In this talk we investigate the geometry and topology of $f$-extremal domains in a manifold with negative sectional curvature. A $f$-extremal domain is a domain that supports a positive solution to the overdetermined elliptic problem
\[
\begin{cases}
\Delta u+f(u)=0&\textrm{in }\Omega,\\
u>0&\textrm{in }\Omega,\\
u=0&\textrm{on }\partial\Omega,\\
\langle\nabla u,\vec\nu\rangle_M=\alpha&\textrm{on }\partial\Omega,
\end{cases}
\]
where $\Omega$ is an open connected domain in a complete Hadamard $n$-manifold $(M, g)$ with boundary $\partial\Omega$ of class $C^2$ , $f$ is a given Lipschitz function, $\langle\cdot,\cdot\rangle_M$ is the inner product on $M$ induced by the metric $g$, $\vec\nu$ the unit outward normal vector of the boundary $\partial\Omega$ and $\alpha$ a non-positive constant.

We will show narrow properties of such domains in a Hadamard manifolds and characterize the boundary at infinity. We give an upper bound for the Hausdorff dimension of its boundary at infinity. Later, we focus on $f$−extremal domains in the Hyperbolic Space $\mathbb{H}^n$ . Symmetry and boundedness properties will be shown. Hence, we are able to prove the Berestycki-Caffarelli-Nirenberg Conjecture in $\mathbb{H}^2$ . Specifically:

$\textbf{Theorem:}$ Let $\Omega\subset\mathbb{H}^2$ a domain with properly embbeded $C^2$ connected boundary such that $\mathbb{H}^2\setminus\Omega$ is connected. If there exists a (strictly) positive function $u\in C^2(\Omega)$ that solves the equation
\[
\begin{cases}
\Delta u+f(u)=0&\textrm{in }\Omega,\\
u>0&\textrm{in }\Omega,\\
u=0&\textrm{on }\partial\Omega,\\
\langle\nabla u,\vec\nu\rangle_M=\alpha&\textrm{on }\partial\Omega,
\end{cases}
\]
where $f : (0, +\infty) \rightarrow \mathbb{R}$ satisfies $f (t)\ge\lambda t$ for some constant $\lambda$ satisfying $\lambda > 1/4$, then $\Omega$ must be a geodesic ball and $u$ radially symmetric.

If time permits, we will generalize the above results to more general OEPs.