In this paper we study the geometry and the topology of unbounded domains in the Hyperbolic Space \(\mathbb {H}^n\) supporting a bounded positive solution to an overdetermined elliptic problem. Under suitable conditions on the elliptic problem and the behaviour of the bounded solution at infinity, we are able to show that symmetries of the boundary at infinity imply symmetries on the domain itself. In dimension two, we can strengthen our results proving that a connected domain \(\Omega \subset \mathbb {H}^2\) with \(C^2\) boundary whose complement is connected and supports a bounded positive solution u to an overdetermined problem, assuming natural conditions on the equation and the behaviour at infinity of the solution, must be either a geodesic ball or, a horodisk or, a half-space determined by a complete equidistant curve or, the complement of any of the above example. Moreover, in each case, the solution u is invariant by the isometries fixing \(\Omega \).

在本文中，我们研究了双曲空间\（\ mathbb {H} ^ n \）中无界域的几何形状和拓扑，它支持超定椭圆问题的有界正解。在椭圆问题和无穷大有界解的行为的适当条件下，我们能够证明无穷大的边界对称性暗示着域本身的对称性。在维数二，我们可以加强我们的结果证明了一个连通域\（\欧米茄\子\ mathbb {H} ^ 2 \）与\（C ^ 2 \）边界，其补连接，并支持有界正解ü对于超定问题，假设方程上的自然条件和解的无穷大行为，则必须是测地线或全息盘，或者是由完全等距曲线确定的半空间，或者是任意上面的例子。而且，在每种情况下，解u通过固定\（\ Omega \）的等距不变。