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Bending deformations of complex hyperbolic surfaces. (English) Zbl 0891.53055
W. M. Goldman’s local rigidity theorem in dimension 2 [Lect. Notes Math. 1167, 95-117 (1985; Zbl 0575.57027)] asserts that every nearby discrete representation $$\rho\colon G\rightarrow PU(2,1)$$ of a cocompact lattice $$G\subset PU(1,1)$$ stabilizes a complex geodesic in complex hyperbolic space $$H^2_\mathbb{C}$$. In [Invent. Math. 88, 495-520 (1987; Zbl 0627.22012)], W. M. Goldman and J. T. Millson proved that the same holds for small deformations of cocompact lattices $$G\subset PU(n-1,1)$$ in higher dimensions $$n\geq 3$$. Due to a theorem of C.-B. Yue [Ann. Math., II. Ser. 143, 331-355 (1996; Zbl 0843.22019)], this local rigidity is even global for complex hyperbolic $$n$$-manifolds homotopy equivalent to their closed complex totally geodesic hypersurfaces in dimensions $$n\geq 3$$. In the paper under review, the author shows that, in contrast to the rigidity of complex hyperbolic $$n$$-manifolds from this class, complex hyperbolic (Stein) manifolds homotopy equivalent to their closed totally real geodesic surfaces are not rigid.
Let $$M$$ be a complex hyperbolic surface homotopy equivalent to a Riemann surface $$S_p$$ of genus $$p>1$$, that is $$M=H^2_\mathbb{C} / G$$, $$G$$ being a discrete torsion free isometry group $$G\subset PU(2,1)$$ isomorphic to the fundamental group $$\pi_1(S_p)$$. The author proves that if $$M=H^2_\mathbb{C} / G$$ is the quotient corresponding to the embedding of $$\pi_1(S_p)$$ as a lattice acting on totally real geodesic 2-planes in $$H^2_\mathbb{C}$$, i.e., $$G\subset PO(2,1)\subset PU(2,1)$$, then for any simple closed geodesic $$\alpha\subset S_p=H^2_\mathbb{R} / G$$ there is a bending deformation of the group $$G$$ along $$\alpha$$, induced by $$G$$-equivariant quasiconformal homeomorphisms of the complex hyperbolic space and its Cauchy-Riemann structure at infinity. As a consequence, the existence of a real analytic embedding $$B^{9p-9}\hookrightarrow {\mathcal T}(M)$$ of a real $$9(p-1)$$-ball into the Teichmüller space of such a complex hyperbolic surface $$M$$ is shown.

##### MSC:
 53C56 Other complex differential geometry 32Gxx Deformations of analytic structures 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds 22E40 Discrete subgroups of Lie groups
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