For the continuity, we just claim that the function $g^0$ constructed in the proof
 is continuous with respect to $\lambda^0$, $a_1^0$ and $a_2^0$.
 Clearly, $g^0$ is not uniquely determined by \eqref{eq:initial-orth}, \eqref{eq:initial-unstable} and \eqref{eq:initial-size}.
 \end{remark}
 \begin{remark}
 Condition \eqref{eq:initial-orth} is exactly \eqref{eq:orth} with $\big(\zeta, \mu, \theta, \lambda\big) = \big(-\frac{\pi}{2}, 1, 0, \lambda^0\big)$.
 Hence, if we consider the solution $u(t)$ of \eqref{har} with initial data $u(T) = -iW + W_{\lambda^0} + g^0$
 and decompose it according to \eqref{eq:decompose}, then $g(T) = g^0$ and the initial values of the modulation parameters are
 $\big(\zeta(T), \mu(T), \theta(T), \lambda(T)\big) = \big({-}\frac{\pi}{2}, 1, 0, \lambda^0\big)$.
 \end{remark}
 \begin{proof}
 The proof is same with \cite[Lemma 4.1]{Jacek:nls}, which mainly use implicit function theorem to obtain the orthogonal decomposition. Here we omit.
 \end{proof}
 In the remaining part of this section, we will analyze solutions $u(t)$ of \eqref{har}
 with the initial data $u(T) = -iW + W_{\lambda^0} + g^0$,
 where $g^0$ is given by the previous lemma.

\begin{proposition}
 \label{prop:shooting}
 Let $|T_0|$ be large enough. For all $T < T_0$ there exist $\lambda^0, a_1^0, a_2^0$ satisfying \eqref{eq:initial-assum}
 such that the solution $u(t)$ with the initial data $u(T) = -iW + W_{\lambda^0} + g^0$ exists on the time interval $[T, T_0]$
 and for $t\in[T, T_0]$ the bounds \eqref{eq:bootstrap-better-zeta}, \eqref{eq:bootstrap-better-mu}, \eqref{eq:bootstrap-better-theta}, \eqref{eq:bootstrap-better-g},
 \begin{align}
 \big|\lambda(t) - \kappa|t|^{-\frac{2}{N-6}}\big| &\leq \frac 12 |t|^{-\frac{5}{2(N-6)}}, \label{eq:bootstrap-better-lambda} \\
 |a_1^+(t)| &\leq \frac 12 |t|^{-\frac{N}{2(N-6)}}, \label{eq:bootstrap-better-a1p} \\
 |a_2^+(t)| &\leq \frac 12 |t|^{-\frac{N}{2(N-6)}} \label{eq:bootstrap-better-a2p},
 \end{align}
 hold.
 \end{proposition}
 \begin{proof}
 The proof is similarly to \cite[Proposition 4.8]{Jacek:nls}, thus we omit.
 \end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:deux-bulles}]
 Let $T_0 < 0$ be given by Proposition~\ref{prop:shooting} and let $T_0, T_1, T_2, \ldots$
 be a decreasing sequence tending to $-\infty$.
 For $n \geq 1$, let $u_n$ be the solution given by Proposition~\ref{prop:shooting}.
 Inequalities \eqref{eq:bootstrap-better-zeta}, \eqref{eq:bootstrap-better-mu}, \eqref{eq:bootstrap-better-theta},
 \eqref{eq:bootstrap-better-lambda} and \eqref{eq:bootstrap-better-g} yield
 \begin{equation}
 \label{eq:uniform}
 \Big\|u_n(t) - \Big({-}iW + W_{\kappa|t|^{-\frac{2}{N-6}}}\Big)\Big\|_\cE \lesssim |t|^{-\frac{1}{2(N-6)}},
 \end{equation}
 for all $t \in [T_n, T_0]$ and with a constant independent of $n$.
 Upon passing to a subsequence, we can assume that $u_n(T_0) \wto u_0 \in \cE$.
 Let $u$ be the solution of \eqref{har} with the initial condition $u(T_0) = u_0$.
 The weak stability Lemma~\ref{cor:weak-cont} implies that $u$ exists on the time interval $({-}\infty, T_0]$
 and for all $t \in ({-}\infty, T_0]$ there holds $u_n(t) \wto u(t)$.
 Passing to the weak limit in \eqref{eq:uniform} finishes the proof.
 \end{proof}

\begin{remark}
		More precisely, we will prove that
		\begin{equation*}
			\Big\|u(t) - \Big({-}iW + W_{\kappa|t|^{-\frac{2}{N-6}}}\Big)\Big\|_\mathcal{E} \leq C_0|t|^{-\frac{1}{2(N-6)}},
		\end{equation*}
		for some constant $C_0 > 0$.
	\end{remark}
	\begin{remark}
		We construct here \emph{pure} two-bubble, that is the solution approaches a superposition of two ground states, with no energy transformed into radiation.
		By the conservation of energy and the decoupling of the two bubbles, we necessarily have $E(u(t)) = 2E(W)$.
	\end{remark}
	\begin{remark}
		For energy-critical wave and NLS equations, similar objects were constructed by the first author in \cite{Jacek:wave} and \cite{Jacek:nls}. In the following work, we will consider dimension $N = 6$ and we expect an analogous result, with an exponential concentration rate.
	\end{remark}
	\begin{remark}
	Solutions of similar kind for the energy-critical heat equation were constructed in \cite{dPMW, SWZ}.
	\end{remark}
		\begin{remark}
		Two-bubble solutions of a different kind were recently constructed in \cite{JK:wmaptwobub} for the energy-critical wave maps equation: the solution blows up in finite time, with the scales of both bubbles converging to zero simultaneously.
	\end{remark}

	\subsection{Outline of the proof}
	The overall structure is similar to the earlier work of the first author on the energy-critical NLS equations \cite{Jacek:nls}. The difference between Hartree equation here with NLS in \cite{Jacek:nls}  is the nonlocal interaction, which is more complex to analyze. The main tools are the modulation analysis, bootstrap argument and topological argument.

	The paper is organized as follows. We study solutions of \eqref{har} close to a sum of two bubbles:
	\begin{equation}
		u(t) = \mathrm e^{i\zeta(t)}W_{\mu(t)} + \mathrm e^{i\theta(t)}W_{\lambda(t)} + g(t).
	\end{equation}
	One should think of $\zeta(t)$ as being close to $-\frac{\pi}{2}$, $\mu(t) \simeq 1$,
	$\theta(t) \sim 0$, $\lambda(t) \ll 1$ and $\|g(t)\|_\mathcal{E} \ll 1$.

	In Section \ref{sec:variational}, we first recall some useful lemmas. We also state the spectral properties  of the linearized operator $Z_{\theta, \lambda}$ around $\mathrm e^{i\theta}W_\lambda$ and coercivity of the energy near a two-bubble. These are very important to estimate $g$, which is the infinite-dimensional part. We will use the energy conservation to deal with this.

	In Section \ref{sec:mod}, we estimate the modulation parameters. For this reason, we impose the orthogonality conditions which make the terms linear in $g$ in the modulation equations disappeared. There is essentially a unique choice of such orthogonality conditions. In Lemma~\ref{lem:mod} we establish bounds on the evolution of the modulation parameters under some bootstrap assumptions. The goal is to improve these bounds, thus closing the bootstrap. In order to improve the bound on $\|g\|_\mathcal{E}$ to close the bootstrap, we establish Lemma \ref{lem:proper} to control the unstable components.

