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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.

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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