  Let $X, Y$ be complete, simply connected Riemannian surfaces with pinched
negative curvature $-b^2 \leq K \leq -1$. We show that if $f : \partial X \to
\partial Y$ is a Moebius homeomorphism between the boundaries at infinity of
$X, Y$, then $f$ extends to an isometry $F : X \to Y$. This can be viewed as a
generalization of Otal's marked length spectrum rigidity theorem for closed,
negatively curved surfaces, in the sense that Otal's theorem asserts that if
$X, Y$ admit properly discontinuous, cocompact, free actions by groups of
isometries and the boundary map $f$ is Moebius and equivariant with respect to
these actions then it extends to an isometry. In our case there are no
cocompactness or equivariance assumptions, indeed the isometry groups of $X, Y$
may be trivial.

Differential Geometry