	In Section \ref{sec:boot}, we establish the bootstrap estimates and prove the main result Theorem \ref{thm:deux-bulles}. First, we use implicit function theorem to obtain the modulation decomposition. Next, combining with virial correction, which is used to improve the bound \eqref{eq:bootstrap-theta} on $\theta(t)$, we can close the bootstrap of modulation parameters. Adding the virial correction allows us to gain a small constant on the right hand side of \eqref{eq:mod-th}, which is decisive for closing the bootstrap. The linear instabilities of the flow can be handled by using a classical topological argument based on the Brouwer fixed point theorem. Finally, we prove Theorem \ref{thm:deux-bulles}.

	\subsection{Acknowledgments}
    		J. Jendrej was supported by  ERC project INSOLIT (No. 101117126).
            G. Xu was supported  by  NSFC (No. 12371240, No. 12431008).

To prove Theorem~\ref{thm:deux-bulles}, the paper follows “the overall structure” of \cite{Jacek:nls}, with the additional difficulty that “the nonlocal interaction … is more complex to analyze.” The “main tools are the modulation analysis, bootstrap argument and topological argument.”

The solution is studied “close to a sum of two bubbles” via the decomposition
\[
 u(t)=e^{i\zeta(t)}W_{\mu(t)}+e^{i\theta(t)}W_{\lambda(t)}+g(t),
\]
with \(\zeta(t)\) near \(-\pi/2\), \(\mu(t)\simeq 1\), \(\theta(t)\sim 0\), \(\lambda(t)\ll 1\), and \(\|g(t)\|_{\mathcal E}\ll 1\).

Key steps indicated in the outline:
1) (Section~\ref{sec:variational}) Recall “useful lemmas,” state “spectral properties of the linearized operator \(Z_{\theta,\lambda}\)” around \(e^{i\theta}W_\lambda\), and establish “coercivity of the energy near a two-bubble.” These are used “to estimate \(g\), which is the infinite-dimensional part,” and “we will use the energy conservation to deal with this.”

2) (Section~\ref{sec:mod}) “Estimate the modulation parameters” by imposing “orthogonality conditions which make the terms linear in \(g\) in the modulation equations disappeared,” with “essentially a unique choice” of such conditions. Under bootstrap assumptions, Lemma~\ref{lem:mod} gives “bounds on the evolution of the modulation parameters,” aiming to “improve these bounds, thus closing the bootstrap.” To improve the \(\|g\|_{\mathcal E}\) bound, Lemma~\ref{lem:proper} is established “to control the unstable components.”

3) (Section~\ref{sec:boot}) “Establish the bootstrap estimates and prove the main result Theorem~\ref{thm:deux-bulles}.” First, use the “implicit function theorem to obtain the modulation decomposition.” Then, “combining with virial correction” (used “to improve the bound … on \(\theta(t)\)”), one can “close the bootstrap of modulation parameters”; “adding the virial correction allows us to gain a small constant on the right hand side of \eqref{eq:mod-th}, which is decisive for closing the bootstrap.” The “linear instabilities of the flow” are handled via “a classical topological argument based on the Brouwer fixed point theorem.” “Finally, we prove Theorem~\ref{thm:deux-bulles}.”

To prove the main theorem, the paper follows “the overall structure” of \cite{Jacek:nls}, with the additional difficulty that “the nonlocal interaction … is more complex to analyze.” The “main tools are the modulation analysis, bootstrap argument and topological argument.”

The solution is studied “close to a sum of two bubbles” via the decomposition
\[
 u(t)=e^{i\zeta(t)}W_{\mu(t)}+e^{i\theta(t)}W_{\lambda(t)}+g(t),
\]
with \(\zeta(t)\) near \(-\pi/2\), \(\mu(t)\simeq 1\), \(\theta(t)\sim 0\), \(\lambda(t)\ll 1\), and \(\|g(t)\|_{\mathcal E}\ll 1\).

Key steps indicated in the outline:
1) (Section~\ref{sec:variational}) Recall “useful lemmas,” state “spectral properties of the linearized operator \(Z_{\theta,\lambda}\)” around \(e^{i\theta}W_\lambda\), and establish “coercivity of the energy near a two-bubble.” These are used “to estimate \(g\), which is the infinite-dimensional part,” and “we will use the energy conservation to deal with this.”

2) (Section~\ref{sec:mod}) “Estimate the modulation parameters” by imposing “orthogonality conditions which make the terms linear in \(g\) in the modulation equations disappeared,” with “essentially a unique choice” of such conditions. Under bootstrap assumptions, we first use the following lemma.
\begin{lemma}
		\label{lem:mod}
		Let $c > 0$ be an arbitrarily small constant. Let $T_0 < 0$ with $|T_0|$ large enough (depending on $c$)
		and $T < T_1 \leq T_0$. Suppose that for $T \leq t \leq T_1$ there holds
		\begin{align}
			\big|\zeta(t) + \frac{\pi}{2}\big| &\leq |t|^{-\frac{3}{N-6}}, \label{eq:bootstrap-zeta} \\
			|\mu(t) - 1| &\leq |t|^{-\frac{3}{N-6}}, \label{eq:bootstrap-mu} \\
			|\theta(t)| &\leq |t|^{-\frac{1}{N-6}}, \label{eq:bootstrap-theta} \\
			\big|\lambda(t) - \kappa|t|^{-\frac{2}{N-6}}\big| &\leq |t|^{-\frac{5}{2(N-6)}}, \label{eq:bootstrap-lambda} \\
			\|g\|_\cE &\leq |t|^{-\frac{N-1}{2(N-6)}}. \label{eq:bootstrap-g}
		\end{align}
		Then
		\begin{align}
			\label{eq:mod-zeta}
			|\zeta'(t)| &\leq c|t|^{-\frac{N-3}{N-6}}, \\
			\label{eq:mod-mu}
			|\mu'(t)| &\leq c|t|^{-\frac{N-3}{N-6}}, \\
			\label{eq:mod-l}
			\Big|\lambda'(t) - \frac{3C_2}{\|W\|_{L^2}^2} \lambda(t)^\frac{N-4}{2}\Big| &\leq c|t|^{-\frac{2N-7}{2(N-6)}}, \\
			\label{eq:mod-th}
			\Big|\theta'(t)+\frac{C_3}{\|W\|_{L^2}^2}\theta(t)\lambda(t)^\frac{N-6}{2}-\frac{K(t)}{\lambda(t)^2\|W\|_{L^2}^2}\Big| &\leq c|t|^{-\frac{N-5}{N-6}},
		\end{align}
		for $T \leq t \leq T_1$, where
		\begin{equation}
			\label{eq:kappa}
			\kappa := \Big(\frac{2C_1}{3(N-6)C_2}\Big)^{\frac{2}{N-6}},
		\end{equation}
		\begin{equation}
			\label{eq:K-def}
			K := -\big\la \eee^{i\theta}\Lambda W_\lambda, f(\eee^{i\zeta}W_{\mu} + \eee^{i\theta}W_{\lambda} + g) - f(\eee^{i\zeta}W_\mu + \eee^{i\theta}W_\lambda) - f'(\eee^{i\zeta}W_\mu + \eee^{i\theta}W_\lambda)g \big\ra.
		\end{equation}
	\end{lemma}
This gives “bounds on the evolution of the modulation parameters,” aiming to “improve these bounds, thus closing the bootstrap.” To improve the \(\|g\|_{\mathcal E}\) bound, one establishes the following auxiliary lemma “to control the unstable components.”
\begin{lemma}
		\label{lem:proper}
		Under assumptions of Lemma~\ref{lem:mod}, for $t \in [T, T_1]$ there holds
		\begin{align}
			\big| \dd t a_1^+(t) - \frac{\nu}{\mu(t)^2}a_1^+(t)\big| &\leq \frac{c}{\mu(t)^2}|t|^{-\frac{N}{2(N-6)}}, \label{eq:proper-1p} \\
			\big| \dd t a_1^-(t) + \frac{\nu}{\mu(t)^2}a_1^-(t)\big| &\leq \frac{c}{\mu(t)^2}|t|^{-\frac{N}{2(N-6)}}, \label{eq:proper-1m} \\
			\big| \dd t a_2^+(t) - \frac{\nu}{\lambda(t)^2}a_2^+(t)\big| &\leq \frac{c}{\lambda(t)^2}|t|^{-\frac{N}{2(N-6)}}, \label{eq:proper-2p} \\
			\big| \dd t a_2^-(t) + \frac{\nu}{\lambda(t)^2}a_2^-(t)\big| &\leq \frac{c}{\lambda(t)^2}|t|^{-\frac{N}{2(N-6)}}, \label{eq:proper-2m}
		\end{align}
		with $c \to 0$ as $|T_0| \to +\infty$.
	\end{lemma}

