A Riemannian manifold $(X,g)$ is *hyperbolic* if the sectional curvatures are constant and negative. A theorem of Mostow says that these manifolds are determined by their fundamental group.



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> **Theorem** (Real Mostow Rigidity) If $X$ and $Y$ are closed, hyperbolic $n$-manifolds with $n \ge 3$ and $\pi\_1 X \simeq \pi\_1 Y$ then there is a Riemannian homothety $\varphi:X \simeq Y$.
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A Riemannian homothety is a diffeomorphism $\varphi:X \simeq Y$ with $\varphi^\*g\_Y = \lambda \cdot g\_X$ for some $\lambda > 0$. 


There is a clear analogue of the hypothesis of Mostow rigidity in the Kahler setting. Namely, a Kahler manifold $(X,\omega,J)$ is *complex-hyperbolic* if the holomorphic sectional curvatures are constant and negative. For a definition of the holomorphic sectional curvatures, see (for instance) *Hyperbolic Complex Spaces* by Kobayashi.


**Main Question:** My question is whether or not the following Kahler analogue is true.



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> **Theorem (?)** (Complex Mostow Rigidity) If $X$ and $Y$ are closed, complex-hyperbolic $2n$-manifolds with $\pi\_1 X \simeq \pi\_1 Y$ then there is a Kahler homothety $\varphi:X \simeq Y$.
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A dimensional restriction may be necessary here, for instance $n \ge 2$ or $n \ge 3$. Again, I would like to know if this theorem is true and if so, where I can find a reference.



Mostow rigidity for complex hyperbolic manifolds
The general statement of Mostow-Prasad rigidity cited from <http://repository.ias.ac.in/36364/1/36364.pdf> is as follows.


Let $G$ (resp. $G^\prime$) be a semi-simple analytic group and $\Gamma$ (resp. $\Gamma^\prime$) an irreducible lattice in $G$ (resp. $G^\prime$). Assume that $G, G^\prime$ have trivial centers and no compact factors and $G$ is not locally isomorphic to $SL(2,{\mathbb R})$. Then any isomorphism $\theta\colon \Gamma\to\Gamma^\prime$ extends to an analytic isomorphism of $G$ onto $G^\prime$.


The case of complex hyperbolic manifolds corresponds to $G=G^\prime=SU(n-1,1)$. An analytic isomorphism is an isometry and Kähler.


