  We investigate the geometry and topology of extremal domains in a manifold
with negative sectional curvature. An extremal domain is a domain that supports
a positive solution to an overdetermined elliptic problem (OEP for short). We
consider two types of OEPs.
  First, we study narrow properties of such domains in a Hadamard manifold and
characterize the boundary at infinity. We give an upper bound for the Hausdorff
dimension of its boundary at infinity and how the domain behaves at infinity.
This shows interesting relations with the Singular Yamabe Problem.
  Later, we focus on extremal domains in the Hyperbolic Space $\mathbb H ^n$.
Symmetry and boundedness properties will be shown. In certain sense, we extend
Levitt-Rosenberg's Theorem \cite{LR} to OEPs, which suggests a strong relation
with constant mean curvature hypersurfaces in $\mathbb H ^n$. In particular, we
are able to prove the Berestycki-Caffarelli-Nirenberg Conjecture under certain
assumptions either on the boundary at infinity of the extremal domain or on the
OEP itself.
  Also a height estimate for solutions on extremal domains in a Hyperbolic
Space will be given.