3) (Section~\ref{sec:boot}) “Establish the bootstrap estimates and prove the main result.” First, use the “implicit function theorem to obtain the modulation decomposition.” Then, “combining with virial correction” (used “to improve the bound … on \(\theta(t)\)”), one can “close the bootstrap of modulation parameters”; more precisely, “adding the virial correction allows us to gain a small constant on the right hand side of the equation
\begin{align}
			\label{eq:mod-zeta}
			|\zeta'(t)| &\leq c|t|^{-\frac{N-3}{N-6}}, \\
			\label{eq:mod-mu}
			|\mu'(t)| &\leq c|t|^{-\frac{N-3}{N-6}}, \\
			\label{eq:mod-l}
			\Big|\lambda'(t) - \frac{3C_2}{\|W\|_{L^2}^2} \lambda(t)^\frac{N-4}{2}\Big| &\leq c|t|^{-\frac{2N-7}{2(N-6)}}, \\
			\label{eq:mod-th}
			\Big|\theta'(t)+\frac{C_3}{\|W\|_{L^2}^2}\theta(t)\lambda(t)^\frac{N-6}{2}-\frac{K(t)}{\lambda(t)^2\|W\|_{L^2}^2}\Big| &\leq c|t|^{-\frac{N-5}{N-6}},
		\end{align}
which is decisive for closing the bootstrap.” The “linear instabilities of the flow” are handled via “a classical topological argument based on the Brouwer fixed point theorem.” This completes the proof of the main theorem.

\label{thm:deux-bulles}
		There exists a solution $u: (-\infty, T_0] \to \mathcal{E}$ of
		\begin{equation}
		\label{har}
		i \partial_t u + \Delta u  + \left( |x|^{-4}*|u|^2\right)u =0,\quad f(u) :=  \left( |x|^{-4} * |u|^2\right)u, \quad
		(t,x)\in \bR \times \bR^N.
	\end{equation}
		such that
		\begin{equation}
			\label{eq:mainthm}
			\lim_{t\to -\infty}\Big\|u(t) - \Big({-}iW + W_{\kappa|t|^{-\frac{2}{N-6}}}\Big)\Big\|_\mathcal{E} = 0,
		\end{equation}
		where $\kappa$ is the explicit constant
		\begin{equation}
			\label{eq:kappa}
			\kappa := \Big(\frac{2C_1}{3(N-6)C_2}\Big)^{\frac{2}{N-6}},
		\end{equation}
		.

Question:
Assume \(N\ge 7\), and let \(\mathcal E\) be the space of radial functions in \(\dot H^1(\mathbb R^N;\mathbb C)\). Consider the focusing energy-critical Hartree equation
\[
i\partial_t u+\Delta u+\big(|x|^{-4}*|u|^2\big)u=0,\qquad (t,x)\in\mathbb R\times\mathbb R^N.
\]
Let \(W\) denote a ground state (bubble), i.e. a solution of
\[
-\Delta W-\big(|x|^{-4}*|W|^2\big)W=0,
\]
and let its rescaling be \(W_\lambda(x)=\lambda^{-\frac{N-2}{2}}W(x/\lambda)\). Also set
\[
\kappa:=\Big(\frac{2C_1}{3(N-6)C_2}\Big)^{\frac{2}{N-6}}.
\]
Which existence statement holds for the Hartree flow under these assumptions?

There exists a time \(T_0\) and a solution \(u:(-\infty,T_0]\to\mathcal E\) of \(i\partial_t u+\Delta u+(|x|^{-4}*|u|^2)u=0\) such that
\[
\lim_{t\to-\infty}\Big\|u(t)-\Big(-iW+W_{\kappa |t|^{-\frac{2}{N-6}}}\Big)\Big\|_{\mathcal E}=0.
\]

Choices:
B.
There exists a time \(T_0\) and a solution \(u:(-\infty,T_0]\to\mathcal E\) of \(i\partial_t u+\Delta u+(|x|^{-4}*|u|^2)u=0\) such that
\[
\lim_{t\to-\infty}\Big\|u(t)-\Big(-iW+W_{\kappa |t|^{-\frac{2}{N-4}}}\Big)\Big\|_{\mathcal E}=0.
\]
C.
There exists a time \(T_0\) and a solution \(u:(-\infty,T_0]\to\mathcal E\) of \(i\partial_t u+\Delta u+(|x|^{-4}*|u|^2)u=0\) such that
\[
\lim_{t\to-\infty}\operatorname{dist}_{\mathcal E}\big(u(t),\{ -iW+W_\lambda:\ \lambda>0\}\big)=0.
\]
D.
There exists a time \(T_0\) and a solution \(u:(-\infty,T_0]\to\mathcal E\) of \(i\partial_t u+\Delta u+(|x|^{-4}*|u|^2)u=0\) such that
\[
\lim_{t\to-\infty}\Big\|u(t)-\Big(e^{i\alpha}W+W_{\kappa |t|^{-\frac{2}{N-6}}}\Big)\Big\|_{\mathcal E}=0
\]
for every phase \(\alpha\in\mathbb R\).
E.
There exists a time \(T_0\) and a solution \(u:(-\infty,T_0]\to\mathcal E\) of \(i\partial_t u+\Delta u+(|x|^{-4}*|u|^2)u=0\) such that
\[
\lim_{t\to-\infty}\Big\|u(t)-\Big(W+W_{\kappa |t|^{-\frac{2}{N-6}}}\Big)\Big\|_{\mathcal E}=0.
\]


