Bending and stamping deformations of hyperbolic manifolds
A conformal structure on an n-manifold M is defined as an \((S^ n,Moeb_ n)\)-structure, i.e. a maximal atlas \((U_ i,f_ i)\), where the transition functions of charts \(f_ i\) or \(f_ j^{-1}\), \(U_ i\cap U_ j\neq \emptyset\), are Möbius transformations on the conformal sphere, i.e. compositions of reflections in subspheres. In this paper hyperbolic manifolds \(M=H^ n/G\) of finite volume are investigated, with \(\pi_ 1(M)=G\subset Isom H^ n\cong SO(n+1,1),\) \(n\geq 3\). Investigating the space C(M) of marked uniformized conformal structures of M is equivalent to the study of conjugacy classes of Kleinian representations \[ Hom(\pi_ 1(M)=G,Moeb_ n)/Moeb_ n=Hom(G,SO(n+1,1))/SO(n+1,1). \] The bending deformation along a totally geodesic hypersurface is motivated by Thurston's ``Micky Mouse example'' in dimension 2. The author compares some interpretations of bending besides his geometric construction. The constructions of \textit{C. Kourouniotis} [Math. Proc. Camb. Philos. Soc. 98, 247-261 (1985; Zbl 0577.53041)] and of the author [Complex analysis and applications. Proc. Conf., Varna/Bulg. 1985, 14-28 (1986; Zbl 0624.30045)] yield an embedding of a ball into C(M) whose dimension equals the number of non-intersecting totally geodesic hypersurfaces in M. The stamping deformation along the intersection of totally geodesic surfaces in M by ``pea-pod'' group is introduced by the author [Ann. Global Anal. Geom. 6, No.3, 207-230 (1988; Zbl 0662.53039)]. His new 3- dimensional construction, however, disproves the Kourouniotis' conjecture that the spatial deformations of \(H^ n/G\) are exactly the bendings. The author remarks that his stamping deformation of \(M=H^ 3/G\), vol \(M<\infty\) along a closed geodesic can also be realized in the case of closed manifold as well as in the case of 4-dimensional manifolds. In higher dimension the existence of similar stamping along a submanifold of codimension 2 remains open.
scientific article; zbMATH DE number 4157409
scientific article