\label{thm:main}
      Let $(\M_1,g_1)$ and $(\M_2,g_2)$ be two  n-dimensional smooth connected Riemannian manifolds with  boundary. Let $\omega_1\subset \M_1$ and $\omega_2\subset\M_2$ be open and connected. For $j=1,2$, let $\mathcal N_j$ be the local source-to-final value map
      \eqref{eq:def_N} with $(M,g)=(\M_j,g_j)$
      and $\omega=\omega_j$.
      In addition, suppose that there is a diffeomorphism $\bm{\Phi}:\omega_1\to \omega_2$ such that
      \begin{equation}\label{NN}
          \cN_1\bm{\Phi}^\ast=\bm{\Phi}^\ast \cN_2.
      \end{equation}
       Then $(\M_1,g_1)$ and $(\M_2,g_2)$ are isometric.

\begin{theorem}
  \label{thm:main}
      Let $(\M_1,g_1)$ and $(\M_2,g_2)$ be two  n-dimensional smooth connected Riemannian manifolds with  boundary. Let $\omega_1\subset \M_1$ and $\omega_2\subset\M_2$ be open and connected. For $j=1,2$, let $\mathcal N_j$ be the local source-to-final value map
      \eqref{eq:def_N} with $(\M,g)=(\M_j,g_j)$
      and $\omega=\omega_j$.
      In addition, suppose that there is a diffeomorphism $\bPhi:\omega_1\to \omega_2$ such that
      \begin{equation}\label{NN}
          \cN_1\bPhi^\ast=\bPhi^\ast \cN_2.
      \end{equation}
       Then $(\M_1,g_1)$ and $(\M_2,g_2)$ are isometric.
  \end{theorem}

\begin{equation}
    \label{eq:NS1}
    \begin{cases}
      \p_t\bu-\Delta_H\bu+R(\bu) + d p=\bf,\ \text{in } [0,T]\times \M,\\
      d^* \bu=0,\ \text{in } [0,T]\times \M,\\
      \bu|_{[0,T]\times\p\M}=0,\\
      \bu=0,\ p=0,\ \text{in }\{0\}\times\M.
    \end{cases}
  \end{equation}

\begin{align}
  \label{eq:def_P}
    \P_i:C_0^\infty((0,T);&C_0^\infty\Omega^1(\omega_i))\to L^2\Omega^1(\omega_i),\\
    \P_i(\bf)&=\bu_i(T,\cdot)|_{\omega_i},
  \end{align}

The Navier–Stokes equations are among the most fundamental mathematical models in fluid dynamics.
In this work, we consider the geometric Navier--Stokes equation on a Riemannian manifold $(M, g)$, see \eqref{eq:NS1} below. When $M$ is a Euclidean domain and $g = c dx^2$, with $c > 0$ a constant, then \eqref{eq:NS1} can be written as
the classical Navier-Stokes equations on $\Omega$ with no-slip boundary condition
\begin{equation}
\begin{cases}
 \p_t \mathbf{u} + \mathbf{u} \nabla \mathbf{u} - \mu \Delta \mathbf{u} + \nabla p = \mathbf{f},\text{ in }[0,T]\timesM\\
 \nabla\cdot \mathbf{u} = 0,\text{ in }[0,T]\timesM\\
 \mathbf{u}=0,\text{ on }[0,T]\times \partialM,\\
 \mathbf{u}(0,\cdot)=\bu_0.
\end{cases}
\end{equation}
Here $\mathbf{u}: [0,T]\times\Omega\to \mathbb{R}^n$ is the velocity field of the fluid, $p: [0,T]\times\Omega\to \mathbb R$ is a scalar function representing the pressure, and
$\mu = c^{-1}$ is the coefficient of kinematic viscosity.

The generalization on a Riemannian manifold enables the study of fluid dynamics in complex geometric settings, thereby opening new perspectives for both theoretical analysis and practical applications \cite{VlKh,EMa,MaRaSh,Sh}. The formulation of the Navier–Stokes equations on manifolds is motivated by several important areas, including:
1) geophysical fluid dynamics \cite{SaT,Val}, where the use of Riemannian geometry allows one to model the curvature of the Earth’s surface;
2) General Relativity and cosmology \cite{FMR}, modeling relativistic fluids, such as those present in neutron stars, black hole accretion disks, and the early universe;
3)~fluid dynamics on biological membranes and microfluidic devices~\cite{STONE}.

In all these contexts, the geometric structure of the manifold plays a central role in characterizing the physical properties of fluids and their surrounding media (e.g., density and viscosity). Motivated by these considerations, we investigate the inverse problem of identifying a Riemannian manifold from measurements of the fluid velocity field, associated with a source-to-final value map for the Navier–Stokes equations.

Inverse problems and, more broadly, identification problems related to fluid motion remain poorly understood. Only a limited number of studies have addressed this class of inverse problems, despite their strong physical motivation and mathematical significance. Among the few existing contributions devoted to inverse problems for the Stokes and Navier–Stokes equations, we may cite works concerning the determination of viscosity from boundary or internal measurements \cite{FDJN,HLiWa,ImYa,LaUWa,LiWa,Liu}, the identification of objects immersed in a fluid \cite{CaCo1,BCZ,CaCo2} as well as inverse source problems \cite{BGKN,CIPY,ILY}. To the best of our knowledge, no prior studies have addressed inverse problems for the Navier–Stokes equations beyond the Euclidean setting.

\subsection{Notations and main results}
 Let $M$ be a n-dimensional, $n\geq2$, smooth connected Riemannian manifold with boundary and $\omega\subset M$ be a nonempty open subset.
 We denote the space consisting of smooth sections of exterior $k$-form bundles on $M$ and $\omega$ by $\Omega^k(M)$ and $\Omega^k(\omega)$, respectively. Let
 \begin{equation}
 \ast:\ \Omega^k (M)\to \Omega^{n-k}(M)
 \end{equation}
 be the Hodge star operator (see e.g. \cite[section 14.1a]{frankel} for details). Then we define the codifferential operator $d^*: \Omega^k(M)\to \Omega^{k-1}(M)$ as
 \begin{equation}
 \label{eq:def_codifferential}
 d^*=(-1)^{n(k+1)+1}\ast d\ast.
 \end{equation}
 We denote the Hodge Laplacian $-dd^*-d^*d$ by $\Delta_H$.
 Using the metric $g$, for every 1-form $u=u_idx_i$ on $M$ we associate the vector field $u_*=g^{ij}u_j\partial_{x_i}\in TM$. In the same way, for every vector field $v=v_i\partial_{x_i}\in TM$ we associate the 1-form $v^*=g_{ij}v_jdx_i$. With these notations, we write $R(w)=(\nabla_{w_*}w_*)^*$,
where $\nabla$ denotes the Levi-Civita connection on $(M,g)$. Several works have been devoted to the study of the following geometric Navier-Stokes system
 \begin{equation}
 \label{eq:NS1}
 \begin{cases}
 \p_t\mathbf{u}-\Delta_H\mathbf{u}+R(\mathbf{u}) + d p=\mathbf{f},\ \text{in } [0,T]\times M,\\
 d^* \mathbf{u}=0,\ \text{in } [0,T]\times M,\\
 \mathbf{u}|_{[0,T]\times\partialM}=0,\\
 \mathbf{u}=0,\ p=0,\ \text{in }\{0\}\timesM.
 \end{cases}
 \end{equation}
 Without being exhaustive, one can refer to \cite{EMa,AA91,CP92,TW93,CRT99} for more details.

We denote by $\mathbb B_r$ the set defined by
 $$\mathbb B_r:=\{\mathbf{g} \in \cC_\alpha(M):\ \supp(\mathbf{g})\subset (0,T]\times\omega,\ \|\mathbf{g}\|_{ \cC_\alpha(M)}\leq r\},$$
where we refer to Section \ref{sec:Nonlinear_NS} for the definition of the space $\cC_\alpha(M)$ and $\norm{\cdot}_{ \cC_\alpha(M)}$. We prove in Theorem \ref{thm:NS} that
there exists $\delta>0$, depending on $(M,g)$ and $T$, such that for all $\mathbf{f}\in \mathbb B_\delta$, problem \eqref{eq:NS1} admits a solution $(\mathbf{u},p)=(\bu_\mathbf{f},p_\mathbf{f})$ with $\mathbf{u}\in \mathcal C_\alpha$. The solution is unique in the sense that $\mathbf{u}$ is unique and $p$ is unique up to a map depending only on $t\in[0,T]$. Then, we can define the local source-to-final value map
\begin{align}
\label{eq:def_N}
 \mathcal N:\mathbb B_\delta&\to L^2\Omega^1(\omega),\\
 \mathcal N(\mathbf{f})&=\bu_\mathbf{f}(T,\cdot)|_\omega,
 \end{align}
 with $\bu_\mathbf{f}\in \cC_\alpha(M)$ solving \eqref{eq:NS1}.
 Our main result can be stated as follows.

To the best of our knowledge, Theorem \ref{thm:main} is the first result establishing the recovery of the general geometric structure of a manifold from local velocity measurements, at the final time $t=T$, associated with solutions of the Navier-Stokes equations. By contrast, the existing literature is confined to the identification of the viscosity coefficient from measurements related to the stationary Navier-Stokes equations \cite{HLiWa,LaUWa,LiWa}.

According to the above discussion Theorem \ref{thm:main} follows from the following theorem.
 \begin{theorem}
 \label{thm_main:hyperbolic}
 Suppose there is a diffeomorphism $\bPhi:\omega_1\to\omega_2$ such that
 \begin{equation}
 \bPhi^\ast\Lambda_{2,\omega_2}=\Lambda_{1,\omega_1}\bPhi^\ast.
 \end{equation}
 Then $(\M_1,g_1)$ and $(\M_2,g_2)$ are isometric.
 \end{theorem}
 The remainder of this section is devoted to the proof of Theorem \ref{thm_main:hyperbolic} using the BC method. We extend the BC method to the auxiliary hyperbolic system \eqref{eq:problem_hypo}. This extension is not a direct application of existing results, due to the presence of the divergence-free constraint. Accordingly, each step of the BC method must be adapted. These adaptations are carried out below.

\subsection{Reconstruction of metric in the accessible domain}
 \begin{proposition}
 \label{prop:dist_omega}
 Suppose there is a diffeomorphism $\bPhi:\omega_1\to\omega_2$ such that $\diam_1(\omega_1)< T_{\bPhi}(\omega_1)$ and $\bPhi^\ast\Lambda_{2,\omega_2}=\Lambda_{1,\omega_1}\bPhi^\ast$, then for any points $x,y\in \omega_1$, there holds $d_{g_1}(x,y)=d_{g_2}(\bPhi(x),\bPhi(y))$. Moreover, $\bPhi$ is an isometry.
 \end{proposition}
 \begin{proof}
 Let $\varepsilon>0$ be small enough that $B_1(x,\varepsilon), B_1(y,\varepsilon)\subset \omega_1$ and $B^c_1(x,\varepsilon)$, $B^c_1(y,\varepsilon)$ are connected. Then we write
 \begin{equation}
 \mathcal{B}_k(p,\varepsilon):=C_0^\infty((0,\infty);C_0^\infty\Omega^1(B_k(p,\varepsilon))),\ k=1,2,
 \end{equation}
 and
 \begin{equation}
 \bPhi_\ast (\cB_1(x,\varepsilon)):=C_0^\infty((0,\infty);C_0^\infty\Omega^1(\bPhi(B_1(x,\varepsilon)))).
 \end{equation}
 We define
 \begin{align}
 t_{1,\varepsilon} := \inf\{t>0\mid \exists \bf\in \mathcal{B}_1(x,\varepsilon)\text{ s.t. } \supp(\bu_1^\bf(t,\cdot))\cap B_1(y,\varepsilon)\neq \emptyset \}
 \end{align}
 \begin{align}
 t_{2,\varepsilon} := \inf\{t>0\mid \exists \bf\in \mathcal{B}_2(\bPhi(x),\varepsilon)\text{ s.t. } \supp(\bu_2^\bf(t,\cdot))\cap B_2(\bPhi(y),\varepsilon)\neq \emptyset \},
 \end{align}
 and
 \begin{align}
 \widetilde{t_{2,\varepsilon}} := \inf\{t>0\mid \exists \bh\in \bPhi_\ast (\cB_1(x,\varepsilon))\text{ s.t. } \supp(\bu_2^\bh(t,\cdot))\cap \bPhi(B_1(y,\varepsilon))\neq \emptyset \}.
 \end{align}
 As $\bPhi:\omega_1\to \omega_2$ is a diffeomorphism, we can find two positive functions $r(\varepsilon),R(\varepsilon)$ such that $\lim_{\varepsilon\to 0}r(\varepsilon)=\lim_{\varepsilon\to 0}R(\varepsilon)=0$, and
 \begin{equation}
 B_2(\bPhi(p),r(\varepsilon))\subset \bPhi(B_1(p,\varepsilon))\subset B_2(\bPhi(p),R(\varepsilon))
 \end{equation}
 for both points $p=x,y$. Hence, $t_{2,R(\varepsilon)}\leq \widetilde{t_{2,\varepsilon}}\leq t_{2,r(\varepsilon)}.$

Next we show the local reconstruction can be glued together.\par
 \begin{lemma}
 \label{lm:glue}
 Let $U_1,U_2\subset \M_1^\inter$ be open sets with isometries
 \begin{equation}
 \bPhi:U_1\to\bPhi(U_1),\ \bPsi: U_2\to \bPsi(U_2).
 \end{equation}
 Suppose $\Lambda_{1,U_1}\bPhi^\ast=\bPhi^\ast \Lambda_{2,\bPhi(U_1)}$, $\Lambda_{1,U_2}\bPsi^\ast=\bPsi^\ast \Lambda_{2,\bPsi(U_2)}$ and there exists an open set $\cV\subset U_1\cap U_2$ such that $\bPhi|_\cV=\bPsi|_\cV$ and $U_1\cap U_2$ is $\cT$-exhaustive by $\cV$. Then there holds
 \begin{equation}
 \label{eq:lm_6.14}
 \bPhi|_{U_1\cap U_2}=\bPsi|_{U_1\cap U_2}.
 \end{equation}
 \end{lemma}
 \begin{proof}
 Let us show that
 \begin{equation}
 \label{eq:lm_6.14_1}
 \bPhi|_{E^{(1)}_{\cT}(\cV,U_1\cap U_2)}=\bPsi|_{E^{(1)}_{\cT}(\cV,U_1\cap U_2)}.
 \end{equation}
 Let $y\in U_1\cap U_2\cap M(\cV,\cT)$ be arbitrary. To get a contradiction, we assume that $\bPhi(y)\neq \bPsi(y)$. Let $z\in \cV$ be such that $d_{g_1}(y,z)<\cT$. For simplicity, we denote $d_{g_1}(y,z)$ by $s$. Since $\bPhi$ is an isometry, we have $d_{g_2}(\bPhi(y),\bPhi(z))=s<\cT$. Then we can choose a small enough $0<\varepsilon<\frac{1}{2}(\cT-s)$ such that $B_2(\bPhi(y),2\varepsilon)\subset \bPhi(U_1)$, $B_2(\bPsi(y),2\varepsilon)\subset \bPsi(U_2)$, $B_2(\bPhi(y),2\varepsilon)\cap B_2(\bPsi(y),2\varepsilon)=\emptyset$, and $\M_2\setminus B_2(\bPhi(y),\varepsilon)$ and $\M_2\setminus B_2(\bPsi(y),\varepsilon)$ are connected.
 By approximate controllability, we may take $\bf\in C_0^\infty((T-\varepsilon,T);C_0^\infty\Omega^1(B_2(\bPhi(y),\varepsilon)))$ such that $\bu_2^{\bf}(T,\cdot)\neq 0$. For any
 \begin{align}
 \bh\in C_0^\infty((T-\varepsilon,T);C_0^\infty\Omega^1(B_2(\bPsi(y),\varepsilon))),
 \end{align}
 due to the finite speed of propagation, we have
 \begin{equation}
 \supp(\bu_2^\bf(T,\cdot))\cap \supp(\bu_2^\bh(T,\cdot))=\emptyset.
 \end{equation}
 Since
 \begin{align}
 \bPhi^\ast: C_0^\infty((T-\varepsilon,T);C_0^\infty\Omega^1(B_2(\bPhi(y),\varepsilon)))\to C_0^\infty((T-\varepsilon,T);C_0^\infty\Omega^1(B_1(y,\varepsilon))),\\
 \bPsi^\ast: C_0^\infty((T-\varepsilon,T);C_0^\infty\Omega^1(B_2(\bPsi(y),\varepsilon)))\to C_0^\infty((T-\varepsilon,T);C_0^\infty\Omega^1(B_1(y,\varepsilon)))
 \end{align}
 are vector space isomorphism, we can choose $\bPsi^\ast \bh = \bPhi^\ast \bf$. Let $\delta>0$ be such that $B_1(z,\delta)\subset \cV$. Then we have $B_2(\bPhi(y),2\varepsilon)\subset M_2(B_2(\bPhi(z),\delta),s+2\varepsilon-\delta)$. Since
 $$T_2(B_2(\bPhi(z),\delta))\geq T_{\bPhi}(B_1(z,\delta))\geq \cT-\delta>s+2\varepsilon-\delta,$$
 by the approximate controllability, there exists a sequence
 $$\{\bphi_n\}_{n=1}^\infty\subset C_0^\infty((T-(s+2\varepsilon-\delta),T);C_0^\infty\Omega^1(B_2(\bPhi(z),\delta))$$
 such that
 $\bu_2^{\bphi_n}(T,\cdot)\to \bu_2^{\bf}(T,\cdot)$ as $n\to \infty$. Since $$B_2(\bPhi(y),2\varepsilon)\cap B_2(\bPsi(y),2\varepsilon)=\emptyset,$$
 there holds
 \begin{equation}
 \lim_{n\to \infty}\inner{\bu_2^{\bh}(T,\cdot),\bu_2^{\bphi_n}(T,\cdot)}=0,
 \end{equation}
 and
 \begin{equation}
 \lim_{n\to \infty}\inner{\bu_2^{\bf}(T,\cdot),\bu_2^{\bphi_n}(T,\cdot)}=\norm{\bu_2^{\bf}(T,\cdot)}_{L^2\Omega^1(\M_2)}>0.
 \end{equation}
 Then for large enough $N$ we write $\bpsi=\bphi_N$, and we have
 \begin{equation}
 \label{eq:glue_contradiction}
 \inner{\bu_2^{\bf-\bh}(T,\cdot),\bu_2^{\bpsi}(T,\cdot)}>0.
 \end{equation}
 Notice that $\bPhi^\ast \bpsi=\bPsi^\ast \bpsi$ as $\bPhi|_\cV=\bPsi|_\cV$. According to Lemma \ref{lm:norm_equal} and the fact that $\bPsi^\ast \bh = \bPhi^\ast \bf$, we have
 \begin{align}
 \inner{\bu_2^{\bf-\bh}(T,\cdot),\bu_2^{\bpsi}(T,\cdot)}&=\inner{\bu_2^{\bf}(T,\cdot),\bu_2^{\bpsi}(T,\cdot)}-\inner{\bu_2^{\bh}(T,\cdot),\bu_2^{\bpsi}(T,\cdot)}\\
 &=\inner{\bu_1^{\bPhi^\ast\bf}(T,\cdot),\bu_1^{\bPhi^\ast\bpsi}(T,\cdot)}-\inner{\bu_1^{\bPsi^\ast\bh}(T,\cdot),\bu_1^{\bPsi^\ast\bpsi}(T,\cdot)}\\
 &=\inner{\bu_1^{\bPhi^\ast\bf-\bPsi^\ast\bh}(T,\cdot),\bu_1^{\bPhi^\ast\bpsi}(T,\cdot)}=0.
 \end{align}
 This is in contradiction with \eqref{eq:glue_contradiction} and we have proved that \eqref{eq:lm_6.14_1} holds as $y\in E^{(1)}_{\cT}(\cV,U_1\cap U_2)$ is arbitrary. Since $U_1\cap U_2$ is $\cT$-exhaustive by $\cV$, there exists $N>0$ such that $U_1\cap U_2=E^{(N)}_{\cT}(\cV,U_1\cap U_2)$. Replacing $E^{(k)}_{\cT}(\cV,U_1\cap U_2)$ by $E^{(k+1)}_{\cT}(\cV,U_1\cap U_2)$ iteratively in above argument for $k=0,1,\cdots,N-1$, we can obtain \eqref{eq:lm_6.14}.
 \end{proof}
 \begin{proof}[Proof of Theorem \ref{thm:main}]
 Provided $\bPhi:\omega_1\to \omega_2$ is a diffeomorphism and $\Lambda_{1,\omega_1}\bPhi^\ast=\bPhi^\ast \Lambda_{2,\omega_2}$. we have $\bPhi$ is an isometry by Proposition \ref{prop:dist_omega}.

\begin{remark}
      By making $\omega_1$ and $\omega_2$ smaller, connectedness can be assumed without loss of generality.
  \end{remark}
\subsection{Comparision to previous literature}

To the best of our knowledge, Theorem \ref{thm:main} is the first result establishing the recovery of the general geometric structure of a manifold from local velocity measurements, at the final time $t=T$, associated with solutions of the Navier-Stokes equations. By contrast, the existing literature is confined to the identification of the viscosity coefficient from measurements related to the stationary Navier-Stokes equations \cite{HLiWa,LaUWa,LiWa}.

Our proof of Theorem \ref{thm:main} is based on application of the Boundary Control (BC) method, which remains one of the most powerful tools for solving inverse coefficient problems for partial differential equations (PDEs) in the time domain. The BC method was first introduced by Belishev in \cite{Beli} and thereafter further developed for solving inverse coefficient determination problems related to the wave equation in Euclidean domains \cite{ABI92,B87,B90,BK89,BK87,BKu89,BK91}. Later, this method was extended to manifold reconstruction \cite{BeKu}. It has been shown that the BC method can be applied to other classes of scalar PDEs \cite{FGKU,HLYZ,KaKuLaMa,KLLY,KOSY}.

The method appears to be less powerful in the context of systems of PDEs, however, inverse problems for the Maxwell \cite{BeIs,KuLaSo} and Dirac \cite{LK09} systems has been solved using it. Further, in \cite{KOP18} the BC method was generalized for abstract wave equations associated to connection Laplacians on vector bundles (for example, a linearized equation satisfied by the Higgs field on the pre-quantum level). To the best of our knowledge, this work constitutes the first application of the BC method to the Stokes system. The approach consists in first establishing a rigorous correspondence between the data for the Stokes system and those for an auxiliary hyperbolic Stokes problem through spectral data, as developed in Section 5. The BC method is then implemented for this auxiliary hyperbolic Stokes problem in Section 7.

It is important to emphasize that the extension of the BC method to this setting cannot be derived from existing results in the literature. The principal obstruction arises from the divergence-free constraint inherent to the Stokes system, which introduces substantial technical and structural difficulties at every stage of the method. In Section 7, we provide a detailed exposition of the adaptation of the BC method to this framework, and we show how the difficulties encountered at each step can be resolved by means of carefully tailored geometric and analytic arguments.

In inverse problems concerning recovery of geometry on manifolds with boundary, measurements are typically taken on the boundary and modeled by the Dirichlet-to-Neumann map or the associated Cauchy data \cite{AKKLT04,KK98,Oksanen2014,KaKuLaMa}. By contrast, inverse problems based on the local source-to-solution map have been predominantly studied on closed manifolds. In particular, it is shown in \cite{Saksala2018,HLYZ,SS25} that the local source-to-solution map for various linear evolution equations determines the underlying manifold up to an isometry. A related result for the wave equation, where the source and observations are supported in disjoint sets, is established in \cite{LNOY24}.
In this work, we consider the local source-to-solution map on manifolds with boundary.

The approach adopted in Sections 4 and 5 to address the nonlinearity relies on a linearization method that has proved effective in the analysis of numerous inverse problems for nonlinear PDEs. This method, originally introduced in \cite{I93,IS94} in the context of parabolic and elliptic partial differential equations, consists in establishing that the first-order linearization of the data coincides with the data associated with an appropriate linearized equation. In contrast with the case of classical scalar partial differential equations (see, for instance, \cite{L25} for a comprehensive review), the application of the linearization method to the Navier–Stokes system involves both the velocity field $\mathbf{u}$ and the pressure $p$. This necessitates a suitable representation of the pressure $p$, which is introduced in Theorem \ref{thm:NS}, together with the corresponding linearization result stated in Proposition \ref{linearization}. To the best of our knowledge, the only existing works employing techniques closely related to this linearization approach for the Navier–Stokes system are \cite{LaUWa,LiWa}, where the analysis is restricted to the Euclidean setting and to stationary Navier–Stokes equations in dimensions two and three. By contrast, our analysis is carried out in H\"older spaces, which allows for an extension to higher-dimensional settings.

Following the linearization step, in Section 5 we make use of analytic and representation properties of solutions to the linear Stokes system in order to establish a connection between the local source-to-final-value map and a source-to-solution map associated with an auxiliary hyperbolic system. The inverse problem for this auxiliary hyperbolic system is subsequently analyzed by means of the BC method. This data transfer technique by mean of spectral data has previously been employed in the study of inverse coefficient problems for various scalar evolution partial differential equations \cite{CK,HLYZ,KaKuLaMa,KLLY,KOSY}. As far as we know, the present work constitutes the first application of this methodology to Stokes systems.

\subsection{Outline of the paper}

The outline of the paper is as follows. In Section \ref{sec:elliptic}, we establish regularity results for the stationary Stokes system on manifolds and analyze properties of the associated spectral data, which prepare the terrain for the study of the inverse problem. Section \ref{sec:Nonlinear_NS} is devoted to the existence, uniqueness and appropriate regularity of the solution to the geometric Navier-Stokes equation \eqref{eq:NS1}. In Section \ref{sec:parabolic}, we consider the non-stationary Stokes equation, which arises as the linearization of \eqref{eq:NS1}, and show that the data $\mathcal N$ determines the corresponding restricted source-to-solution operator $\mathcal P$, defined as \eqref{eq:def_P}, for the non-stationary Stokes equation. Moreover, we show that $\mathcal P$ is equivalent to the localized spectral projections of the stationary Stokes system, which in turn allows the data to be converted into the restricted source-to-solution operator for an auxiliary hyperbolic Stokes system \eqref{eq:problem_hypo}. Section \ref{sec:hyperbolic} analyzes \eqref{eq:problem_hypo} and develops several technical tools, including conditional finite speed of propagation and approximate controllability. The new feature of these tools is that they hold only on a finite time interval determined by the underlying geometry. Combining these techniques with an adapted BC, our main result Theorem \ref{thm:main} is proved in Section \ref{sec:mian_proof}.

The introduction states that the proof of Theorem~\ref{thm:main} is based on the Boundary Control (BC) method. The approach is to (i) address the nonlinearity by a linearization method (Sections 4--5), establishing that the first-order linearization of the data coincides with the data for an appropriate linearized equation, which requires a suitable representation of the pressure $p$ (Theorem~\ref{thm:NS}) and yields a linearization result (Proposition~\ref{linearization}); then (ii) use analytic/representation properties of solutions to the linear Stokes system to connect the local source-to-final-value map $\mathcal N$ to a (restricted) source-to-solution operator $\mathcal P$ for the non-stationary Stokes equation, and show $\mathcal P$ is equivalent to localized spectral projections of the stationary Stokes system; this (iii) allows converting the data into the restricted source-to-solution operator for an auxiliary hyperbolic Stokes system \eqref{eq:problem_hypo} through spectral data (Section 5); finally (iv) analyze this auxiliary hyperbolic problem using tools developed in Section~\ref{sec:hyperbolic} (including conditional finite speed of propagation and approximate controllability, valid on a finite time interval determined by the geometry) and then implement an adapted BC method (Section 7 / Section~\ref{sec:mian_proof}) to conclude the manifold is determined up to an isometry.

The introduction states that the proof of the main theorem is based on the Boundary Control (BC) method. The approach is to (i) address the nonlinearity by a linearization method (Sections 4--5), establishing that the first-order linearization of the data coincides with the data for an appropriate linearized equation, which requires a suitable representation of the pressure $p$. We first use the following theorem.
\begin{theorem}
    \label{thm:NS}
There exists $\delta>0$, depending on $(\M,g)$ and $T$, such that for all $\bf\in \mathbb B_\delta$, problem \eqref{eq:NS1} admits a unique solution $(\bu,p)=(\bu_\bf,p_\bf)$, $p$ being unique up to a map depending only on $t\in[0,T]$, with $u_\bf\in \mathcal C_\alpha$, $dp_\bf\in C^{\frac{\alpha}{2}}([0,T];C^{\alpha}\Omega^1(\M))$ and with $p_\bf$ a solution of \eqref{eq:p}. Moreover, the map $\mathbb B_\delta\ni\bf\mapsto (u_\bf,dp_\bf)\in \cC_\alpha(M)\times L^2([0,T]; L^2\Omega^1(\M))$ is $C^1$.
  \end{theorem}
This yields the following linearization result.
\begin{proposition}\label{linearization}
For $i=1,2$ and $\bf\in C^\infty_0([0,T];C^\infty_0\Omega^1(\omega_i))$, we fix $\delta_{i,\bf}=\frac{\delta}{1+\|\bf\|_{\mathcal C_\alpha(M_i)}}$ with $\delta$ introduced in Theorem \ref{thm:NS}. Then the map
$$(-\delta_{i,\bf},\delta_{i,\bf})\ni s\mapsto \mathcal N_i(s\bf)\in L^2\Omega^1(\omega_i)$$
is $C^1$ and we have
\begin{equation}\label{lin1}
    \partial_s\mathcal N_i(s\bf)|_{s=0}=\mathcal P_i(\bf).
\end{equation}

\end{proposition}
Then (ii) one uses analytic/representation properties of solutions to the linear Stokes system to connect the local source-to-final-value map $\mathcal N$ to a (restricted) source-to-solution operator $\mathcal P$ for the non-stationary Stokes equation, and show $\mathcal P$ is equivalent to localized spectral projections of the stationary Stokes system; this (iii) allows converting the data into the restricted source-to-solution operator for an auxiliary hyperbolic Stokes system
\begin{equation}
    \label{eq:problem_hypo}
    \begin{cases}
      (\p_t^2-\Delta_H)\bu + d p=\bf,\ \text{in } \RR^+\times \M,\\
      d^* \bu=0,\ \text{in } \RR^+\times \M,\\
      \bu|_{[0,T]\times\p\M}=0,\\
      \bu=0,\ p=0,\ \text{in }\RR^-\times\M.
    \end{cases}
  \end{equation}
through spectral data (Section 5); finally (iv) one analyzes this auxiliary hyperbolic problem using tools developed later (including conditional finite speed of propagation and approximate controllability, valid on a finite time interval determined by the geometry) and then implements an adapted BC method (Section 7 / later in the paper) to conclude the manifold is determined up to an isometry.

\label{thm:main}
      Let $(\M_1,g_1)$ and $(\M_2,g_2)$ be two  n-dimensional smooth connected Riemannian manifolds with  boundary. Let $\omega_1\subset \M_1$ and $\omega_2\subset\M_2$ be open and connected. For $j=1,2$, let $\mathcal N_j$ be the local source-to-final value map
      \begin{align}
\label{eq:def_N}
    \mathcal N:\mathbb B_\delta&\to L^2\Omega^1(\omega),\\
    \mathcal N(\bf)&=\bu_\bf(T,\cdot)|_\omega,
  \end{align}
with $(M,g)=(\M_j,g_j)$
      and $\omega=\omega_j$.
      In addition, suppose that there is a diffeomorphism $\bm{\Phi}:\omega_1\to \omega_2$ such that
      \begin{equation}\label{NN}
          \cN_1\bm{\Phi}^\ast=\bm{\Phi}^\ast \cN_2.
      \end{equation}
       Then $(\M_1,g_1)$ and $(\M_2,g_2)$ are isometric.,

Question:
Let \((\mathcal M_1,g_1)\) and \((\mathcal M_2,g_2)\) be two smooth connected \(n\)-dimensional Riemannian manifolds with boundary, and let \(\omega_1\subset \mathcal M_1\) and \(\omega_2\subset \mathcal M_2\) be open connected subsets. For each \(j=1,2\), consider the geometric Navier--Stokes system on \([0,T]\times \mathcal M_j\) for a time-dependent 1-form \(\mathbf u\) and pressure \(p\):
\[
\begin{cases}
\partial_t \mathbf u-\Delta_H\mathbf u+R(\mathbf u)+dp=\mathbf f,\\
d^*\mathbf u=0,\\
\mathbf u|_{[0,T]\times \partial \mathcal M_j}=0,\\
\mathbf u=0,\ p=0 \text{ at } t=0,
\end{cases}
\]
where \(\Delta_H=-dd^*-d^*d\) is the Hodge Laplacian and \(R(\mathbf u)=(\nabla_{\mathbf u_*}\mathbf u_*)^*\). For some \(\delta>0\), let \(\mathbb B_\delta\) denote the admissible source 1-forms supported in \((0,T]\times \omega_j\) with \(\|\mathbf f\|_{C_\alpha(\mathcal M_j)}\le \delta\), and for each such \(\mathbf f\) let \(\mathbf u_{\mathbf f}\) be the corresponding solution. Define the local source-to-final value map
\[
\mathcal N_j(\mathbf f)=\mathbf u_{\mathbf f}(T,\cdot)|_{\omega_j}\in L^2\Omega^1(\omega_j).
\]
Assume there exists a diffeomorphism \(\boldsymbol\Phi:\omega_1\to \omega_2\) such that, with \(\boldsymbol\Phi^*\) denoting pullback of time-dependent 1-forms,
\[
\mathcal N_1\,\boldsymbol\Phi^*=\boldsymbol\Phi^*\mathcal N_2,
\]
i.e. for every admissible source on \(\omega_2\), the observed final 1-form on \(\omega_1\) agrees after pullback. Which conclusion about \((\mathcal M_1,g_1)\) and \((\mathcal M_2,g_2)\) holds?

The manifolds \((\mathcal M_1,g_1)\) and \((\mathcal M_2,g_2)\) are isometric; equivalently, there exists a diffeomorphism \(F:\mathcal M_1\to \mathcal M_2\) such that \(F^*g_2=g_1\).

Choices:
B. The manifolds \((\mathcal M_1,g_1)\) and \((\mathcal M_2,g_2)\) have isometric neighborhoods of \(\omega_1\) and \(\omega_2\); more precisely, there exist open sets \(U_j\subset \mathcal M_j\) with \(\omega_j\subset U_j\) and a diffeomorphism \(F:U_1\to U_2\) such that \(F|_{\omega_1}=\boldsymbol\Phi\) and \(F^*g_2=g_1\) on \(U_1\).
C. The diffeomorphism \(\boldsymbol\Phi:\omega_1\to \omega_2\) is an isometry between the observed regions; that is, \(\boldsymbol\Phi^*g_2=g_1\) on \(\omega_1\).
D. The manifolds \((\mathcal M_1,g_1)\) and \((\mathcal M_2,g_2)\) are isometric provided the intertwining identity \(\mathcal N_1\,\boldsymbol\Phi^*=\boldsymbol\Phi^*\mathcal N_2\) holds for all smooth compactly supported sources on \((0,T]\times\omega_2\), without any smallness restriction \(\|\mathbf f\|_{C_\alpha(\mathcal M_2)}\le \delta\).
E. There exists a constant \(c>0\) and a diffeomorphism \(F:\mathcal M_1\to \mathcal M_2\) extending \(\boldsymbol\Phi\) such that \(F^*g_2=c\,g_1\); in particular, the manifolds are determined only up to a global conformal scaling by the local source-to-final value map.